There are several ways in which transition systems and their relatives appear in the literature relevant to us.

Examples 2.4. Let (X, U, T) be a transition system.

We can reformulate the notion of a history information space introduced by LaValle (2006) as follows. In this context, X is an external state space that characterizes the robot's configuration, velocity, and environment, U is an action space, f is a state transition mapping that produces a next state from a current state and action, h is a sensor mapping that maps states to observations, and Y is a sensor observation space. As in LaValle (2006), for each x ∈ X, let Ψ(x) be a finite set of “nature sensing actions” and for each x ∈ X and u ∈ U let Θ(x, u) be a finite set of “nature actions.” Let X_Ψ = {(x, ψ) ∣ ψ ∈ Ψ(x)} and let h:X_Ψ → Y be a “sensor mapping” where Y is a set called the “observation space.” Let X_Θ = {(x, u, θ) ∣ θ ∈ Θ(x, u)} and let f:X_Θ → X be the “transition function.” The following definition is an adaptation from LaValle (2006).

Definition 2.5. A valid history I-state for X, Ψ, Θ, f is a sequence (u₀, y₀, …, u_k−1, y_k−1) of length 2k for which there exist x̄=(x0,…,xk-1), ψ̄=(ψ0,…,ψk-1) and θ̄=(θ0,…,θk-2) such that for all i < k we have

In this case we say that (u₀, y₀, …, u_k−1, y_k−1) is witnessed by x̄, ψ̄ and θ̄.

Now let I be the set of all valid history I-states for X, Ψ, Θ, f. For all k ∈ ℕ, all x̄∈Xk-1, all ψ̄=(ψ0,…,ψk-1) and all θ̄=(θ0,…,θk-2), let Ik(x̄,ψ̄,θ̄) be the set of all valid paths (u₀, y₀, …, u_k−1, y_k−1) witnessed by x̄, ψ̄, and θ̄. Now let T⊆I×(U×Y)×I be defined by

Here, x^⌢y is the concatenation of sequences x and y. Then (I,U×Y,T) is the history I-space transition system.

Suppose for each x, y ∈ X there is at most one u ∈ U with x→uy. Let

and let l:E_T → U be defined so that l((x, y)) is the unique u such that x→uy. Then (X, E_T, l, x₀) with x₀ ∈ X is a passive I-state graph as in O'Kane and Shell (2017, Def 1).

The following definition is more of a notational than mathematical value.

Definition 2.6. Let X=(X,U,T) be a transition system. If for all (x, u) ∈ X × U there is a unique y ∈ X with (x, u, y) ∈ T, then we denote the function (x, u) ↦ y by τ, and write (X, U, τ) instead of (X, U, T). In this case we call X an automaton. Note that usually in computer science literature an automaton is finite and also has an initial state and a set of accepting states, but we do not have those in our definition.

For automata we also use the notation x * u = τ(x, u) and if ū = (u₀, …, u_k−1), then x * ū is defined by induction for k > 1 as follows: x * (u₀, …, u_k−1) = (x * (u₀, …, u_k−2)) * u_k−1.

Examples 2.7. Automata and transition systems can model agent-environment and related dynamics.

Next, we will define a sensorimotor system, which is a special case of a transition system. Following the tenet (EA1) that “environment is inseparable from the body which is inseparable from the brain,” our sensorimotor systems can model any part of the environment-body-brain coupling. The model that describes the environment differs from the one that describes the agent merely in the type of structure it possesess, but not in an essential mathematical way.

SM-systems can be thought of as a partial specification of (some part of) the brain-body-environment coupling. Physicalist determinism demands that under full specification² we are left with a deterministic system. A specification is partial when it leaves room for unknowns in some, or all, parts of the system.

Definition 2.8. A sensorimotor system (or SM-system) is a transition system (X, U, T) where U = S × M for some sets S and M, which we call in this context the sensory set and the motor set, respectively.

The interpretation is that if x→(s,m)y, then s is the sensation that either occurs at x, or along the transition to the next state y, and m the motor action which leads to the transition. We will show later how SM-systems can be connected together (Definition 2.22) to form coupled systems. Sometimes an SM-system is modeling a brain-body totality, and other times it is modeling body-environment totality. A coupling between these two will model the brain-body-environment totality. This is a flexible framework which enables enactivist-style analysis. We do not assume that the agent “knows” the effect of a given m ∈ M or that the “meaning” of a given s ∈ S. The sets S and M are purely mathematical sets denoting the interface between the agent and the environment from the third person perspective.

In fact, the sensory and motor components can be decoupled which might be more natural from the mathematics' point of view in some cases. The following shows that we can look at it both ways.

Definition 2.9. An asynchronous SM-system is a transition system (X, U, T) such that there exist partitions U = S ∪ M and X = X_s ∪ X_m such that for all (x, u, y) ∈ T we have

Thus, the state space of a sequential SM-system contains separate sensory states and motor states.

Definition 2.10. Suppose E is an equivalence relation on a set X. We say that a map f:X → X is E-preserving if for all x, y ∈ X, we have xEy ⇔ f(x)Ef(y).

