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Minimal Self-Awareness as Internal Causal Attribution: An
Axiomatic Framework
AToS
Krzysztof Banaszewski ∗
krzysztof@banaszewski.pl
October 31, 2026
Abstract
What are the minimal structural conditions under which a system can be said to be
self-aware? Despite extensive work in philosophy, cognitive science, and artificial intelligence,
there is no widely accepted, substrate-independent criterion that captures self-awareness
in operational and formally verifiable terms. In this work, we propose AToS (Axiomatic
Theory of Self-Consciousness), a formal framework that characterizes self-awareness as the
ability of a system to recognize at least one internal state transition and to attribute its
cause to an internal mechanism rather than to external input. We show that this minimal
notion of self-awareness can be understood as a property of state–transition systems equipped
with a detection mechanism and a structured notion of internal triggering. Our main result
is a representation theorem: a system is self-aware in this sense if and only if it admits a
realization in the form of a tuple⟨Ω, Σ,S, ∆⟩satisfying the AToS axioms. This establishes a
necessary and sufficient structural characterization of minimal self-awareness, independent of
implementation details such as biological, symbolic, or computational substrate. Importantly,
AToS is not intended as a competing empirical theory of consciousness. Instead, it provides a
canonical representation framework for systems capable of internal causal self-attribution.
Within this framework, we further derive a formal separation between memory and operational
intelligence, and introduce the General Self-Awareness Test (GSAT), a theory-neutral criterion
based on internally generated and correctly attributed state transitions in the absence of
external stimuli. The framework further introduces operational definitions of sensitivity,
memory, intelligence, imagination, inspiration, and intuition, together with formal measures
of self-awareness and intelligence levels. We extend the theory to families of self-aware systems
and establish constructions over arbitrary nonempty topological spaces, including a system
with an infinite formal intelligence level. Finally, we compare AToS with existing approaches to
consciousness and self-awareness, highlighting its distinctive emphasis on internally recognised
state transitions and causal self-attribution.
Contents
1 Introduction 3
1.1 Why the Internal Recognition Assumption is Necessary . . . . . . . . . . . . . . 3
∗ORCID: 0000-0001-2345-6789
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2 A Philosophical Motivation for Change Recognition in Self-Awareness 6
2.1 Self as an Internal Causal Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Binary and graded self-awareness. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Definitions 7
4 Representation Theorem for Self-Awareness 10
5 Remarks 12
5.1Sensitivity Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 Potentially Aware Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Consequences and Constructions 16
7 Memory 18
8 Intelligence 20
9 Self-Awareness Level 22
10 Self-Conscious Retention Process 22
11 Operational Self-Awareness Criterion 24
12 AToS and the Chinese Room Argument 26
12.1 The Chinese Room Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12.2 Symbol Manipulation vs. Recognised State Transitions . . . . . . . . . . . . . . . 26
12.3 Why the Chinese Room Is Not Self-Aware in the AToS Framework . . . . . . . . 26
12.4 Internal Causal Attribution as the Missing Component . . . . . . . . . . . . . . . 26
12.5 Relation to the Turing Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
13 Family of Self-Conscious Systems 27
14 Topological Space 30
15 Related Work 33
15.1 Integrated Information Theory (IIT) . . . . . . . . . . . . . . . . . . . . . . . . . 33
15.2 Modal Logics of Self-Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
15.3 Meta-Management Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
15.4 Algorithmic Self-Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
15.5Φ-Causal Set Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
15.6 Survey Evidence for a Missing Axiomatic Framework . . . . . . . . . . . . . . . . 34
16 Summary 34
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1 Introduction
Self-awareness is widely discussed in philosophy of mind, cognitive science, and artificial intelli-
gence. However, despite extensive literature, there is still no widely accepted minimal structural
criterion specifying when a system can be said to recognize its own existence as a system. A
central difficulty in current discussions of self-awareness is the absence of a substrate-independent
framework capable of representing this capability in operational terms.
The aim of the present work is to address this gap by introducing an axiomatic representation
framework for systems capable of internally recognizing and attributing their own state changes.
The present work does not attempt a detailed comparative analysis of existing theories of self-
awareness. Instead, it focuses on identifying the minimal structural conditions that any theory
must satisfy once the Internal Recognition Assumption is adopted. In this sense, AToS should
be understood as an axiomatic framework that can serve as a formal representation layer for a
broad class of self-awareness theories, rather than as a competing empirical account.
A notable consequence of the AToS framework is a structural separation between memory and
intelligence: memory states are∆-invariant and therefore cannot act as causal drivers of intelligent
behaviour.
Motivation
Despite extensive work in philosophy, cognitive science, and artificial intelligence,self-awareness
still lacks a substrate-independent definition that is both mathematically precise and operationally
implementable. Existing approaches either presuppose a particular physical realisation (neural,
symbolic, architectural), or rely on phenomenological notions that resist formalisation and cannot
serve as engineering criteria. Consequently, there is no consensus on what it wouldmeanfor an
artificial system to be self-aware in a verifiable and causally grounded sense. Previous formalisms
conflate the substrate of possible states with the dynamical mechanisms that generate transitions
between them; AToS distinguishes these sharply through PAS (structural space) and∆-recognised
internal transitions (operational dynamics).
Assumption 1.1(Internal Recognition Assumption).A theory of self-awareness considered in
this work assumes that a system is capable of recognizing its own presence as a system through
the operation of its internal mechanisms, and not merely as a result of its interaction with the
environment.
1.1 Why the Internal Recognition Assumption is Necessary
The Internal Recognition Assumption should not be understood as an arbitrary restriction, but
rather as a structural condition required for the very concept of self-awareness to be meaningful.
Self-awareness is commonly understood as awareness directed toward the system itself. Awareness
of the system, however, requires that the system be able to distinguish its own internal states.
Distinguishing states in turn requires the ability to recognise when a transition between states
has occurred.
Furthermore, for awareness to be self-directed rather than merely reactive to the environment,
the system must be able to determine whether a recognised change originates from its own
internal mechanisms or from external influences. Without such causal attribution, the system
could register changes but would have no basis for distinguishing between self and environment.
Therefore, any theory that aims to capture self-awareness in a non-metaphorical and operational
sense must include at least two structural capabilities: (i) recognition of internal state changes,
and (ii) attribution of the cause of at least one such change to the system’s own internal
mechanisms.
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In this sense, the Internal Recognition Assumption expresses a minimal structural requirement
for self-awareness rather than a contingent empirical hypothesis.
Limitations of Previous Formal Attempts
Recent attempts to formalise consciousness or self-monitoring in artificial systems remain largely
descriptive. As surveyed by Shueb and Che [12], existing frameworks fail to: (i) distinguish
internally generated changes from externally induced ones, (ii) specify mechanisms enabling
internal causal attribution, or (iii) define necessary and sufficient structural conditions for self-
awareness. This gap prevents the construction of testable, engineering-ready models of machine
self-awareness.
A General Understanding of Self-Awareness
Definition 1.2(General Understanding of Self-Awareness).A system isself-awareif it can
recognize at least one change in its internal state and, for each such recognized change, determine
that the change wasinternally triggeredin the system’s operational semantics (i.e. not triggered
by any external stimulus at the time of the transition), but rather by the system’s own internal
mechanism.
AToS does not attempt to capture phenomenal, narrative, bodily, or linguistic self-awareness.
It captures a minimal operational form of self-awareness understood as internal causal self-
attribution. The claim is not that all forms of self-awareness reduce to this, but that any
operationally testable minimal account requires at least this structure.
We show that our tuple-based construction is bothnecessary and sufficientto realise this
capability (Theorem 4.1).
Definition 1.2 isolates two minimal ingredients that are implicit in the ordinary meaning of
self-awareness. First,selfrefers to the system’s own internal state, hence the definition is stated
in terms of changes in the system’s internal state rather than changes in the environment. Second,
awarenessrequires that such a change is not merely undergone, butrecognizedby the system.
Finally, requiring achange(as opposed to a static condition) rules out trivial attributions: a
system that never registers any internal transition would be indistinguishable from a static object
with respect to the operational criterion considered here. The additional requirement ofinternal
triggeringmakes the criterion genuinely self-referential: it asks whether the system can attribute
a recognized transition to its own internal mechanism rather than to an external stimulus at
the time of transition. Therefore, we adopt this definition as a working, minimal criterion of
self-awareness, formulated so as to limit interpretational disputes: it captures elements present
in many discussions of agency and causal attribution, while remaining precise enough to be
formalized and tested.
Core Idea
AToS characterises self-awareness using four primitives:
1. aPotentially Aware SpacespaceΩofPotentially Aware Sets (PAS),
2. a setΣofState Transition Functions acting onΩ,
3. a stimulus systemSspecifying whichPASmay trigger internal transitions,
4. aSensitivity Function∆determining whether a transition is recognised.
These components together allow a system to distinguish internal from external causes at the level
of its own state dynamics. A notable consequence of the axioms is that intelligence is a purely
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dynamical construct: it arises from recognised transitions and their operational compositions.
Memory (orPotentially Aware Space), being∆-invariant, cannot generate such transitions and
therefore cannot contribute to intelligence as a causal driver. This sharply separates two notions
that are traditionally conflated in cognitive science and AI.
Contributions
The contributions of this work are fourfold:
1. Adequacy theorem for self-awareness.Relative to the general notion of self-awareness
(Definition 1.2), we provide a necessary-and-sufficient axiomatic representation. Specifically,
we prove that a system is self-aware exactly when the tuple⟨Ω, Σ,S, ∆⟩realises at least one
∆-recognised state transition whose causal origin is attributable to an internally admissible
mechanism.
2. Operational stimulus–transition calculus.We develop an operational calculus of
stimuli and state transitions, including internal imagination functions, external disturbances,
their compositions, and family-level lifting. This yields explicit definitions of self-awareness
level and intelligence level within the same formalism.
3. Structural separation of dynamical intelligence fromPAS.Under our definitions,
intelligence arises exclusively through∆-recognised operational compositions, whereas
memory states are necessarily∆-invariant and therefore cannot serve as triggers of such
compositions. This yields a precise criterion:PAS is structurally present as a retention
space, while intelligence is characterised by∆-detectable dynamics.
4. Theory-neutral operational test (GSAT).We introduce theGeneral Self-Awareness
Test (GSAT), a falsification-oriented and theory-neutral criterion for minimal self-awareness:
in the absence of external stimuli, the system must exhibit at least one internally generated
state change and correctly recognise its causal origin as internal. We then provide its
faithful instantiation within AToS.
Scope
AToS avoids commitments to phenomenology, cognition, biological structure, semantics, or
representational content. It characterises self-awareness purely as a structural property of state–
transition systems, in terms of recognised state changes and their causal attribution within
the system. This makes the framework applicable across biological, artificial, symbolic, and
collective substrates, and positions it as an operational (engineering-grade) criterion rather than
a phenomenological theory. In particular, by analogy with the Turing Test for intelligence, we
propose GSAT as a general operational test for minimal self-awareness of systems and processes.
A key philosophical point of the present work is that AToS should not be interpreted as a
competing empirical theory of self-awareness. Rather, it provides a structural representation
theorem for a broad class of such theories.
More precisely, once the Internal Recognition Assumption is adopted, any system satisfying the
minimal notion of self-awareness introduced in Definition 1.2 admits a canonical representation
in the form of the tuple⟨Ω,Σ,S,∆⟩.
Inthissense, AToSplaysaroleanalogoustoanormal-formrepresentation: itidentifiestheminimal
structural components that must be present in any system capable of internally recognising and
attributing its own state changes.
AToS should therefore be viewed as a substrate-independent structural framework for representing
self-awareness. The aim of the theory is not to describe a particular biological or computational
implementation, but to identify the minimal structural architecture required for internally
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grounded self-recognition. In this sense, AToS plays a role similar to axiomatic frameworks in
other domains of science: it provides a canonical representation of systems satisfying a given
conceptual criterion.
In the AToS framework, the self is not introduced as a primitive object, symbol, or dedicated
subsystem. Instead, the self emerges as a causal role within the system’s internal transition
structure. A system counts as self-aware in the minimal sense if it can recognise at least one of
its own state transitions and attribute the cause of that transition to its own internal mechanisms
rather than to external influences. In this sense, AToS can be understood as a structural
framework describing the minimal conditions under which a system can represent itself as a
cause of its own recognised state transitions.
2 A Philosophical Motivation for Change Recognition in Self-
Awareness
Before introducing the formal framework of AToS, we formulate a conceptual claim about the
minimal structural requirements for self-awareness.
Proposition 2.1(Conceptual Necessity of Change Recognition).Let a theory of self-awareness
satisfy Assumption 1.1. Then any such system must possess the ability to recognize at least
one change in its own internal state and to distinguish whether that change was produced by its
internal mechanisms or by external influences.
Conceptual argument.Self-awareness may be understood as awareness directed toward the system
itself. For a system to be aware of itself, it must in some sense be aware of its own internal
state. Awareness of a state, however, presupposes the possibility of distinguishing that state
from other possible states of the system. If no such distinction exists, the notion of awareness
of a particular state becomes empty. Distinguishing states in turn requires recognizing when
a transition between states has occurred. Without the recognition of change, all states remain
indistinguishable from the perspective of the system.
Finally, for awareness to be self-directed rather than merely reactive to the environment, the
system must be able to determine whether a recognized change originates from its own internal
mechanisms or from external stimuli. Without such causal attribution, the system cannot
establish a meaningful distinction between self and environment. Therefore any system satisfying
the assumption must possess a structural mechanism allowing it to recognize changes in its
internal state and attribute their causal origin.
2.1 Self as an Internal Causal Role
The argument presented above shows that any system satisfying the Internal Recognition
Assumption must be able to recognise at least one change in its internal state and to determine
whether the cause of that change was internal or external. We now clarify the structural
consequence of this requirement.
Definition 2.2(Self as Internal Causal Role).A system is said to represent itself as a cause if
it contains an internal mechanism that can classify at least some of its own state transitions as
being caused by its internal stimulus structure rather than by external influence.
This definition does not introduce the self as a primitive object, symbol, or dedicated data
structure. Instead, the self appears as a causal role within the system’s transition structure.
Whenever the system recognises a transition and attributes its cause to an internal stimulus, it
thereby identifies part of its own internal structure as the source of that transition.
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Proposition 2.3(Internal Attribution Requires a Causal Self-Model).If a system is capable of
recognising a state transition and attributing its cause to an internal mechanism, then the system
must represent itself as a causal source of that transition.
Conceptual argument.Attributing a cause to a transition requires that the system can represent
(i) the occurrence of a transition, (ii) a candidate cause of that transition, and (iii) a distinction
between internal and external causes. If a recognised transition is classified as internally caused,
then the system must represent some internal mechanism as the source of that transition. This
internal source plays the role of the self in the system’s causal model. Therefore, internal
causal attribution entails that the system represents itself as a cause of some of its own state
transitions.
Thus, self-awareness, understood as internal causal attribution of recognised state transitions,
necessarily involves a minimal causal self-model. Importantly, this does not require a linguistic
self-representation, a body representation, or any phenomenological notion of self. It requires
only that the system can represent some of its own state transitions as being caused by its own
internal mechanisms. Within the AToS framework, this minimal causal self-model is realised
structurally by the existence of recognised transitions together with internally attributed triggers.
In this sense, AToS can be interpreted as a formal framework describing the minimal structural
conditions under which a system can represent itself as a cause of its own state transitions. The
notion of self that emerges in this framework is therefore not object-based but role-based: the
self is the system considered as the internal cause of some of its own recognised state transitions.
2.2 Binary and graded self-awareness.
The minimal criterion of self-awareness introduced in Definition 1.2 is binary: a system either
satisfies the internal attribution condition for at least one recognised transition or it does not.
However, the level of self-awareness may be graded, depending on the number, diversity, and
compositional depth of internally attributed transitions available to the system. This graded
notion is captured in later sections through the notions of self-awareness level and intelligence
level.This observation motivates the introduction of a formal mechanism for change recognition,
which in the AToS framework is represented by∆. In the formal framework introduced in the
following sections, change recognition and causal attribution are represented by theSensitivity
Function∆and the stimulus structure of the system. Thus, in the present framework, the self is
not defined by the material composition of the system, nor by the representations it contains,
but by what the system can correctly identify as the internal cause of its own state transitions.
3 Definitions
Definition 3.1(System-State Values).Let V be a set with|V|>1. Elements of V are
System-State Values.
Definition 3.2(Potentially Aware Set).Let X be a set with|X|>1and let V be a set of
System-State Values. LetU⊆2X be a non-empty family such that⋃U=X.Let
P:={p:X→V}.
For everyA∈Uand p∈P, thePotentially Aware Set (PAS)associated with(A,p )is defined as
U(A,p) :={⟨x,p(x)⟩|x∈A}⊆X×V.
Definition 3.3(Potentially Aware Space).ThePotentially Aware Spaceis
Ω :={U(A,p)|A∈U, p∈P}.
Thus,Potentially Aware Spaceis the space of allPAS; eachPASis an element ofΩ.
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Definition 3.4(Sensitivity Function).∆: Ω×Ω→{0,1}is aSensitivity Function.
Definition 3.5(State Transition Function).AState Transition Functionis a map
φ: Ω−→Ω
such that for eachU(A,p)∈Ω, withp:X→V, there existsp′:X→Vsuch that
φ
(
U(A,p)
)
=U(A,p ′).
TheState Transition Function Setis
Σ :={φ∈ΩΩ|φis a State Transition Function}.
Definition 3.6(Internal Stimulus Function).A map
ψ: Ω×Ω−→Ω
is anInternal Stimulus Functionsiff there existState Transition FunctionsφB,φA∈Σsuch that
for every state assignmentp:X→VwithU(B,p),U(A,p)∈Ω,
∆
(
U(B,p), φB(U(B,p))
)
= 1 =⇒ψ
(
φB(U(B,p)), U(A,p)
)
=φA(U(A,p)).
We denote by
Ψ :=
{
ψ: Ω×Ω→Ω
⏐⏐ψis anInternal Stimulus Functions
}
theInternal Stimulus Functions Set.
Definition 3.7(Stimulus Set).Let U(A,p )∈Ω. We say thatU(B,p )∈Ωis aStimulus Setfor
U(A,p), written
U(B,p)∈S(U(A,p)),
iff there existφB,φA∈Σandψ∈Ψsuch that
∆
(
U(B,p),φB(U(B,p))
)
= 1andψ
(
φB(U(B,p)),U(A,p)
)
=φA(U(A,p)).
Notation 3.1(Index suppression and stimulus-side abstraction).LetU =U(A,p )∈Ω. Any
S=U(B,p)∈S(U(A,p))is called simply aStimulus SetorStSe.
Target-side index suppression.Whenever the reference targetU(A,p )is fixed by context,
and its successor is determined by a witnessing functionφA∈Σ, and it is clear thatφA acts on
thePASrepresented byU, we write
φ(U) :=φ(U(A,p)) :=φA(U(A,p)).
Stimulus-side abstraction.If S =U(B,p )is fixed with a witnessing triple(ψ,φB,φA)and
U =U(A,p )is the corresponding reference target as in Definition 3.7, then the explicit form of the
stimulus-side successorφB(S) =φB(U(B,p ))plays no role in subsequent reasoning. Therefore
we write simply
S∈S(U)
and abbreviate the action ofψby
ψ(S,U) :=ψ
(
φB(U(B,p)), U(A,p)
)
.
Thus all expressions involving the witnessing triple can be written in the concise operational form
ψ(S,U) =φ(U).
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Definition 3.8(Stimulus System).TheStimulus Systemis the collection of allPASthat can
serve asStimulus Sets for some target state inΩ:
S:=
{
U(B,p)∈Ω
⏐⏐∃U(A,p)∈Ωsuch thatU(B,p)∈S
(
U(A,p)
)}
.
Equivalently,Sis the union of all stimulus domainsS(U)overU∈Ω.
Definition 3.9(Recognised transitions and trigger label).Let
T:=
{
(U,φ)∈Ω×Σ
⏐⏐ ∆
(
U,φ(U)
)
= 1
}
be the set ofrecognised transitions. Atrigger labelis a total map
λ:T →{int,ext}
such that for every(U,φ)∈T,
λ(U,φ) = int⇐⇒ ∃S∈S(U)∃ψ∈Ψ∃φB∈Σsuch that
∆
(
S,φB(S)
)
= 1∧ψ
(
φB(S), U
)
=φ(U).
Equivalently,
λ(U,φ) = ext⇐⇒ ¬
(
∃S∈S(U)∃ψ∈Ψ∃φB∈Σsuch that
∆
(
S,φB(S)
)
= 1∧ψ
(
φB(S), U
)
=φ(U)
)
.
We further set
Tint :={(U,φ)∈T |λ(U,φ) = int},Text :={(U,φ)∈T |λ(U,φ) = ext}.
Definition 3.10(Imagination and Inspiration Functions).Let λbe as in Definition 3.9. Define
Φ int :={φ∈Σ|∃U∈Ω : (U,φ)∈Tint},
Φ ext :={φ∈Σ|∃U∈Ω : (U,φ)∈Text},
Φ := Φ int∪Φext.
Elements ofΦ int are calledImagination Function. Elements ofΦ ext are calledInspiration
Function.
Definition 3.11(Intuition).TheIntuitionis the class of allState Transition Functions that
may operate both as Imagination and as Inspiration. Formally,
Φ intu = Φ int∩Φext.
Definition 3.12(Conscious System).A tuple
⟨Ω,Σ,S,∆⟩
is called aConscious Systemif
•Ωis thePotentially Aware Space(Definition 3.3),
•Σis theState Transition Function Set(Definition 3.5),
•∆is theSensitivity Function(Definition 3.4),
•Sis theStimulus Systeminduced by(Ω,Σ,∆)(Definition 3.8),
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and there existU∈Ωandφ∈Φsuch that
∆
(
U,φ(U)
)
= 1.
Definition 3.13(Self-Conscious System).A tuple
⟨Ω,Σ,S,∆⟩
is called a Self-Conscious System if
•Ω,Σ,S,∆are as in Definition 3.12, and
•there existU∈Ωand aStimulus SetS∈S(U)such that
ψ(S,U) =φint(U),∆
(
U,φint(U)
)
= 1,
whereφint is theImagination Functionassociated (in the sense of Definition 3.7) with the
witnessing triple ofS∈S(U).