There is a natural correspondence between SM-systems and their asynchronous counterpart:

Theorem 2.11. Let SM and aSM be the classes of SM-systems and asynchronous SM-systems, respectively. There are functions F:SM → aSM and G:aSM → SM such that

Proof. See Appendix B.               □

Another type of a system, which is in a similar way equivalent to a special case of an SM-system, is a state-labeled transition system which we will introduce next, and prove a similar result, Lemma 2.19.

The amount of information specified in a given SM-system depends on which part of the brain-body-environment system we are modeling. At one extreme, we specify the environment's dynamics down to the small detail and leave the brain's dynamics completely unspecified. In this case the SM-system will have only one sensation corresponding to each state and the transition to the next state will be completely determined by knowing the motor action. This is, in a sense, the environment's perspective. At the other extreme, we specify the brain completely, but leave the environment unspecified. We “don't know” which sensation comes next, but we “know” which motor actions are we going to apply. This is in a sense the perspective of the agent. The first extreme case is the perspective often taken in robotics and other engineering fields when either specifying a planning problem (Ghallab et al., 2004; Choset et al., 2005; O'Kane and LaValle, 2008), or designing a filter (Hager, 1990; Thrun et al., 2005; LaValle, 2012; Särkkä, 2013) (also known as sensor fusion). This is why we call SM-systems of that sort quasifilters (Definition 2.12). The other extreme is the perspective of a policy. The policy depends on sensory input, but the motor actions are determined (by the policy). This is why we call the SM-systems of the latter sort quasipolicy. The “quasi-” prefix is used because both are weaker and more general notions than those that appear in the literature; see Remarks 2.20 and 2.21.

Another way to look at this is the dichotomy between virtual reality (VR), and robotics. In virtual reality, scientists are designing the (virtual) environment for an agent whereas in robotics they are typically designing an agent for an environment. In the former case the agent is partially specified: the type of embodiment is known (S and M are known) and some types of patterns of embodiment are known (eye-hand coordination). However, the specific actions to be taken by the agents are left unspecified. The job of the designer is to specify the environment down to the smallest detail, so that every sequence of motor actions of the agent yields targeted sensory feedback. The VR-designer is designing a quasifilter constrained by the partial knowledge of the agent's embodiment and internal dynamics. The case for the robot designer is the opposite. She has a partial specification of the robot's intended environment and usually works with a complete specification of the robot's mechanics. She is designing a quasipolicy. For VR-designers the agent is a black box; for roboticists the agent is a white box (Suomalainen et al., 2020) (unless the task is to reverse engineer an unknown robot design). For the environment, the roles are reversed. A similar dichotomy can be seen between biology (in which the agent is a black box) and robotics (in which it usually is a white box).

All the definitions in this section are new.

Definition 2.12. Suppose that (X, S × M, T) is an SM-system with the property that for all x₁ ∈ X there exists s_x1 ∈ S such that for all x₂ ∈ X and all (s, m) ∈ S × M we have that x1→(s,m)x2 implies s = s_x1. Then, (X, S × M, T) is a quasifilter.

In a quasifilter the sensory part of the outgoing edge is unique. The dual notion (quasipolicy) is when the motor part is unique:

Definition 2.13. Suppose that (X, S × M, T) is an SM-system with the property that for all x ∈ X there exists m_x ∈ M such that for all y ∈ X and all (s, m) ∈ S × M we have that x→(s,m)y implies m = m_x. Then, (X, S × M, T) is a quasipolicy.

Before explaining the connections between quasifilter and a filter and quasipolicy and a policy, let us define projections of the sensorimotor transition relation to “motor” and to “sensory”:

Definition 2.14. Given an SM-system (X, S × M, T), let

These are called the motor and the sensory projections, respectively of the sensorimotor transition relation. They are also called the motor transition relation and the sensory transition relation, respectively. The corresponding transition systems (X, M, T_M) and (X, S, T_S) are called the motor and the sensory projection systems.

Definition 2.15. Given a transition system (X, U, T), and x ∈ X, let O_T(x) ⊆ U be defined as the set OT(x)={u∈U∣(∃y∈X)(x→uy)}. Combining this notation with the one introduced in Example 2.4(2), given x, y ∈ X, we have

For a transition relation T ⊆ X × (S × M) × X, define its transpose by T^t ⊆ X × (S × M) × X such that T^t = {(x, (m, s), y) ∣ (x, (s, m), y) ∈ T}. Note that (T^t)^t = T. For a subset of a Cartesian product A ⊆ S × M, let A₁ be the projection to the first coordinate A₁ = {s ∈ S∣(∃m ∈ M)((s, m) ∈ A)} and A₂ the projection to the second one: A₂ = {m ∈ M∣(∃s ∈ S)((s, m) ∈ A)}.

Mathematically coupling of two transition systems is symmetric [see Theorem 2.24(3)], but from the cognitive perspective there is (usually) an asymmetry between the agent and the environment (which can be evident from some specific properties of the agent and of the environment). Because of the partial symmetry, many properties of an agent can dually be held by the environment and vice versa. The following proposition highlights the duality between quasipolicy and quasifilters: reversing the roles of the environment and the agent.

Proposition 2.16. For an SM-system X=(X,S×M,T) the following are equivalent:

Similarly, X is a quasipolicy if and only if O_TM(x) = (_OT(X))1 is a singleton for each x ∈ X.