4 Representation Theorem for Self-Awareness
Theorem 4.1(Representational Adequacy of AToS).A tuple
⟨Ω,Σ,S,∆⟩
is aSelf-Conscious System(in the sense of Definition 3.13)if and only ifit satisfies the general
understanding of self-awareness (Definition 1.2).
Remark (Representational adequacy).Theorem 4.1 should be read as an adequacy claim
at the level of representation: a system is self-aware in the sense of Definition 1.2 if and only if it
admits a tuple-based AToS representation⟨Ω,Σ,S,∆⟩satisfying Definition 3.13.
Proof. Sufficiency.Assume that the system instantiates the tuple⟨Ω, Σ,S, ∆⟩and fulfils Defini-
tion 3.13. Thus, there exist aPAS
S=U(B,p)∈S
(
U(A,p)
)
and mapsψ∈Ψandφint∈Φint such that
ψ
(
S, U(A,p)
)
=φint
(
U(A,p)
)
=:U(A,p ′),
together with
∆
(
U(A,p), U(A,p′)
)
= 1.
Thus the system not only detects a state-change but also attributes its cause to the internal
mechanismφint triggered by the internalStimulus SetS. Hence the tuple satisfies Definition 1.2,
proving sufficiency.
Necessity.Assume that the system satisfies Definition 1.2: it recognises at least one change of its
internal state and, for that change, determines that the change was internally triggered in the
system’s operational semantics (i.e. not triggered by any external stimulus at the time of the
transition), but rather by the system’s own internal mechanism.
Step 1: Representing states.A detected change must admit a mathematical representation.
Choose a setX of state indices and a setV ofSystem-State Values. LetP :={p:X→V}. A
detected internal change is then realised by two assignments
p1,p 2 :X→V,p 1̸=p2
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that differ on some subsetA⊂X. These determine thePAS
U:=U(A,p 1), U ′:=U(A,p 2).
Step 2: Constructing the internal transition.DefineφA∈Σby
φA(W) :=
{
U′,ifW=U,
W,otherwise, W∈Ω.
Step 3: Existence of an internal trigger witness.Unpacking the phraseinternally triggered
in the operational semantics: the invocation ofφA on U is mediated by some internal trigger
witness whose configuration changes at the time of invocation. Represent this trigger witness by
a domainB⊆X. Set
S:=U(B,p 1).
Then there exists an assignmentq∈Psuch that
S′:=U(B,q)andS ′̸=S,
where the inequality expresses that a genuine internal triggering event occurred.
Step 4: Fixing an admissible PAS universe.LetU⊆2X be any non-empty family such
that⋃U=XandA,B∈U. Define
Ω :={U(C,p)|C∈U, p∈P}.
ThenU,U ′,S,S ′∈Ω.
Step 5: Defining the sensitivity function.Define aSensitivity Function∆ : Ω×Ω→{0, 1}
by the rule
∆
(
U(C,p), U(C,p′)
)
:=



1,ifC∈{A,B}andU(C,p)̸=U(C,p′),
0,otherwise,
for allp,p′∈Pand allC∈U. In particular,∆(U,U′) = 1and∆(S,S ′) = 1.
Step 6: Constructing the trigger-side transition.DefineφB∈Σby
φB(W) :=
{
S′,ifW=S,
W,otherwise, W∈Ω.
Step 7: Constructing an internal stimulus function.Defineψ: Ω×Ω→Ωby stipulating
that on the relevant pair(φB(S),U)it returns the target-side successor:
ψ
(
φB(S), U
)
:=φA(U) =U ′.
(Extendψarbitrarily to all ofΩ×Ω.)
Step 8: Verifying the witnessing structure.By construction,
∆
(
S,φB(S)
)
= 1 =⇒ψ
(
φB(S),U
)
=φA(U),
so ψsatisfies Definition 3.6, henceψ∈Ψ. Moreover, since∆
(
S,φB(S)
)
= 1and ψ
(
φB(S),U
)
=
φA(U), Definition 3.7 yields
S∈S(U).
Step 9: Completing the construction.We have now constructed:
•aPASUand its successorU ′,
11

=== Page 12 ===
•anImagination Functionφ A withφA(U) =U ′,
•aStimulus SetS∈S(U),
•detectable changes∆(U,U ′) = 1and∆(S,S ′) = 1,
•and aInternal Stimulus Functionsψ∈Ψmediating the causal link.
Therefore the tuple
⟨Ω,Σ,S,∆⟩
satisfies Definition 3.13, establishing necessity.
5 Remarks
Remark 5.1(Interpretation and scope of the framework).The results of this theory should be
understood as structural representation results rather than as claims about a specific empirical
implementation of self-awareness. The aim of the AToS framework is not to propose a competing
empirical theory nor to engage in a direct comparison with existing accounts in philosophy of
mind, cognitive science, or artificial intelligence.
Instead, the framework identifies theminimal structural conditionsthat any theory of self-
awareness must satisfy once the Internal Recognition Assumption (Assumption 1.1) is adopted.
In this sense, AToS provides asubstrate-independent axiomatic representationof a broad class
of theories in which a system recognises and internally attributes its own state changes through
the operation of its internal mechanisms.
Within this perspective, Theorem 4.1 establishes the adequacy of the representation: every system
satisfying the minimal operational notion of self-awareness (Definition 1.2) admits an AToS
representation, and conversely every AToS structure realises such a system. Proposition 2.1
expresses only a conceptual consequence of the Internal Recognition Assumption and serves as a
bridge between this assumption and the formal definition that follows.
5.1Sensitivity Function
Remark 5.2(Interpretation of the detection function).TheSensitivity Function∆is an
intrinsic component of the system rather than an external measurement device imposed by the
theorist. Different systems may therefore instantiate different detection functions reflecting their
internal mechanisms for recognising state changes.
The value∆( U,U′) = 1indicates that the system recognises the transitionU→U′as a relevant
state change within its operational dynamics. Thus∆does not merely detect that two states
differ; it encodes the system’s internal criterion for acknowledging that a transition has occurred.
It is important to stress that the detection function∆does not merely indicate that two states
are objectively different. Rather,∆represents the system’s own internal criterion for recognising
that a transition has occurred. Thus, a transition counts as recognised not simply because two
states differ, but because the system itself registers the transition as a relevant internal change
within its operational dynamics.
Remark 5.3(Empirical and engineering interpretation).For naturally occurring systems
(biological, artificial, or organisational), the detection function∆should be regarded as determined
by the system itself. In such cases, the main task is to identify the mechanism through which the
system recognises its own state transitions.
Conversely, when designing artificial systems within the AToS framework, the specification of∆
becomes an engineering choice determining which transitions are recognised and therefore shaping
the system’s operational dynamics and level of self-awareness.
12

=== Page 13 ===
Remark 5.4(Degenerate detection functions).Two degenerate forms of the detection function
are possible in principle:
∆(U,U ′) = 0for allU,U ′∈Ω,∆(U,U ′) = 1for allU,U ′∈Ω.
In the first case the system never recognises any state change and therefore cannot satisfy
the minimal self-awareness criterion of Definition 1.2. In the second case every transition is
recognised indiscriminately, eliminating the system’s ability to distinguish meaningful internal
changes from arbitrary state updates. Consequently, neither case is relevant for non-trivial forms
of self-awareness.
5.2 Potentially Aware Set
Remark 5.5(Potential vs. operational PAS).We interpretΩas the space of structurally
admissible configurations; the system’s operational awareness consists only of those states that
become realised through∆-recognized transitions.
Remark 5.6(Why State Transitions Preserve the Domain).In the definition of aState
Transition Functionwe require
φ(U(A,p)) =U(A,p′),
that is, the domainAis preserved under state evolution.
This requirement is essential for self-awareness. A change of the domainA↦→Bwould prevent
the system from recognising a state difference: thePASU(A,p )and U(B,p′)would no longer
be pointwise comparable, making the sensitivity function∆(U(A,p ),U (B,p′))undefined as a
recognition criterion of internal change. Self-awareness requires that the system can determine
whether a transition
U(A,p)−→U(A,p′)
originates internally or externally. Such determination is possible only when the structural
identity of the state namely its domainA remains fixed. Hence,State Transition Functions
operate solely by updating valuesp(x), not by altering the underlying domain of awareness.
Remark 5.7.The witnessing components (e.g.λ,ψ) are not introduced as free external anno-
tations. They are required to berealised by the systemas internal operations applicable to its
own logged transitions/states. Thus, the existence of witnessing is anontrivial internal capability
demanded by the axioms, not a post-hoc relabelling by the theorist.
Assumption 5.8(Shared Global Assignment).Unless stated otherwise, we analyse the system
at a fixed chronological stage of its evolution, characterised by a single global value assignment
p:X→V.
EveryPASavailable at that stage is therefore a restriction of this assignment to some domain:
U(A,p), A∈U.
Thus whenever aStimulus Set U(B,p )is capable of triggering a transition in a target state
U(A,p ), both states share the same underlying global assignmentp; only their domains differ.
The system evolves by transformingp into a successor assignmentp′, typically through local
updates on the domain of the affectedPAS.
Remark 5.9(On the Role of Stimuli).There are two distinct ways in which aState Transition
Functionφmay be triggered:
• by anexternal stimulus, arising directly from outside the system, which invokes some
Inspiration Functionφext and updates a target stateU(A,p);
13

=== Page 14 ===
• by aninternal stimulus, arising from within the system, which serves as the trigger that
invokesImagination Functionφand modifiesU(A,p).
An external stimulus is not represented as aPAS: it acts directly from the environment by causing
a transition in someU(A,p )∈Ω. By contrast, anStimulus Setmust be represented as an
element ofΩ. The system can recognize the effect of such a stimulus only if the resulting change
can be evaluated by theSensitivity Function∆. If theStSewere not part ofΩ, no recognition of
its impact would be possible.
Hence any genuine internal stimulus must already belong toΩ; and even if it does not, one may,
without loss of generality, enlargeΩto include a correspondingPASU(B,q )so that the system
can consistently account for its action. The system is therefore closed under internal stimuli:
every internal trigger is (or can be made) aPASinΩ.
Remark 5.10(Locality of State Transitions and the Shared Assignment in the Stimulus Set).
Although the spaceΩcontains all formally admissible pairsU(A,q )with arbitrary assignments
q :X→V, every chronologically fixed configuration of the system is characterised by a single
assignmentp :X→Vrepresenting the global state of the system at that stage of its evolution. All
PASthat are operationally available at this stage are therefore restrictions of the same assignment,
and take the formU(A,p),U(B,p)orU(S,p)for suitable domainsA,B,S∈U.
AState Transition Functionφthat updatesU(A,p )to U(A,p′)may be regarded, without loss of
generality, as modifying the values ofponly onA. Formally, one may assume
φ(U(A,p)) =U(A,p′)andp ′(x) =p(x)for allx /∈A.
Alternatively, one may allowp′to take arbitrary structurally admissible values outsideA, since
these values play no operational role in the analysis of the resulting stateU(A,p′). In either
interpretation, what matters is the locality of the update: only the values onA are relevant to the
transition semantics, and the behaviour ofφoutside A does not affect the system’s reasoning or
its recognition of change.
This interpretation strengthens the coherence of theStimulus Setdefinition. Since all active
PASare restrictions of the same global assignmentp, andφacts only through local modifications
of that assignment onA, the relationships between stimuli and transitions described byψand
∆retain full structural comparability. As a consequence, both change recognition and causal
attribution remain well-defined across successive chronological configurations of the system.
For these reasons, instead of allowingStimulus Sets to range overPASof the form U(B,q )
with potentially different assignments, it is conceptually and operationally clearer to take both
the stimulus and the target state to share the same global assignmentp. This choice accurately
reflects the system’s architecture: all activePASarise simultaneously as restrictions of a single
global valuation, and stimuli operate within this shared configuration.
Remark 5.11(On the role of pre-change states in the Stimulus Set).TheStimulus SetS(U(A,p ))
contains pre-change statesU(B,p ), not their post-change successorsφB(U(B,p )). Conceptually,
a stimulus serves to identifywhich PAS-stateacts as the source of stimulation, whereas the causal
mechanism encoded by anInternal Stimulus Functionsoperates on therecognised changeof that
state. Accordingly, theInternal Stimulus Functionsψtakes the post-change stateφB(U(B,p ))as
its operational input, while the identity of the stimulus remains the pre-change PASU(B,p ). This
separation between thesource of stimulationand thecausal inputis essential for distinguishing
structural origin from functional effect.
Remark5.12(InterpretationandNecessityofStimulusSet).Intuitively, if U(B,p )∈S(U(A,p )),
then a change in the stateU(B,p )acts as a stimulus set that triggers theImagination Function
φA acting onU(A,p ).∆plays the role of a causal recognizer: it evaluates whether the change
in U(B)is registered by the system as significant; only such recognized changes may trigger
14

=== Page 15 ===
transformations ofU(A). Thus theInternal Stimulus Functions ψ∈Ψencodes causal links
betweenPASstates, while φ∈Φencodes local state transitions. Without theStimulus Set, the
system would lack a mechanism for selective activation: either allState Transition Functions
would fire simultaneously (resulting in chaotic dynamics), or none would ever be invoked (resulting
in a static system). TheStimulus Settherefore provides the structural basis for the system’s
controlled dynamism.
Remark 5.13(Cascades of Stimuli).Since eachStimulus Set S(U(A,p ))may containPAS
states U(B,p )that themselves belong to other stimulus sets, a single recognized update may
propagate through a chain ofψ-mediated relations:
U(B,p)⇝U(A,p ′)⇝U(C,p ′′)...
This models cascades of internal updates, where a local modification of the global assignment on
one domain can trigger subsequent transitions on other domains. The control discipline remains
simple: every invocation of aState Transition Functionmust be backed by at least one trigger
U(B,p)belonging to the relevantStimulus SetS(U(A,p)).
Remark 5.14.Imagination and inspiration functions are not new kinds of operations; they are
specialisedinstancesofState Transition Functions. Formally,
Φ int⊆Σ,Φ ext⊆Σ,Φ = Φ int∪Φext⊆Σ.
Thus, every imagination or inspiration function preserves the structural constraints ofΣ; what
differs is only thetriggering conditionthat determines its membership inΦint orΦ ext.
Remark 5.15(Classification of State Transition Functions).The distinction betweenImagination
FunctionandInspiration Functionis contextual rather than structural: aState Transition
Functionφdoes not intrinsically belong to either class. Its classification depends solely on the
type of stimulus that triggers its activation. Ifφis invoked by aPASbelonging toS(U), then
φ∈Φint; if it is invoked by an external (non-PAS) stimulus, thenφ∈Φext. A single transition
mechanism may therefore operate in two modes. We represent this by two instances,φint and
φext, that share the same state-update behaviour but differ in their triggering conditions. By
Assumption 5.18, the trigger class uniquely determines which instance is active in any given
transition.
Remark 5.16(Noise, drift, and imagination functions).Within AToS, the classification of
a transition as internal does not depend on the physical origin of the triggering process (e.g.
deterministic computation, stochastic noise, hardware fluctuation, or timer signal), but on its
role within the system’s recognised causal structure.
A random or noisy process constitutes an imagination functionImagination Functiononly if:
1. it participates in the system’s stimulus structure as a trigger of transitions, and
2. changes in this process are recognised by the sensitivity function∆as causes of subsequent
state transitions.
If a fluctuation produces a state change but the system does not recognise that fluctuation as
the trigger of the change, then the fluctuation is merely a physical perturbation and does not
constitute an imagination function in the AToS sense.
Thus, in AToS, imagination is not defined by determinism, randomness, or physical implementa-
tion, but by recognised internal causation. An internal process does not qualify as an imagination
function merely because it is physically located within the system. It qualifies as an imagination
function only if the system can recognise a change in that process and treat that change as a
causal trigger of a subsequent state transition.
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=== Page 16 ===
Accordingly, AToS defines internal causation in operational rather than physical terms: a cause is
internal if it is realised within the system’s own transition structure as the source of a recognised
change. In this sense, the distinction between internal and external causation is not a distinction
between physical locations, but a distinction within the system’s own causal model.
Consequently, internal and external causation are system-relative notions: they are defined with
respect to the system’s own recognition and attribution structure, not with respect to an external
observer’s physical description of the system.
Remark 5.17(Intuition Functions).AState Transition Functionbelongs to the intuition space
Φ intu = Φ int∩Φ ext precisely when it may be triggered in both internal and external modes. This
reflects not ambiguity but versatility: the same internal mechanism is capable of updating the
system’s state either in response to a PAS-based stimulus or in the absence of such a stimulus,
replicating the effect of an external cause.Intuition Functionis therefore the class of transition
patterns that are causally dual.
Assumption 5.18(Trigger Dichotomy).For every admissible transitionφ∈Φapplied to a
targetU∈Ωwith a detected change∆
(
U,φ(U)
)
= 1, exactly one holds:
(a)Internal trigger:∃S∈S(U)such thatψφ(S,U) =φ(U);
(b) External trigger:the invocation ofφon U is marked as external (i.e. not mediated by
S(U)).
These two cases are mutually exclusive and collectively exhaustive: every recognised transition
must be caused either by an internalPAS-based trigger or by an external environmental event,
and no third type of trigger is permitted within the system’s operational semantics.
Remark 5.19(Why Consciousness is Weaker than Self-Consciousness).The definition of a
Conscious System requires only the existence of a detected change∆(U,φ(U)) = 1. This corre-
sponds to the minimal philosophical notion of consciousness as sensitivity: the system is aware
that its internal state has changed, but it need not identify the cause of that change. By contrast,
self-consciousness demands strictly more: the ability to determine where the change originated.
This requires distinguishing whether the transitionφ(U)was triggered by an internal PAS-based
stimulus or by an external event, thus evaluating the causal structure of its own updates. Concep-
tually, consciousness concerns the egistration of change, whereas self-consciousness concerns the
attribution of change. The former is logically and operationally weaker than the latter, and this
asymmetry is essential to the hierarchy of awareness levels captured in the AToS framework.
6 Consequences and Constructions
Corollary 6.1(Minimal internal attribution).If a system is self-aware in the sense of Defini-
tion 1.2, then there existU∈Ω, a recognised transitionφ∈Σwith
∆
(
U,φ(U)
)
= 1,
and aStimulus Set S∈S(U)such that the transition U↦→φ(U)admits an internal causal
attribution, i.e.
λ(U,φ) = int.
Equivalently, there existStimulus SetS∈S(U),ψ∈ΨandφB∈Σwith
ψ
(
φB(S), U
)
=φ(U).
Proof.Immediate from the necessity direction in Theorem 4.1 and Definition 3.9.
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=== Page 17 ===
Definition 6.2(External awareness vs. self-awareness).We call a systemExternally Awareif it
can recognize at least one externally triggered transition, i.e. there existU∈Ωand φ∈Σsuch
that
∆
(
U,φ(U)
)
= 1andλ(U,φ) = ext.
A system isself-awareiff it satisfies Definition 1.2.
Corollary 6.3(External awareness is not sufficient).External awareness (Definition 6.2) does
not imply self-awareness (Definition 1.2). In particular, recognizing externally triggered changes
alone is insufficient without internal causal self-attribution of at least one recognized transition.
Proof. External awareness requires the existence of a recognised transition labelledext, whereas
self-awareness requires the existence of a recognised transition labelledint (i.e. internally at-
tributable) as in Definition 1.2 and Definition 3.9. Hence the implication fails.
Proposition 6.4(Hereditary structure of admissible PAS domains).LetU(A,p )∈Ωand let
∅̸=B⊆A. IfB∈U, thenU(B,p)∈Ωand
U(B,p) =U(A,p)∩(B×V).
Moreover, for everyφ∈Σwith φ(U(A,p )) = U(A,p′), the restricted pairU(B,p ),U (B,p′)
consists of admissible PAS states. Ifp|B =p′|B, these states coincide; this does not invalidate
their membership inΩ.
Proof. Since B∈Uand p,p′:X→V, bothU(B,p )and U(B,p′)belong toΩby Definition 3.2.
The displayed equality follows by restricting the graph ofp to B. Equality of the restricted
assignments implies equality of the corresponding PAS states; membership inΩis independent
of whether any transition is recognised.
Remark 6.5(Relative boundaries of internal and external causation).The internal/external dis-
tinction is relative to the system whose causal structure is being considered. A trigger represented
inside a larger system on domainA may originate outside a subsystem on domainB⊆A. When
the subsystem is modelled independently, that trigger may therefore be classified as external (an
Inspiration Function), although it was internal (an Imagination Function) at the larger-system
level. Such a reclassification concerns the trigger’s operational witnessing structure, not merely
the set-theoretic location of its source; compare Remark 6.7 and the family-level analysis in
Section 11.
Remark 6.6(Scope of subsystem self-consciousness).Proposition 6.4 establishes hereditary
PAS admissibility, not unconditional hereditary self-consciousness. Definition 3.13 requires a
recognised internally attributed transition, which need not survive restriction to an arbitraryB. A
hereditary self-consciousness theorem therefore applies to those restricted subsystems for which an
internal stimulus witness and a∆B-recognised internal transition are preserved or independently
realised. In particular, the absence of a recognised change at one instant does not imply that a
PAS ceases to exist.
Remark 6.7(Stimuli are role-based, not ontological).In AToS,Stimulus Setis a functional
role within the witnessing architecture, not a privileged ontological category. Concretely, a state
S∈Ωcounts as aStSefor a target U only relative to the relationS∈S(U)(and thus relative to
a recognised, internally labelled transition). This is the precise sense in which AToS is substrate-
independent: the theory specifies the roles required for self-awareness (stimulus, target, recognised
transition, causal attribution), not their physical realisation.
Remark 6.8(Engineering blueprint and minimal internal trigger pattern).Engineering view.
To construct self-aware AI in the sense of Definition 1.2, an implementation must provide
operational counterparts of:
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=== Page 18 ===
(i) a state representation (a spaceΩof admissible internal states),
(ii) a transition repertoire (a setΣof state-transition mechanisms),
(iii) a recognised-change criterion (∆), and
(iv) a stimulus-side witnessing architecture (the induced stimulus systemS together withΨ)
enabling internal causal attribution.