Proof. A straightforward consequence of all the definitions.               □

It will become convenient in the coming framework to assign labels to the states. The elements x of the state space X are already named; thus, our labeling can be more properly considered as a relabeling via a function h:X → L, in which L is an arbitrary set of labels. This allows partitions to be naturally induced over X by the preimages of h. Intuitively, this will allow the state space X to be characterized at different levels of “resolution” or “granularity.” Thus, we have the following definition:

Definition 2.17. A state-relabeled transition system (or simply labeled transition system) is a quintuple (X, U, T, h, L) in which h:X → L is a labeling function and (X, U, T) is a transition system.

We think of state-relabeled to be a more descriptive term, but we shorten it in the remainder of this paper to being simply labeled.

Remark 2.18. In an analogy to Definition 2.6, a labeled transition system is a labeled automaton, if T happens to be a function; in other words, for all (x, u) ∈ X × U there is a unique y ∈ X with (x, u, y) ∈ T. In this case we may denote this function by τ:(x, u) ↦ y and work with the labeled automaton (X, U, τ, h, L). For example, the temporal filter in Section 2.1 is a labeled automaton.

The isomorphism and bisimulation relations are defined similary as for transition systems, but in a label-preserving way.

One intended application of a labeled transition system (X, U, T, h, L) is that h is a sensor mapping, L is a set of sensor observations, and U is a set of actions. Thus, actions u ∈ U allow the agent to transition between states in X while h tells us what the agent senses in each state. We intend to show that this can be seen as a special case of an SM-system by proving a theorem similar to Theorem 2.11, but stronger, namely these corresponces preserve isomorphism:

Lemma 2.19. Let F be the class of quasifilters, P the class of quasipolicies, and L the class of labeled systems. Then there are one-to-one maps

such that

Proof. See Appendix B               □

Remark 2.20. Let X=(X,S×M,T) be a quasifilter and X′=LTSF(X)=(X,M,TM,h,S) as in Lemma 2.19. Suppose further that for each x, y ∈ X there is at most one u ∈ U with x→uy. Let

Then (X, M, E_T, x₀) coincides with the definition of a filter (O'Kane and Shell, 2017, Def 3). If it is also an automaton, meaning that above we replace “at most one” by “exactly one,” then every sequence of motor actions (m₀, …, m_k−1) determines a unique resulting state x_k−1 ∈ X. This is analogous, and can be proved in the same way, as the fact that each sequence of sensory data determines a unique resulting state in Remark 2.21 below.

Remark 2.21. Usually, a policy is a function which describes how an agent chooses actions based on its own past experience. Thus, if M is the set of motor commands and S is the set of sensations, a policy is a function π:S^* → M where S^* is the set of finite sequences of sensory “histories”; see for example (LaValle, 2006). Now, suppose that an SM-system X=(X,S×M,T) is a quasipolicy in the sense of Definition 2.13 and let x ↦ m_x be as in that Definition. Assume further that X is an automaton (Section 2.1) and let τ:X × (S × M) → X be the corresponding transition function so that for all x ∈ X and (s, m) ∈ S × M we have (x, (s, m), τ(x, (s, m))) ∈ T. Let x₀ ∈ X be an initial state. We will show how the pair (X,x0) defines a function π:S^* → M in a natural way. Let s̄=(s0,…,sk-1)∈Sk be a sequence of sensory data. If k = 0, and so s̄=( )=∅, let π(s̄)=mx0. If k > 0, assume that π(s₀, …, s_k−2) and x_k−1 are both defined (induction hypothesis). Then let x_k = τ(x_k−1, (m_xk−1, s_k−1)) and π(s₀, …, s_k−2, s_k−1) = m_xk. The idea is that because of the uniqueness of m_x, a sequence of sensory data determines (given an initial state) a unique path through the automaton X.

The central concept of this work pertaining to all principles (EA1)–(EA5) is the coupling of SM-systems. We define coupling, however, for general transition systems with the understanding that our most interesting applications will be for SM-systems where U₀ = U₁ = S × M. The idea is that in every transition there is a sensory component and a motor component. The set S could be thought of as all possible events that trigger afferent nervous signals, or their combinations. The elements of M are those events that are triggered by efferent nervous signals. This is an abstract space and in transitioning from one state to another some subset of S × M is “active.” If we know little of what kind of sensory data the agent receives during the transition, then that transition will occupy a subset of S × M whose projection to the S-coordinate is large. If, on the other hand we know a lot, and can specify the exact sensory data, then the projection to the S-coordinate is small. Vice versa, if we do not know which motor actions lead from one state to another, then the projection of the corresponding subset to the M-coordinate is large etc. This was made more precise in Section 2.4. The fact that the transition consists of pairs (s, m) where s is a sensory input and m is a motor command does not mean that the agent is equipped with the semantics of what m “means,” or what it “does” in the world. The effect of m is “computed” by the environment and the agent only receives the next “s” as the feedback. It might have been more intuitive, but more cumbersome to make this definition in terms of functions that map events of the environment to sensory stimuli and internal events of the nervous system to motor actions, and further functions that map the motor actions to the actual events in the environment, etc., but from the point of view of essential mathematical structure these extra identifications wouldn't add anything qualitatively new.

Definition 2.22. Let X0=(X0,U0,T0) and X1=(X1,U1,T1) be two transition systems. The coupled system X0*X1 is the transition system (X, U, T) defined as follows: X = X₀ × X₁, U = U₀ ∩ U₁, and

Equivalently, for all ((x0,x1),(y0,y1))∈(X0×X1)2 we have

(recall the T~ notation from Example 2.4(2)).