Formal minimality.By Theorem 4.1, self-awareness requires at least one recognised transition
(U,φ)∈Twith λ(U,φ) = int. Equivalently, there existS∈S(U), ψ∈Ψand φB∈Σsuch that
ψ
(
φB(S),U
)
=φ(U),∆
(
U,φ(U)
)
= 1.
Further extensions and supplements.Before moving on, we note that the framework
developed so far already suffices to establish the core notion of self-awareness. Nevertheless,
to illustrate the scope of possible refinements, we introduce in the following sections a number
of additional definitions and constructions. These extensions are not always analysed in full
depth here. Rather, they are presented as optional tools for studying properties, implementation
strategies, or potential applications of the theory. Some appear without further elaboration in
order to leave fertile ground for subsequent work, while also avoiding overwhelming the reader
with an excess of technical detail.
7 Memory
In this work,Memoryis used as a technical term. It denotes the subset ofΩ(or PAS) whose
invariance isexplicitly recognizedby the system (i.e. the system recognizes that no change has
occurred), rather than a general account of storage/retrieval in cognitive architectures. Accordingly,
results such as “memory cannot be a stimulus” are consequences of this technical meaning.
Remark 7.1(On unrecognised changes).Consider the query:“find aPotentially Aware Set
whose change you did not recognise.”At first sight this appears paradoxical. Since the system
recognises a change precisely when
∆(U,U ′) = 1,
it may seem impossible for the system to identify any case in which∆(U,U′) = 0. One might
therefore suspect that the query is internally inconsistent. However, in the AToS framework the
application of aState Transition Functionφ∈Σto aPAS U∈Ωneed not result in a recognised
change. The system may apply a transitionU↦→φ(U)such that
∆(U,φ(U)) = 0,
meaning that no change has been registered. This expresses the absence ofrecognisedchange, not
necessarily the absence of any underlying structural modification of thePAS. Such a situation
is perfectly consistent: the system can be aware of the existence ofU while also registering that
its value remains constant under all applied transitions. What is impossible, however, is for
the system to present aspecific transition U→U′that both (i) occurred and (ii) remained
unrecognised. Once the system is able to name a transitionU→U′, that transition is, by
definition, recognised, and therefore satisfies∆(U,U′) = 1. Thus the only admissible responses
to the query are states for which no recognised change occurs, not unnoticed transitions. The
apparent paradox therefore dissolves once we distinguish between (1) the possibility of aPAS
undergoing no recognised change and (2) the impossibility of reporting a concrete unrecognised
transition.
Definition 7.2(Memory).Let(Ω , Σ,S, ∆)be aSelf-Conscious System. TheMemoryof the
system is the subset
M:=
{
U(A,p)∈Ω
⏐⏐∀φ∈Σ : ∆
(
U(A,p), φ
(
U(A,p)
))
= 0
}
.
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=== Page 19 ===
In words:M consists precisely of thosePotentially Aware Setthat are∆-invariant under all
admissible state-transition functionsφ∈Σ. No internal or external operation of the system
produces a recognized change in such states; they therefore serve as the system’s stable memory.
Remark 7.3(On the meaning of Memory in AToS).The termMemoryis used in this work as
a technical notion specific to the AToS framework. It should not be confused with the standard
meaning of memory in computer science. In many discussions in philosophy of mind, memory
refers to information that the system has stably registered as part of its internal knowledge or
experience. Even if such information gradually degrades or becomes less precise over time, the
system typically does not recognise this degradation as a state change. From the perspective of
the system’s awareness, such information therefore behaves as a stable reference.
In contrast, in computer science and artificial systems, memory usually denotes a storage medium
(e.g. RAM, databases, logs) whose contents may be continuously updated and whose changes are
typically detected by the system.
Within the AToS framework, the notion of Memory is closer to the first interpretation. Memory
consists of those PAS elements whose invariance is recognised by the system, i.e. states for
which no transition produces a∆-recognised change. Conversely, the dynamically updated storage
structures typical for artificial systems are conceptually closer to the Potentially Aware SpaceΩ
itself.
Remark 7.4(Why the quantification over allφ∈Σis well-defined).The definition of Memory
requires that for everyU(A,p )∈Ωand every state-transition function φ∈Σ, the expression
φ(U(A,p ))is well-defined. This is not an additional assumption: it follows directly from
Definition 3.5. By that definition, everyφ∈Σis atotalmap
φ: Ω→Ω,
meaning thatφis defined on everyPASU∈Ω. Moreover, φpreserves the domain of eachPAS:
for everyU(A,p)∈Ω, there exists a uniquely determinedp′:X→Vsuch that
φ(U(A,p)) =U(A,p′).
Thus eachState Transition Functionacts by modifying only the value-assignmentp while keeping
the structural domainA fixed. Consequently, everyφ∈Σis mathematically applicable to every
U∈Ω, regardless of whether it is ever invoked operationally by an internal or external stimulus.
The universal quantification
∀φ∈Σ
in Definition 7.2 is therefore fully justified: Memory contains precisely thosePASthat remain
∆-invariant under all admissible transition mechanisms of the system.
Remark 7.5(Existence and Interpretation of Memory).ASelf-Conscious System, in the
strict axiomatic sense of AToS, is not required to possess a non-empty Memory in order to
satisfy Definition 3.13. The ability to attribute at least one internally caused and recognised
state-transition already suffices for self-consciousness. Accordingly, both situations,
M=∅andM̸=∅,
are fully consistent with the axioms.
Nothing in the framework requires that the system ever revisit or reactivate a memory state. APAS
may belong toM even if no internal or external stimulus ever addresses it again. What matters
is that the system retains the potential ability via any admissibleState Transition FunctioninΣ
to verify, should such a stimulus eventually occur, that the state has remained unchanged. Thus
Memory comprises precisely thosePASthat are stably preserved under all transition mechanisms,
19

=== Page 20 ===
irrespective of whether the system ever engages with them operationally. In this sense, Memory is
not a catalogue of actively accessed states but a set of structurally persistent configurations of the
system: invariant under all admissible dynamics, available for future verification, and providing
the stable background against which recognised changes acquire meaning.
8 Intelligence
Definition 8.1(Operational Composition of State Functions).LetS∈S(U(A,p ))be aStimulus
Setsuch that its update is detected,
S′:=φS(S),∆(S,S ′) = 1.
Assume that this recognised change triggersφ1∈Φon the target state, producing
φ1
(
U(A,p)
)
=U(A,p ′).
If the updated stateU(A,p′)belongs to theStimulus Setof anotherPAS,
U(A,p′)∈S(U(B,p′)),
and if the correspondingInternal Stimulus Functionsψ∈ΨandState Transition Function
φ2∈Φsatisfy
ψ
(
U(A,p′), U(B,p′)
)
=φ2
(
U(B,p′)
)
=:U(B,p ′′),
thenφ1 andφ2 are said to beoperationally composable, written
φ2 ⊚φ1.
The composite transition mapsU(B,p )↦→U(B,p′′)and is causally rooted in the initial recognised
stimulusS.
Definition 8.2(Intelligence).TheIntelligenceof the system is the smallest family ofState
Transition Function’s closed under operational composition. Formally,
I:=⟨Φ⟩⊚,
where⟨Φ⟩⊚ denotes the closure ofΦunder all finite operational compositions
φn ⊚···⊚φ1,
each arising from a chronologically valid chain of recognised stimuli
S1 ⇝S 2 ⇝···⇝Sn.
ThusIntelligenceis the totality of state-transformations that the system can construct through
iterated, stimulus-guided internal reasoning.
Remark 8.3(Operational Composition vs. Naive Function Composition).In contrast to naive
algebraic composition of functions, the notion ofIntelligenceincludes only those transformation
chains that the system canactually realisethrough a sequence
StSe−→PAS−→StSe,
activated by recognised internal changes. Operational composition requires that each step in a
chain be triggered by an internal stimulus whose state-change has been detected by the Sensitivity
function∆. Only such a recognised stimulus can activate an appropriate Imagination Function,
20

=== Page 21 ===
leading to a further state-transition. Mere formal composability of functions is therefore insuffi-
cient; a causal link grounded in the system’s architecture is required.
Consequently, the space ofIntelligenceneed not form a classically closed algebra under ordinary
function composition. Its structure may be partially non-transitive and only fragmentarily con-
nected: only those pairs of transitions are operationally composable that can be embedded into
a chronologically valid chain of recognised stimuli. Other pairs, though formally composable as
maps, are excluded because no admissible stimulus sequence can activate them in the required
order.
This reflects a fundamental aspect of intelligence: a system may count as intelligent even if it
cannot compose arbitrary pairs ofState Transition Functions, provided that it can construct
meaningful and extendable reasoning chains arising from its own internal dynamics and causal
architecture.
Lemma 8.4(∆-Invariance of Memory).Let U be a memory state inM. Then for everyState
Transition Functionφ∈Σ, the system registers no change:
∆
(
U,φ(U)
)
= 0.
Proof. This follows directly from Definition 7.2, which stipulates that memory states are invariant
under all internal and externalState Transition Functions. Hence no application ofφto U can
produce a∆-detectable difference.
Theorem 8.5(Memory Cannot Serve as a Stimulus).No memory stateU∈Mcan serve as a
Stimulus Setfor anyPotentially Aware SetU(A,p).
Proof. AStimulus Set S forU(A,p )requires the existence of aState Transition FunctionφS∈Σ
such that
∆(S,φS(S)) = 1.
By Lemma 8.4, every memory stateU∈Msatisfies
∆(U,φ(U)) = 0
for allφ∈Σ. Hence no memory state can fulfil the triggering requirement forStimulus Sets.
Therefore, memory states cannot function as stimuli.
Theorem 8.6(Memory Excluded from Operational Composition).No memory state can
participate as an intermediate state in any operational composition ofState Transition Functions
forming Intelligence.
Proof.Operational composition requires a chronologically valid sequence of recognised stimuli:
S1 ⇝S 2 ⇝···⇝Sn,
where eachSi satisfies
∆(S i,φS
i (Si)) = 1.
By Lemma 8.4, memory states never satisfy∆ = 1. Hence they cannot appear as anySi in such
a sequence. Since every compositionφi+1 ⊚φi requires a stimulus at stepi, no memory element
can participate in the construction of Intelligence.
Remark 8.7(Intelligence Is Dynamical, Not Mnemonic).Intelligence, defined as the closure of
Φunder operational composition, does not require memory states as generative elements. Memory
contributes only structurally, by providing stable information within potentially aware spaces that
undergo∆-detectable change, but never as causal drivers of state-transitions.
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=== Page 22 ===
Proof.Immediate from Theorem 8.6.
Theorem 8.8(Intelligence Does Not Require Knowledge).LetI denote the Intelligence of the
system, defined as the closure ofΦunder operational composition (Definition 8.2). ThenI does
not require any memory state as a generative component. Formally, noU∈Mcan appear in
any operational composition that constructs an element ofI.
Proof.By Lemma 8.4, every memory stateU∈Msatisfies
∆(U,φ(U)) = 0for allφ∈Σ.