Example 2.23. A simple example of coupling is illustrated in Figure 2.

Mathematically the coupling is a product of sorts. If we think of one transition system as “the environment” and the other as “the agent,” then the coupling tells us about all possible ways in which the agent can engage with the environment. The fact that the state space of the coupled system is the product of the state spaces of the two initial systems reflects the fact that the coupled system includes information of “what would happen” if the environment was in any given state while the agent is in any given (“internal”) state.

We immediately prove the first theorem concerning coupling:

Theorem 2.24. Suppose that Xi=(Xi,Ui,Ti) and Xi′=(Xi′,Ui′,Ti′) for i ∈ {0, 1} are four SM-systems. Then the following hold:

Proof. See Appendix B               □

Coupling provides an interesting way to compare SM-systems from the “point of view” of other SM-systems. For example, given an SM-system E one can define an equivalence relation on SM-systems by saying that I~EI′, if E*I=E*I′. If E is the “environment” and I, I′ are “agents,” this is saying that the agents perform identically in this particular environment. Or vice versa, for a fixed I, the relation E*I=E′*I means that the environments are indistinguishable by the agent I.

Remark 2.25. In the definition of coupling we see that the two SM-systems constrain each other. This is seen from the fact that in the definition we take intersections. For example, when an agent is coupled to an environment, it chooses certain actions from a large range of possibilities. In this way the agent structures its own world through the coupling (EA3). To make this notion further connect to enactivist paradigm, we invoke the dynamical systems approach to cognition (Tschacher and Dauwalder, 2003). An attractor in a transition system X=(X,U,T) is a set A ⊆ X with the property that for all infinite sequences

there are infinitely many indices n such that x_n ∈ A. There could be other possible definitions, such as “for all large enough n, x_n ∈ A”. For the present illustration purposes it is, however, irrelevant. It could be the case that A ⊆ X is not an attractor of X, but after coupling with X′=(X′,U′,T′), A × X′ may be an attractor of X*X′. Thus, if X is the environment and X′ is the agent and A is a set of desirable environmental states, then we may say that the agent is well attuned to X, if A was not initially an attractor, but in X*X′, then A × X′ becomes one. It could also be that the agent needs to arrive to A while being in a certain type of an internal state B ⊆ X′, for example, if A is “food” and B is “hungry”. Then it is not important that A × X′ is an attractor, but it is imperative that A × B is one.

As we mentioned before, the information specified in an SM-system depends on which part of the brain-body-environment system we are modeling. In the extreme case we do not specify anything, except for the very minimal information. Consider a body of a robot for which the set of possible actions (or motor commands) is M and the set of possible sensor observations is S. Suppose that is all we know about the robot. We do not know what kind of environment it is in and we do not know what kind of “brain” (a processor or an algorithm) it is equipped with. Thus, we do not know of any constraints the robot may have in sensing or moving. We then model this robot as an unconstrained SM-system:

Definition 2.26. An SM-system (X, S × M, T) is called unconstrained iff for all x ∈ X, we have O_T(x) = S × M; recall Definition 2.15.

Unconstrained systems have the role of a neutral element with respect to coupling (Proposition 2.29). We now show that given all unconstrained SM-systems with shared M and S are mutually bisimulation equivalent:

Proposition 2.27. Suppose that X=(X,S×M,T) and X′=(X′,S×M,T′) are unconstrained systems. Then X~X′.

Proof. See Appendix B               □

There are many intuitions behind the above. An unconstrained system is one where anything could happen: the agent might perform any actions in any order and the environment could provide the agent with any sensory data. Such a world is reminiscent of white noise. Such a system is only interesting from an abstract mathematical perspective, it is in some sense “maximal”. The content of Proposition 2.27 is that such systems are indistinguishable from each other. An unconstrained system has a similar role with respect to all SM-systems as the free group has to other groups, although we haven't made this universality claim precise in the present paper. Intuitively it means that every possible agent-environment combination can be found as a subsystem (or possibly a quotient) of the unconstrained one. The term “unconstrained” refers in particular to that when coupled to other systems, this system doesn't constrain them, so it acts in the same way as 0 in arithmetic addition (Proposition 2.29). The opposite is the fully constrained system (Definition 2.31, Proposition 2.32). In that case, the intuition is the opposite: in environments where nothing happens and actions do not have any effects, any agent is as good as any other and vice versa: agents that don't do anything are equivalent.

Corollary 2.28. The SM-system ε = ({0}, {0} × (S × M) × {0}}) is the unique, up to bisimulation, unconstrained system.

Proposition 2.29. Let ε be as in Corollary 2.28 and let X=(X,S×M,T) be any SM-system. Then X*ε≅X.

Corollary 2.30. If X and X′ are SM-systems and X′ is unconstrained, then X*X′~X.

Proof. By Corollary 2.28 X′~ε, So by Theorem 2.24 we have X*X′~X*ε. However, by Proposition 2.29, X*ε~X; thus, X*X′~X.               □

The opposite of an unconstrained system is a fully constrained one:

Definition 2.31. An SM-system (X, S × M, T) is fully constrained iff T = ∅.