Operational composition requires a chronologically valid sequence of recognised stimuli
S1 ⇝S 2 ⇝···⇝Sn,
where eachSi satisfies∆( Si,φi(Si)) = 1for some φi∈Σ. Since no memory state can satisfy the
stimulus condition∆ = 1, no memory state can appear as anySi in such a sequence. Therefore,
memory states cannot contribute to the generative structure ofI, and Intelligence is constructed
solely from∆-detectable transitions of non-memory states.
Definition 8.9(Intelligence Level).Let C be the set of all admissible finite operational composi-
tions,
c=φn ⊚···⊚φ1, φ i∈Φ,
each step of which is triggered by a recognised stimulus (∆ = 1). For each such compositionc, let
|c|denote its length. TheIntelligence Levelof the system is defined as
L(I) := sup
c∈C
|c|1.
9 Self-Awareness Level
Definition 9.1(Awareness Level).The Awareness Level of the system is
LΩ
∆ :=
⏐⏐{(U,U′)|∆(U,U′) = 1}
⏐⏐.
Definition 9.2(Self-Awareness Level).The Self-Awareness Level of the system is
LΩ
∆,Φ int :=
⏐⏐⏐
{
(U,U′)
⏐⏐∃S∈S(U),∃ψ∈Ψ,∃φ∈Φint :ψ(S,U) =φ(U) =U′,∆(U,U ′) = 1
}⏐⏐⏐.
Remark 9.3.The quantity LΩ
∆,Φ int measures how many distinct state transitions the system
both generates through its internal mechanisms and successfully recognises via∆. It therefore
quantifies the extent to which the system can detect changes that originate from its own internally
triggered dynamics.
10 Self-Conscious Retention Process
Definition 10.1(Self-Conscious Retention Process).Let U(B,p )∈S(U(A,p ))and let φB,φA∈
Φ int. Assume there exists anInternal Stimulus Functionsψsuch that
∆
(
U(B,p),φB(U(B,p))
)
= 1,
ψ
(
φB(U(B,p)),U(A,p)
)
=φA(
U(A,p)
)
:=U(A,p ′),
1The supremum supA of a set A is the least upper bound of A; it coincides with maxA when the maximum
exists.
22

=== Page 23 ===
and
∆
(
U(A,p),U(A,p ′)
)
= 1.
We say the system implementsSelf-Conscious Retentionif there existφB
1 ∈Φ int, p′′:X→V,
and anInternal Stimulus Functionsψ′such that
U(A,p′)∈S
(
φB(U(B,p))
)
,
ψ′(
φB(U(B,p)),U(A,p ′)
)
=φB
1
(
φB(U(B,p))
)
:=U(B,p ′′),
and moreover
∆
(
φB(U(B,p)),U(B,p ′′)
)
= 1.
In words: the stimulus U(B,p )undergoes an internally recognised updateφB(U(B,p )); this
updated state triggers the internal update ofU(A,p )to U(A,p′). The new stateU(A,p′)becomes
a stimulus forφB(U(B,p ))and triggers a further internal update toU(B,p′′), which the system
again recognises as a change.
Remark 10.2(Termination of Retention Loops).The Self-Conscious Retention Process, once
initiated, unfolds as an alternating sequence of internally caused and∆-registered updates between
two potentially aware statesU(A,·)and U(B,·). The dynamics can be schematically expressed as
U(B,q 0)→U(B,q 1)⇒U(A,p 0)→U(A,p 1)⇒
⇒U(B,q1)→U(B,q 2)⇒U(A,p 1)→U(A,p 2)⇒ ···⇒
⇒
{
U(A,p n−1)→U(A,pn),
U(B,q n−1)→U(B,qn)
until
∆(U(A,p n−1),U(A,p n)) = 0or∆(U(B,q n−1),U(B,q n)) = 0.
That is, at each stage the newly produced state on one side becomes a stimulus for the other
side, triggering a further internal update as long as the transition is recognised by the sensitivity
function∆. The retention loop terminates at the first indexn at which∆fails to register the
induced change on either branch of the alternation.
Remark 10.3(Temporal Character of Self-Conscious Retention).Without retention, an act of
self-awareness is essentially instantaneous: the system detects a change, attributes it internally,
and the event is over. By contrast, if Self-Conscious Retention processes are implemented, the
act of self-awareness can unfold over time. In this case, each newly produced state may itself
act as a stimulus for further internal updates, generating a temporally extended chain of self-
referential recognition. This parallels the human mind, where a single act of self-awareness (e.g.
“I am aware that I am thinking”) can evolve dynamically into an extended process of reflection.
Thus, Self-Conscious Retention transforms self-awareness from a single recognition event into
a potentially continuous cognitive process, subject to the detection-based termination condition
∆ = 0; a stable recurrent cycle with∆ = 1can instead persist indefinitely (see Section??).
Remark 10.4(Retention as the Simplest Form of Intelligence).The Self-Conscious Retention
Process constitutes the minimal non-trivial instance of systemic intelligence in the sense of
Definition 8.9. A single∆-registered update realises only self-awareness, but a retention chain
realises anoperational compositionof internal imagination functions: each newly produced state
becomes an operator that transforms the next state. Thus, retention is the simplest mechanism
by which a system expands its reachable space of internal transformations. In this sense, Self-
Conscious Retention is the lowest-level manifestation of intelligence: a constructive process that
extends reasoning beyond isolated updates and creates a multi-step internal computation.
23

=== Page 24 ===
11 Operational Self-Awareness Criterion
In this section we introduce an operational, theory-neutral criterion for minimal self-awareness.
The proposed criterion plays a role analogous to the Turing Test for intelligence, but targets
internal self-causation rather than externally observed behaviour. The Turing Test evaluates
intelligence, and the mirror test evaluates bodily self-recognition, whereas the GSAT evaluates
internal causal self-recognition.
Definition 11.1(General Self-Awareness Test).A system satisfies theGeneral Self-Awareness
Test (GSAT)if and only if, under conditions ofno external stimuli, there exists at least one
instance in which:
1. the system, by virtue of its own internal dynamics, brings about a change in at least one of
its internal states, and
2. the system is capable of recognising the source of this change, correctly classifying it as
internally causedrather than resulting from external influence.
The GSAT is anexistentialcriterion: the existence of at least one such internally generated and
internally recognised state change is sufficient. The test does not require global introspection, a
complete self-model, or full transparency of all system states.
Remark 11.2(AToS instantiation of GSAT).By Theorem 4.1, a system is self-aware in the
general sense (Definition 1.2) if and only if it admits an AToS representation⟨Ω, Σ,S, ∆⟩
satisfying Definition 3.13. Accordingly, theGeneral Self-Awareness Test (GSAT), formulated in
a theory-neutral manner, admits a precise and faithful instantiation within AToS.
In AToS terms, GSAT holds if and only if there exist a target stateU∈Ω, an internal stimulus
S∈S(U)(aStSe), an imagination function Imagination Function∈Φ int, and a sensitivity
function∆such that, in the absence of external stimuli,
ψ(S,U) =Imagination Function(U)and∆
(
U,Imagination Function(U)
)
= 1,
for some internal stimulus functionψ∈Ψ. That is, an internally generated stimulusStSe
triggers anImagination Function, which induces a change in aPAS-state, and this induced
change is normatively recognised by∆as internally caused.
Formally,∆does not merely register the occurrence of a state transition, but constitutes its
recognition as an internally generated event within the system’s decision structure. Thus, GSAT
corresponds in AToS to the existence of a non-trivial internally driven causal loop involvingStSe,
Imagination Function,PAS, and∆.
In AToS, such triggering presupposes that the change of the stimulus itself is∆-recognised; without
∆-recognition at the stimulus side, no imagination function can be operationally invoked.
This establishes GSAT as an operational characterisation of self-awareness whose necessary and
sufficient formal realisation is provided by AToS, without presupposing acceptance of the theory
itself.
Near-miss counterexample (non-self-aware).Consider a closed system with sensors and
with both externally-triggered and internally-triggered state updates, and assume it logs its
own transitions. Suppose, however, that the system lacks any internal mechanism that can
attributea logged change to the internal mechanism as opposed to the external one (i.e. no
witnessing/attribution map that separatesΦint fromΦ ext in the sense required by Def. 1.1). Then
the system may exhibit rich dynamics and even persistent records, yet it fails to be self-aware
in the present sense because it cannot recognize a changeas internally caused. This illustrates
that AToS self-awareness is not implied by sensing, dynamics, or logging alone; the attribution
capability is essential.
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=== Page 25 ===
Remark 11.3(Operational interpretation of external stimuli in GSAT).In practical systems it is
often argued that a complete absence of external stimuli is difficult to realise. Real implementations
typically contain background processes such as clocks, schedulers, interrupts, hardware noise,
or slow physical drift of system parameters. Within the AToS framework such phenomena do
not require a special ontological treatment. AToS does not classify triggers according to their
physical origin (e.g. noise, clock, scheduler), but according to their operational role in the system’s
transition structure. Formally, the distinction between internal and external transitions is
determined solely by the trigger labelλ:T →{int,ext}(Definition 3.9). A recognised transition
is internal precisely when its activation admits a stimulus-side witnessing structure within the
system.
Consequently, phenomena such as noise, timers, or scheduling mechanisms may appear either as
Imagination Functionor asInspiration Function, depending on how they are integrated into the
system’s architecture. If such mechanisms are represented asPASelements participating in the
stimulus systemS(U)and their changes are recognised by∆, they function as ordinary internal
stimuli and therefore belong toΦint.
In this sense, AToS treats these mechanisms as ordinary components of the system’s dynamics
rather than as exceptional disturbances. A system whose only internally recognised transitions
originate from such processes would still satisfy Definition 1.2, although such a system would
typically exhibit only a very limited form of self-awareness.
Remark 11.4(Empirical accessibility of internal attribution).The GSAT criterion is formulated
at the level of the system’s internal causal structure. Accordingly, a distinction must be made
between theexistenceof internal causal attribution and itsempirical accessibilityto an external
observer.
In general, the fact that a system satisfies GSAT is an ontological property of the system’s
transition structure, whereas the ability to verify GSAT is an epistemic matter that depends on
the observer’s access to the system’s internal organisation.
For empirical verification, the mere observation of a state change in the absence of external input
is not sufficient. A state transition may occur for many reasons, including processes that are not
recognised by the system as internal causes. Therefore, empirical evidence for internal attribution
requires at least the following conditions:
1. absence of recognised external stimuli at the time of transition,
2. occurrence of a recognised internal state transition,
3. existence of a system-level mechanism that classifies the source of this transition as internal,
4. structural or interventional confirmation that the triggering process originated within the
system’s recognised causal structure.
In artificial systems, these conditions may be directly testable because internal states, trigger path-
ways, and attribution mechanisms can be made inspectable through logs, architectural constraints,
or controlled interventions. For example, one may experimentally disable all recognised external
inputs and then selectively disable internal triggering modules; if the attributed internal transitions
disappear under such intervention, this provides evidence for genuine internal attribution rather
than unrecognised perturbation.
In biological or poorly accessible systems, such verification may be difficult or impossible in
practice. This limitation, however, reflects the epistemic limits of observation rather than a
limitation of the GSAT criterion itself.
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=== Page 26 ===
12 AToS and the Chinese Room Argument
12.1 The Chinese Room Argument
The Chinese Room argument, introduced by John Searle, is one of the most influential arguments
against the claim that symbol manipulation alone is sufficient for understanding or consciousness.
In the classical formulation, a person inside a room follows a set of syntactic rules to manipulate
Chinese symbols without understanding their meaning. From the outside, the system appears
to understand Chinese, although internally it performs only formal symbol manipulation. The
argument is commonly interpreted as showing that passing the Turing Test or producing intelligent
behaviour is not sufficient for genuine understanding or consciousness. The Chinese Room
therefore distinguishes between behavioural competence and genuine understanding.
12.2 Symbol Manipulation vs. Recognised State Transitions
The AToS framework introduces a different distinction. Instead of focusing on semantics, AToS
focuses on the internal structure of state transitions and their causal attribution. In AToS,
self-awareness is not defined in terms of language, meaning, or behaviour, but in terms of the
system’s ability to:
1. recognize at least one change in its internal state, and
2. attribute the cause of that recognized change to its own internal mechanism rather than to
an external stimulus.
A system may therefore manipulate symbols, produce correct answers, and simulate intelligent
behaviour, while still failing the AToS self-awareness criterion if it lacks internal recognition of
state transitions and internal causal attribution.
12.3 Why the Chinese Room Is Not Self-Aware in the AToS Framework
From the perspective of AToS, the crucial question is not whether the system produces correct
outputs, but whether it possesses an internal mechanism that:
•detects its own internal state transitions, and
•represents at least one of these transitions as being caused by its own internal processes.
A purely syntactic system may implement arbitrarily complex input–output mappings while lacking
any internal causal model of its own state transitions. Such a system would fail the AToS self-
awareness criterion even if it perfectly simulated intelligent behaviour in the sense of the Turing
Test. In this sense, AToS separates three notions that are often conflated:
1. behavioural intelligence,
2. symbol manipulation,
3. internal causal self-awareness.
AToS claims that only the third notion is structurally sufficient for minimal self-awareness.
12.4 Internal Causal Attribution as the Missing Component
The Chinese Room argument shows that symbol manipulation is not sufficient for understanding.
AToS strengthens this conclusion by showing that symbol manipulation is also not sufficient for
self-awareness. What is missing is not semantics but internal causal attribution. In the AToS
framework, the self is not defined as a particular object, representation, or symbol. Instead, the
26

=== Page 27 ===
self appears as a causal role within the system’s transition structure. A system represents itself
as a cause if it can classify at least some of its own recognised state transitions as being caused
by its own internal stimulus structure. Thus, a system that merely manipulates symbols without
representing itself as a causal source of its own state transitions does not satisfy the minimal
structural condition for self-awareness defined in this work.
12.5 Relation to the Turing Test
The Turing Test evaluates behavioural indistinguishability, whereas the AToS criterion evaluates
internal causal structure. These criteria are logically independent. Passing the Turing Test is
neither necessary nor sufficient for self-awareness in the AToS sense: a system may be self-aware
without human-like behaviour, and a system may exhibit human-like behaviour without being
self-aware. The AToS criterion therefore addresses a different question: not whether a system
behaves intelligently, but whether it can recognise and internally attribute its own state transitions.
More broadly, the Turing Test evaluates behaviour, and the Chinese Room argument targets
semantic understanding, whereas the GSAT evaluates internal causal self-recognition. These
three criteria therefore address three different questions:
1. what a system does,
2. what a system understands,
3. whether a system can represent itself as a cause of its own state changes.
In this sense, behavioural intelligence, semantic understanding, and self-awareness correspond
to three distinct explanatory levels. The AToS framework is concerned only with the third level
and proposes internal causal attribution of recognised state transitions as a minimal structural
condition for self-awareness.
Intelligence concerns what a system does, understanding concerns what a system means, and
self-awareness concerns whether a system can recognise itself as a causal source of its own state
transitions.
13 Family of Self-Conscious Systems
Definition 13.1(Family of Self-Conscious Systems).Let S ={Si}i∈Ibe a family ofSelf-
Conscious Systems, each of the form
Si =⟨Ωi,Σ i,S i,∆ i⟩,
defined over a common carrierX, and a common value setV. We define theFamily(meta-system)
S∗=⟨Ω∗,Σ ∗,S ∗,∆ ∗⟩
as follows.
(1) Meta-level carrier and value set.
X∗:=I×X, V ∗:=V.
(2) Meta-level target family.
U∗:=
{
{i}×X
⏐⏐ i∈I
}
.
(3) Meta-level assignment set.
P∗:={p∗:X∗→V∗}={p∗:I×X→V}.
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=== Page 28 ===
(4) Meta-levelPAS.ForA i ={i}×X∈U∗andp∗∈P∗define
U∗(Ai,p∗) :=
{
⟨(i,x), p∗(i,x)⟩
⏐⏐ x∈X
}
⊆X∗×V∗.
(5) Meta-levelPotentially Aware Space.Define
Ω∗:=
{
U∗(Ai,p∗)
⏐⏐ i∈I, p∗∈P∗}
.
(6) Meta-level sensitivity function.Leti∈Iand letU∗(Ai,p∗)∈Ω∗. Define
Πi
(
U∗(Ai,p∗)
)
:=U i
(
A, pi
)
,
whereA∈Ui is the target used in the definition ofΩi, and
pi :X→V, p i(x) :=p∗(i,x).
Define the sensitivity function
∆∗: Ω∗×Ω∗→{0,1}
as follows. For anyi∈Iand anyp∗,p′∗∈P∗, set
∆∗(
U∗(Ai,p∗), U∗(Ai,p′∗)
)
= 1
if and only if there exist assignmentspi,p′
i∈Pi and a targetA∈Ui such that
Ui(A,pi), Ui(A,p′
i)∈Ωi,∆ i
(
Ui(A,pi), Ui(A,p′
i)
)
= 1,
Πi
(
U∗(Ai,p∗)
)
=U i
(
A, pi
)
andΠ i
(
U∗(Ai,p′∗)
)
=U i
(
A, p′
i
)
.
In all other cases, define∆∗arbitrarily.
(7) Meta-level State Transition Functions.The notion of aState Transition Function
extends naturally to a family of self-conscious systems as ameta-level State Transition Function,
defined as follows.Ameta-level state transition functionis a map
φ∗: Ω∗→Ω∗
such that for everyU∗(Ai,p∗)∈Ω∗there existU∗(Ai,p′∗)∈Ω∗, a state transition function
φi∈Σi, and statesUi(A,p), Ui(A,p′)∈Ωi satisfying
φi(
Ui(A,p)
)
=U i(A,p′),
φ∗(
U∗(Ai,p∗)
)
=U ∗(
Ai,p′∗)
,
and
Πi
(
U∗(Ai,p∗)
)
=U i
(
A,p
)
andΠ i
(
U∗(Ai,p′∗)
)
=U i
(
A,p′)
.
The action ofφ∗on all other elements ofΩ∗is left unspecified.
Themeta-level state transition function setis defined as
Σ∗:=
{
φ∗∈(Ω∗)Ω ∗ ⏐⏐φ∗satisfies the conditions specified above
}
.
(8) Meta-level Internal Stimulus FunctionWe define anInternal Stimulus Functionsfor a
family of self-conscious systems as ameta-level Internal Stimulus Functionin the following way.
A map
ψ∗: Ω∗×Ω∗−→Ω∗
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=== Page 29 ===
is a meta-level Internal Stimulus Function iff for everyi∈Ithere exist meta-level state transition
functionsφ∗B,φ∗A∈Σ∗such that for all meta-statesU∗(Ai,p∗)∈Ω∗,
∆∗(
U∗(Bi,p∗), φ∗B(U∗(Bi,p∗))
)
= 1 =⇒
ψ∗(
φ∗B(U∗(Bi,p∗)), U∗(Ai,p∗)
)
=φ∗A(U∗(Ai,p∗)),
whereAi,Bi∈U∗are targets associated with the same agenti, and the induced local states satisfy
Πi
(
φ∗B(U∗(Bi,p∗))
)
=φB(
Πi(U∗(Bi,p∗))
)
,
Πi
(
φ∗A(U∗(Ai,p∗))
)
=φA(
Πi(U∗(Ai,p∗))
)
for someφB,φA∈Σi. We denote by
Ψ∗:=
{
ψ∗: Ω∗×Ω∗→Ω∗⏐⏐ψ∗is a meta-levelInternal Stimulus Functions
}
theMeta-level Internal Stimulus Functions Set.
(9) Meta-level Stimulus SetLetU∗(Ai,p∗)∈Ω∗. We define aStimulus Setfor a family of
self-conscious systems as ameta-level Stimulus SetU∗(Bi,p∗)∈Ω∗forU ∗(Ai,p∗), written
U∗(Bi,p∗)∈S∗(
U∗(Ai,p∗)
)
,
iff there exist meta-level state transition functionsφ∗B,φ∗A∈Σ∗and a meta-level internal
stimulus functionψ∗∈Ψ∗such that
∆∗(
U∗(Bi,p∗), φ∗B(U∗(Bi,p∗))
)
= 1
and
ψ∗(
φ∗B(U∗(Bi,p∗)), U∗(Ai,p∗)
)
=φ∗A(U∗(Ai,p∗)).
Themeta-level Stimulus Systemis the collection of all meta-levelPASthat can serve as meta-level
Stimulus Sets for some target meta-state inΩ∗:
S∗:=
{
U∗(Bi,p∗)∈Ω∗⏐⏐∃U∗(Ai,p∗)∈Ω∗such thatU∗(Bi,p∗)∈S∗(
U∗(Ai,p∗)
)}
.
Equivalently,S∗is the union of all meta-level stimulus domainsS∗(U∗)overU ∗∈Ω∗.
Theorem 13.2(Self-Consciousness of a Family).Let
S∗=⟨Ω∗,Σ∗,S∗,∆ ∗⟩
be a family of self-conscious systems as in Definition 13.1. ThenS∗is aSelf-Conscious System
in the sense of Definition 3.13 if and only if there exist meta-statesU∗(Ai,p∗),U∗(Bi,p∗)∈Ω∗
such that
U∗(Bi,p∗)∈S∗(
U∗(Ai,p∗)
)
,
and there exists a meta-level internal state transition functionφ∗
int∈Σ∗satisfying
ψ∗(
U∗(Bi,p∗), U∗(Ai,p∗)
)
=φ∗
int
(
U∗(Ai,p∗)
)
,
together with
∆∗(
U∗(Ai,p∗),φ∗
int(U∗(Ai,p∗))
)
= 1.
Proof. By Definition 3.13,S∗is aSelf-Conscious Systemiff there existU∗∈Ω∗andS∗∈S∗(U∗)
such that
ψ∗(S∗,U∗) =φ∗
int(U∗)and∆ ∗(
U∗,φ∗
int(U∗)
)
= 1,
whereφ∗
int is the internal transition function associated with the witnessing triple forS∗∈S∗(U∗).
Unfolding Definition 13.1(9), we may writeU∗=U∗(Ai,p∗)and S∗=U∗(Bi,p∗)for some i∈I
andp∗∈P∗, withU∗(Bi,p∗)∈S∗(U∗(Ai,p∗)). This is exactly the stated condition.