Proposition 2.32. Dually to the propositions above, we have that (1) all fully constrained systems are bisimulation equivalent to each other, (2) the simplest example being λ = ({0}, S × M, ∅), and (3) if X=(X,S×M,T) is another SM-system, then X*λ~λ.

All transition systems are in some sense between the fully constrained and the unconstrained, these being the two theoretical extremes.

When considering labelings and their induced equivalence relations, it will be convenient to develop a notion of quotient systems, analogous to quotient spaces in topology. Suppose X=(X,U,T) is a transition system and E is an equivalence relation on X. We can then form a new transition system, called the quotient of X by E in which the new states are E-equivalence classes and the transition relation is modified accordingly.

The following definition of a quotient is standard in Kripke model theory, especially bisimulation theory:

Definition 2.33. Suppose X=(X,U,T) and E are as above. Let X/E = {[x]_E ∣ x ∈ X}, in which each [x]_E is an equivalence class of states x under relation E, and T/E = {([x]_E, u, [y]_E)∣(x, u, y) ∈ T}. Then X/E=(X/E,U,T/E) is the quotient of (X, U, T) by E.

The following definition is inspired by the idea of sensory pre-images, see LaValle (2019), but is also needed for technical reasons.

Definition 2.34. Given any function h:X → L, denote by E^h the inverse-image equivalence: E^h = {(x, y) ∈ X² ∣ h(x) = h(y)}. We will denote the equivalence classes of E^h by [x]_h instead of [x]Eh if no confusion is possible.

The equivalence relation E^h partitions X according to the preimages of h, as considered in the sensor lattice theory of LaValle (2019). The partition of X induced by h directly yields an quotient transition system by applying the previous two definitions:

Definition 2.35. Let X=(X,U,T) be a transition system and h:X → L be any mapping. Then define X/h to be X/Eh where we combine Definitions 2.34 and 2.33.

Proposition 2.36. If h is one-to-one, then X/h≅X.

Proof. h is one-to-one if and only if E^h is equality, in which case it is straightforward to verify that the function x↦[x]Eh is an isomorphism.               □

For h:X → L, the transition system (X/h, U, T/h) is essentially a new state space over the preimages of h. In this case X/h is called the derived information space (as used in LaValle, 2006). More precisely:

Proposition 2.37. Let L′ = ran(h) ⊆ L. Define

Then (X/h, U, T/h) is isomorphic to (L′, U, T′) via the isomorphism f:[x]Eh↦h(x).

Proof. See Appendix B               □

The intuitive meaning of the quotient is the following. There is a Soviet comedy film from the 1970's where the main character ends up in an apartment in Leningrad, while he thinks that he is actually in Moscow. The apartement in Leningrad is identical to his home in Moscow and he cannot distinguish between them. He thinks for a while that he is at his home in Moscow while being in an apartment in Leningrad. Even his key from Moscow worked for the Leningrad apartment. The pun is that in Soviet times all houses were built according to the same blueprint. Now, before he realized his situation, as far as he was concerned, he was in Moscow. He thought he came to the same place in the evening as in the morning, while he actually didn't. The idea of the quotient captures exactly that: We identify those states that “look the same” (the label is the same) even though they are actually different states. In fact, let us look at a cognitive system on several levels of granularity: When I type on my laptop at home or in a cafeteria, my fingers experience the keyboard in (approximately) the same way. As far as my fingers (and associated motor areas) are concerned, we can identify all situations where they are pressing keys on my keyboard. On a higher level, I might be coming home after a 10 h time and experience as if I am in the same place, but we all know that the planet, on which my home is, has moved, so I actually am not in the same place, just like the main character in the movie referenced above.

The following consider the main definition of this work. It is based on the idea of sufficiency in LaValle (2006, Ch.11).

Definition 4.1. Let (X, U, T) be a transition system and E ⊆ X × X an equivalence relation. We say that E is sufficient or completely sufficient, if for all (x, y) ∈ E and all u ∈ U, if (x, u, x′) ∈ T and (y, u, y′) ∈ T, then (x′, y′) ∈ E.

This means that if an agent cannot distinguish between states x and y, then there are no actions it could apply to later distinguish between them. To put it differently, if the states are indistinguishable by an instant sensory reading, then they are in fact indistinguishable even through sensorimotor interaction. This is related to the equivalence relation known as Myhill-Nerode congruence in automata theory.

The equivalence relation of indistinguishability in the context of sensorimotor interactions is at its simplest the consequence of indistinguishability by sensors. Thus, we define sufficiency for labelings or sensor mappings:

Definition 4.2. A labeling h:X → L is called sufficient (or completely sufficient) iff for all x, y, x′, y′ ∈ X and all u ∈ U, the following implication holds:

Proposition 4.3. If (X, U, τ) is an automaton, then h:X → L is sufficient if and only if for all x, y ∈ X and all u ∈ U, we have that if h(x) = h(y), then h(τ(x, u)) = h(τ(y, u)).

Proof. Checking the definitions.               □

The above proposition is saying that when the sensorimotor system is deterministic, then sufficiency is equivalent to predictability.

There is a connection with the classical notion of bisimulation in classical transition systems theory (recall Definition 2.2):

Proposition 4.4. An equivalence relation on a state space of an automaton (X, U, τ) is sufficient if and only if it is an autobisimulation.

Proof. See Appendix B.               □

The above proposition can intuitively be interpreted as saying that a sufficient relation is one where different states with the same label are not only indistinguishable on their own, but are actually indistinguishable even by their consequences. Starting from one of two states with same labels, there is no way to ever find out which one of them it was, no matter how much will the agent investigate its environment, compare to the discussion in the end of Section 2.8.

Proposition 4.5 below is an important proposition on which the idea of derived I-spaces and combinatorial filters builds upon (LaValle, 2006, 2012; O'Kane and Shell, 2017), although as far as the authors are aware, in the literature, only the “if”-direction is mentioned. We say that a transition system (X, U, T) is full, if for all x₁ ∈ X and all u ∈ U there exists at least one x₂ ∈ X with (x₁, u, x₂).

Proposition 4.5. Suppose X=(X,U,T) is a transition system. Let h:X → L be a labeling. Then X/h is an automaton if and only if X is full and h is sufficient.

Proof. See Appendix B.               □

The above proposition brings together the ideas of a quotient, automaton and sufficiency. The idea of the quotient is that indistinguishable states can be in some circumstances considered the same and the idea of an automaton is that it is deterministic. The above proposition says that as far as the agent is concerned, if it equalizes indistinguishable states, then the world looks deterministic from the agent's perspective if and only if the underlying labeling satsifies Definition 4.2.

The sufficiency of an information mapping was introduced in LaValle (2006, Ch 11), and is encompassed by a sufficient labeling in this paper. In the prior context, it has meant that the current sensory perception together with the next action determine the next sensory perception. The elegance with respect to our principle (EA2) is that sufficiency is not saying that the agent's internal state corresponds to the environment's state (as is in representational models). Nor is it saying that the agent predicts the next action. It is saying, rather, that the agent's current sensation together with a choice of a motor command determine the agent's next sensation; and this statement is true only as a statement made about the system from outside, not as a statement which would reside “in the agent.” The sensation may carry no meaning at all “about” what is actually “out there.” However, if the agent has found a way to be coupled to the environment in a sufficient way, then sensations begin to be about future sensation. In this way meaning emerges from sensorimotor patterns. This relates to (EA3) and somewhat touches on the topic of perception (EA5). Furthermore, the property of determining future outcomes is related to (EA4) because that is what skill is. There is no potential to reliably engage with the environment in complex sensorimotor interactions, if the sensations do not reliably follow certain historical patterns.

Thus, the notion of sufficiency is considered by us to be of fundamental importance for enactivist-inspired mathematical modeling of cognition. The violation of sufficiency means that the current sensation-action pair does not correlate with the future sensation, making it harder to be attuned to the environment. Having a different sensation following the same pattern can be seen as a primitive notion of a “surprise.” This can be seen as aligning with the predictive coding and the free energy principle from neuroscience (Rao and Ballard, 1999; Friston and Kiebel, 2009; Friston, 2010), although our framework leaves the space to a clean non-representational interpretation while this is not obvious for these other frameworks. Does the notion of sufficient labelings capture the same ideas in a more general way? This is an open question for further research.

A generalization of sufficiency is n-sufficiency, in which the data of n previous steps is needed to determine the next label. Here, we define an n-chain.

Definition 4.6. An n-chain in X=(X,U,T) is a sequence

such that xi→uxi+1 for all i < n. If n = 0, then by convention c = (x_n). Let E ⊆ X × X be an equivalence relation. Let k < n. We say that two n-chains c = (x₀, u₀, …, x_n−1, u_n−1, x_n), c′=(x0′,u0′,…,xn-1′,un-1′,xn′) are (T, E, k)-equivalent if for all i < k, we have ui=ui′ and (xi,xi′)∈E. An ∞-chain is defined in the same way as n-chain, except the sequences are infinite, without the “last” x_n.

Definition 4.7. For a transition system X=(X,U,T), an equivalence relation E on X is called n-sufficient if there are no two (T, E, n)-equivalent n-chains

such that (xn,xn′)∉E. A labeling h:X → L is called n-sufficient if E^h is n-sufficient (Recall Definition 2.34).

Proposition 4.8. An equivalence relation E is 0-sufficient if and only if there is only one E-equivalence class, and a labeling function h is 0-sufficient if and only if it is constant.

Proof. See Appendix B               □

Proposition 4.9. An equivalence relation E (resp. a labeling h) is sufficient if and only if it is 1-sufficient.

Proof. See Appendix B               □

Proposition 4.10. Suppose n < m are natural numbers. If a labeling h is n-sufficient, then it is m-sufficient. The same holds for equivalence relations.

Proof. See Appendix B               □

This enables us to define the degree of insufficiency:

Definition 4.11. The degree of insufficiency of the labeled automaton X=(X,U,τ,h,L) is defined to be the smallest n such that h is n-sufficient, if such n exists, and ∞ otherwise. Denote the degree of insufficiency of X by degins(X), or degins(h) if only the labeling needs to be specified and X is clear from the context.

The intuition is that the larger the degree of insufficiency of an environment X, the harder it is for an agent to be attuned to it. We talk more about the connection between attunement and sufficiency in the following sections.