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=== Page 30 ===
Remark 13.3.In the above definition, the operatorΠi serves as the sole formal bridge between
the meta-level familyS∗and the individual systemsSi. Both the meta-level sensitivity function
∆∗and the meta-level state transition functionsφ∗are entirely induced by the corresponding local
operators∆ i and φi via this projection. Consequently, the family meta-systemS∗introduces no
new internal dynamics or awareness mechanisms beyond those already present in its constituent
self-conscious systems.
Remark 13.4.
(i) An inter-agent signalSj→Si is internal to the family (sinceSj∈Ω∗) while constituting
an external stimulus for Si; accordingly, localInspiration Functions lift to meta-level
Imagination Functions.
(ii) The choice of the target stateU∈Ωi in(2)is determined by the concrete interaction
pattern and can be formalized via meta-level internal stimulus functionsΨ∗.
(iii) The setΦ∗
ext captures influences exogenous toΩ∗; in the minimal construction one may set
Φ∗
ext =∅.
(iv) The edge relationE∗provides a convenient encoding ofS∗for graph- or network-based
analyses.
(v) The sensitivity function∆ ∗registers an inter-agent stimulus if and only if it induces a
change that is detectable by the target system’s own sensitivity∆Si.
Remark 13.5(Local vs. Meta-level Stimuli).For an individual systemSi, the state of another
system Sj (j̸=i) acts as anexternalstimulus. However, at the level of the family meta-system
S∗, the same interaction is internal, sinceUj∈Ω∗. Thus, inter-agent interactions are external
at the local level while constituting internal stimulus sets at the meta-level.
Proposition 13.6(Self-Consciousness via Inter-Agent Interaction).A family of self-conscious
systems S∗is a self-conscious system in the sense of Definition 3.13 if and only if there exists an
inter-agent interaction that induces a meta-level stimulus setS∗∈S∗(U∗)witnessing a meta-level
internal state transition detectable by∆∗.
Remark 13.7(Limits of Level Aggregation and Prospects for Synergy).Within the present
axiomatic framework, there is no simple, monotonic, or generally valid relationship between the
Awareness Level, the Self-Awareness Level, or the Intelligence Level of a family of self-conscious
systems and the corresponding levels of its constituent systems. A state change that is detected by
an individual system may remain undetectable to other systems or to the family meta-system as
a whole, and operational composability of state-transition functions is inherently system-relative
and constrained by the local stimulus structure. Consequently, level-based properties cannot be
inferred by direct aggregation, maximisation, or summation across agents. At the same time,
this limitation delineates a natural direction for further research: the systematic investigation
of structural conditions and additional mechanisms under which inter-system interactions could
give rise to genuine synergistic effects, potentially enabling amplification or transformation of
awareness- or intelligence-related capacities at the family level.
14 Topological Space
Definition 14.1(Topological Space of Self-Conscious Systems).LetC be the class of allSelf-
Conscious Systems defined on a fixed carrierX and value setV. WriteSi =⟨Ωi, Σi,Si, ∆ i⟩and
letΩ total :=⋃
Si∈CΩi. DeclareB :={Ωi :Si∈C}to be a base and letT be the topologygenerated
byB, i.e. the family of all unions of finite intersections of members ofB that are again unions
of elements ofB. Then(Ω total,T)is called thetopological space of self-conscious systems.
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Theorem 14.2.The structure(Ω total,T)defined in Definition 14.1 is a topological space.
Proof.We verify the three axioms of a topology.
Empty set and total space.By definition, the empty system is included in C, hence
∅= Ω ∅∈T. AlsoΩtotal =⋃
Si∈CΩi belongs toTby construction.
Arbitrary unions.If {Uα}α∈A⊆T, then eachUαis a union of someΩj. Hence⋃
α∈AUαis
still a union ofΩj’s, and thus belongs toT.
Finite intersections.LetU 1 =⋃
j∈J1 Ωj andU 2 =⋃
j∈J2 Ωj be two elements ofT. Then
U1∩U2 =
⋃
j∈J1
⋃
k∈J2
(Ωj∩Ωk).
ButΩ j, Ωk⊆Ω total are themselves Potentially Aware Spaces. By the intersection property
(Definition 3.2), their intersection is again a validΩof some system inC. ThereforeU1∩U2∈T.
The same reasoning extends to finite intersections by induction. Thus all three axioms are
satisfied, and(Ω total,T)is a topological space.
Theorem 14.3(Topological universality of self-aware system construction).Let( T,τ)be
an arbitrary nonempty topological space. A self-aware system in the sense of AToS can be
constructed over(T,τ)by introducing a suitable internal state structure and an internal attribution
function Imagination Function. The construction does not require any particular topology, metric,
geometry, or notion of distance.
Proof. First, suppose thatT contains at least two points. Choose a nonempty proper subset
A⊂Tand let
B=T\A.
Assign to each of the two subsets a binary state, represented by one of two colours. Define the
internal transition function by
φint(a,b) = (1−a,1−b), a,b∈{0,1}.
Thus, every change of the state assigned toA is accompanied by a corresponding change of the
state assigned to its complementB. The construction is independent of the topologyτ. If T
consists of a single point, the distinction between the system and its internal reference is realised
through self-reference: the system serves as its ownStSe. In either case, the construction provides
an internal state structure capable of supporting the self-attribution mechanism required by
AToS.
Theorem 14.4(Existence of a self-aware system with infinite intelligence level).Within the
AToS framework, there exists a self-aware system whose Intelligence Level satisfies
L(I) =∞.
Such a system can be constructed using a sequence of connected components of the complement
of the Cantor set.
Proof.LetC⊂[0,1]be the standard Cantor set and put
In =
( 1
3n+1, 2
3n+1
)
, n∈N 0.
These intervals are pairwise disjoint connected components of[0, 1]\C, and both sequences of
endpoints converge to0. Set
X=
∞⋃
n=0
In.
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Represent a configuration by an assignment
p:X−→{0,1}
that is constant on eachIn. The values0and1represent two colours. Write
Un(p) :=U(I n,p)
for the corresponding PAS. For eachn≥0, define a local state-transition mechanismφn+1 that
reverses the colour onIn+1. Its update of the global assignment is
p′(x) =
{
1−p(x), x∈In+1,
p(x), x /∈In+1.
Equip the system with an internal stimulus structure such that a recognised change onIn triggers
φn+1 onI n+1. In particular, at the relevant configuration,
Un(p)∈S
(
Un+1(p)
)
.
The triggering mechanism records the source of each update internally. Define the detection and
attribution mechanisms on these transitions by
∆
(
Un+1(p),Un+1(p′)
)
= 1, λ
(
Un+1(p),φn+1
)
= int.
Thus eachφn+1 operates as an Imagination Function: its invocation is caused by a recognised
internal change, and the system identifies that change as its source. Starting from a recognised
change onI 0, the system therefore admits the causal cascade
I0 ⇝I 1 ⇝I 2 ⇝···.
Every step after the initiating change is internally triggered, recognised, and internally attributed.
In particular, the system satisfies the AToS criterion of self-awareness. For everyN≥1, the first
Ninternally triggered updates form an admissible finite operational composition
cN =φN ⊚···⊚φ1,|c N|=N.
Consequently, by Definition 8.9,
L(I) = sup
c∈C
|c|≥Nfor everyN≥1.
Hence
L(I) =∞.
Remark 14.5.The conclusion concerns the formal Intelligence Level of AToS. It requires
admissible operational compositions of arbitrarily large finite length, not the completion of
infinitely many transitions in finite time.
Definition 14.6(Hereditary admissible-domain family).An admissible-domain family U is
hereditary on nonempty subsetsif
A∈U,∅̸=B⊆A=⇒B∈U.
This is an additional structural property of a subclass of AToS models; it is not assumed in
Definition 3.2.
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Proposition 14.7(Restriction of a PAS).In a hereditary AToS model, ifU(A,p )∈Ωand
∅̸=B⊆A, then
U(A,p)∩(B×V) =U(B,p)∈Ω.
Proof. The hereditary property makesB admissible. Restriction of the graph ofp toB is exactly
U(B,p)by Definition 3.2.
Remark 14.8(Discrete topology is optional).The discrete topology2X may be placed on any
index setX, but topology and the admissible-domain family are different objects. Our choice
U = 2X\{∅}is a hereditary covering family and does not rely on topological openness. In
particular, no topological property alone supplies an internal stimulus witness; that witness is
constructed above.
15 Related Work
We briefly compare five influential approaches to consciousness or self-monitoring with the present
axiomatic proposal. Our comparison focuses on three structural criteria that are central to the
present framework: (i) whether the approach explicitly separates internal from external causes of
change, (ii) whether it provides a persistent and formally defined memory orPotentially Aware
Spacelog K, and (iii) whether it supports an explicit mechanism for self-extension or creative
novelty (our notion ofWisdom). We briefly compare five influential approaches to consciousness
or self-monitoring with the present axiomatic proposal.
15.1 Integrated Information Theory (IIT)
Tononi’s IIT equates phenomenal consciousness with the amount of integrated information
(Φ) produced by a causal network [1], [2]. While IIT shares our focus on the structure of
state transitions, it tightly couples consciousness to aspecificphysical substrate and does not
distinguish between internal and external sources of change.
15.2 Modal Logics of Self-Knowledge
Gamez [3] proposes a modal calculus capable of expressing nested epistemic claims such as “the
system knows that it knows.” This resonates with our concept ofPotentially Aware Space, though
it lacks an explicit separation between internal and external transformations.
15.3 Meta-Management Architectures
Sloman & Chrisley [5] distinguish reactive, deliberative, and meta-management layers in cognitive
agents. The meta-layer monitors both internal and external events—a conceptual analogue to
our setsΦ int andΦ ext. However, their approach is architectural, not axiomatic.
15.4 Algorithmic Self-Modelling
Schmidhuber’s Gödel machine [6] rewrites its own code if it can prove the change will increase
expected utility. This self-modification aligns conceptually with our notion ofWisdom, though its
foundation is proof-theoretic rather than sensitivity-based.
15.5Φ-Causal Set Formulations
Barrett & Seth [7] recast IIT in set-theoretic terms, yielding a more abstract formulation, but
still centered around measures of integrated information.
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15.6 Survey Evidence for a Missing Axiomatic Framework
A recent comprehensive survey of AI consciousness research by Shueb and Che [12] concludes
that existing approaches—ranging from IIT and global workspace models to formal logics, self-
models, and meta-cognitive architectures—remain descriptive rather than operational. The review
explicitly notes the absence of (i) a causal distinction between internally and externally generated
state transitions, (ii) a persistent and formally definedPotentially Aware Spacelog, and (iii) any
necessary and sufficient criterion for self-awareness. These gaps directly motivate the axiomatic
structure developed in the present work.
16 Summary
The Chinese Room argument demonstrates that behaviour alone is not a sufficient indicator of
understanding. The AToS framework extends this insight by proposing a structural criterion
for self-awareness based on recognised state transitions and internal causal attribution. In this
framework, self-awareness is not a matter of symbol manipulation or behavioural performance,
but of the system’s ability to recognise at least one of its own state changes and to represent
itself as the internal cause of that change. This provides a substrate-independent and operational
criterion for minimal self-awareness that applies equally to biological, artificial, and abstract
systems.
Figure 1:Comparison of AToS with selected approaches to consciousness and self-awareness.
References
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