In this section we prove that the minimal sufficient refinements are always unique (Theorem 4.19). This will follow from a deeper result that the sufficient equivalence relations form a complete sublattice of the lattice of all equivalence relations. This does not hold for n-sufficient equivalence relations for n > 1 (Example 4.20). We will then explore how the minimal sufficient refinements can be thought of as an enactive perceptual construct that emerges from the body-environment, brain-body, and brain-body-environment dynamics. The idea is that a minimal sufficient refiniment corresponds to an optimal attunement of the agent to the base labeling which corresponds to some minimal information that the agent is interested in the environment, such as death or life, danger or safety information. It is “optimal” by minimality and “attunement” by sufficiency. Our Theorem 4.19 states that such attunement is mathematically unique.

Definition 4.12. An equivalence relation E is a refinement of equivalence relation E′, if E ⊆ E′, also denoted E′≤rE. A labeling function h is a refinement of a labeling function h′, if E^h is a refinement of E^h′.

An important interpretation of the concept of a refinement is that a better sensor provides the agent with more information about the environment⁵. Each sensor mapping h induces a partition of X via its preimages, and refinement applies in the usual set-theoretic sense to the partitions when comparing sensors mappings. If a sensor mapping h is a refinement of h′, then it enables the agent to react in a more refined way to nuances in the environment. Using the partial ordering given by refinements, we obtain the sensor lattice (LaValle, 2019).

By a referee's request, let us give a couple of biological examples.

Example 4.13 (First biological example). There is an accepted theory that primates see red color wavelength, because it enables them to distinguish ripe fruit from non-ripe. Assuming this theory is true, it is an example of a refinement which is to some extent “minimal” and to some extent “sufficient” (of course strictly speaking it is neither – in the same way as there is no ideal circle in the physical world). The minimality is seen in this example, because we perceive other things as red, even if it is completely unnecessary (certainly unnecessary to tell the ripeness of fruits). So we are not distinguishing “too much.” On the other hand, perceiving red color is a refinement of ripe/non-ripe which is only detected through stomach ache after the fruit has been already consumed. And it is sufficient in the sense that it is predictive of the original “base” labeling (ripe/non-ripe).

Example 4.14 (Second biological example). Where our eyes look depends on the position of our head as well as the position of our eyes. Despite this, “looking up” (or “left,” “right” etc..) are not ambiguous, even though these can be achieved with virtually infinitely many different head-eye configurations. One way to understand how this invariance could emerge is through minimal sufficient refinements. Suppose at birth, every head-eye configuration is considered as a separate state, but we label them by what we see in any given (stable) situation. A minimal sufficient refinement of that labeling will never distinguish between different states in which the eyes are pointing in the same direction. So then, by learning the minimal sufficient refinements, the agent may learn eye-direction invariance.

Please refer to Appendix A in the Supplementary material for notations and definitions used in this section.

We will prove in this section that if (X, U, τ) is an automaton, the sufficient equivalence relations form a complete sublattice of (E(X),⊆). Given an automaton X=(X,U,τ), denote by EsufU,τ(X)⊆E(X) the set of sufficient equivalence relations on X. When U and τ are clear from the context, we write just Esuf(X)=EsufU,τ(X).

Theorem 4.15. Suppose (X, U, τ) is an automaton and suppose that E⊆Esuf(X) is a set of sufficient equivalence relations. Then ∧E and ∨E are sufficient. Thus, (Esuf(X),⊆) is a complete sublattice of (E(X),⊆).

Proof. See Appendix B.               □

Suppose that a labeling h is very important for an agent. For example, h could be “death or life,” or it could be relevant for a robot's task. Suppose that h is not sufficient. The robot may want to find a sufficient refinement of h. Clearly a one-to-one h′ would do. However, assume that the agent has to use resources for distinguishing between states; thus, the fewer distinctions the better. This motivates the following definition. Recall Definition 4.12 of refinements.

Definition 4.16. Let (X, U, T) be a transition system and E₀ ⊆ X × X an equivalence relation. A minimal sufficient refinement of E₀ is a sufficient equivalence relation E which is a refinement of E such that there is no sufficient E′ with E0≤rE′<rE.

Given a labeling h₀ of a transition system (X, U, T), a minimal sufficient refinement of h₀ is a labeling h such that E^h is a minimal sufficient refinement of Eh0 (recall Definition 2.34).

Example 4.17. Let X=(X,U,τ) be an automaton where X = {0, 1}^*, U = {0, 1} and τ(x, b) = x^⌢b (concatenation of the binary string x with the bit b). Let h(x) = 1 if and only if the number of ones and the number of zeros in x are both prime; otherwise h(x) = 0. Then the only sufficient refinements of h are one-to-one.

Example 4.18. Let X be as above and let h:X → {0, 1} be such that if |x| is divisible by 3, then h(x) = 1; otherwise, h(x) = 0. Then h is not sufficient. Let h′:x ↦ {0, 1, 2} be such that

Then h′ is a minimal sufficient refinement of σ.

Theorem 4.19. Consider an automaton X=(X,U,τ) and let E₀ be an equivalence relation on X. Then a minimal sufficient refinement of E₀ exists and is unique.

Proof. See Appendix B               □

Theorem 4.19 fails, if “automaton” is replaced by “transition system,” or if “sufficient” is replaced by “n-sufficient” for n > 1 (recall Definition 4.7)

Example 4.20 (Failure of uniqueness for n-sufficiency). Let X = {0, 1, 2, 3, 4, 5}, U = {u₀} and

and

Let E₀ be an equivalence relation on X such that the equivalence classes are {0, 1, 3, 4}, {2} and {5}. Then this relation is not 2-sufficient, because (0, u₀, 1, u₀, 2) and (3, u₀, 4, u₀, 5) are (T, E₀, 2)-equivalent, but 2 and 5 are not E₀-equivalent. Let E₁, E₂ ⊆ E₀ be equivalence relations with equivalence classes as follows:

Then E₁ and E₂ are refinements of E₀. They are both 2-sufficient, because there doesn't exist any (T, E₁, 1) or (T, E₂, 1) equivalent 2-chains. They are also both ≤_r-minimal with this property which can be seen from the fact that they are actually ≤_r-minimal refinements of E₀ as equivalence relations (not only as sufficient ones).

Example 4.21 (Failure of uniqueness for transition systems). Let X = {0, 1, 2, 3, 4}, U = {u₀} and T = {(0, u₀, 3), (2, u₀), 4}. Let E₀ be the equivalence relation with the equivalence classes {0, 1, 2}, {3} and {4}. Then E₀ is not sufficient, because (0, 2) ∈ E₀, but (3, 4) ∉ E₀. Let E₁ and E₂ be the refinements of E₀ with the following equivalence classes:

Now it is easy to see that both E₁ and E₂ are sufficient refinements of E₀, and by a similar argument as in Example 4.20 they are both minimal. The reason why this is possible is the odd behavior of the state 2 which doesn't have out-going connections. Such odd states are the reason why the decision problem “Does there exist a sufficient refinement with k equivalence classes?” is NP-complete for finite transition systems (O'Kane and Shell, 2017).

Remark. It is worth noting that Theorems 4.15 and 4.19 do not assume anything about the cardinality of X or of U, other structure on them (such as metric or topology) nor anything about the function τ or the relation E₀. Keeping in mind potential applications in robotics, X and U could be, for instance, topological manifolds, and τ a continuous function, or X could be a closed subset of ℝⁿ, U discrete and τ a measurable function, or any other combination of those. In each of those cases, the sublattice of sufficient equivalence relations is complete, as per Theorem 4.15, and every equivalence relation E₀ on X admits a unique minimal sufficient refinement as per Theorem 4.19.

Recall Definition 2.10 of an equivalence relation preserving function. We say that an equivalence relation E on X is closed under f:X → X if for all x ∈ X, we have (x, f(x)) ∈ E. If E is closed under f, then f is E-preserving: given (x, x′) ∈ E, we have (x, f(x)), (x′, f(x′)) ∈ E, because E is closed under f. Now by transitivity of E we have (f(x), f(x′)) ∈ E, so f is E-preserving.

Definition 4.22. Let f:X → X be a bijection. The induced orbit equivalence relation is the relation E_f on X defined by (x,x′)∈Ef⇔(∃n∈ℤ)(fn(x)=x′), in which fⁿ(x) is defined by induction as: f⁰(x) = x, fⁿ⁺¹(x) = f(fⁿ(x)), fⁿ⁻¹(x) = f⁻¹(fⁿ(x)).

Theorem 4.23. If f is an automorphism of the automaton (X, U, τ), then E_f is a sufficient equivalence relation.

Proof. See Appendix B               □

Theorem 4.24. Let X=(X,U,τ) be an automaton and E be an equivalence relation on X. Suppose f:X → X is an automorphism such that E is closed under f. Let E′ is the minimal sufficient refinement of E. Then E′ is closed under f and E≤rE′≤rEf.

Proof. See Appendix B               □

Example 4.25. Consider the environment which is a one-dimensional lattice of length five, E = {−2, −1, 0, 1, 2}, in which the corners “smell bad”; thus, we have a sensor mapping h:E → S, S = {0, 1} defined by h(n) = 0 ⇔ |n| = 2; see Figure 6A. Consider two agents in this environment. Both are equipped with the same h sensor, but their action repertoires differ. Both have two possible actions. One has actions L= “move left one space” and R= “move right one space,” and the other one has actions T= “turn 180 degrees” and F= “go forward one space.” Let M₀ = {L, R} and M₁ = {T, F}. Thus, these agents have a slight difference in embodiment. Although both of them can move to every square of the lattice in a very similar way (almost indistinguishable from the outside perspective), we will see that the differences in embodiment will be reflected in that the minimal sufficient refinements will produce non-equivalent “categorizations” of the environment. The structures that emerge from these two embodiments will be different. These agents enact different environments, although physically the environments are the same, as congruent with tenet (EA3).

First, we define the SM-systems that model these agents' embodiments in E. The first agent does not have orientation. It can be in one of the five states, and the state space is X₀ = E (Figure 6B). For the second agent, the effect of the F action depends on the orientation of the agent (pointing left or pointing right). Thus, there are ten different states the agent can be in, yielding X₁ = E × {−1, 1} (Figure 6C). The effects of motor outputs are specified completely (L means moving left, and so on), whereas the agent's internal mechanisms are left completely open, so our systems will be quasifilters. According to Remark 2.18, we can work with a labeled automaton instead. Hence, let τ₀:X₀ × M₀ → X₀ be defined by τ₀(x, L) = max(x − 1, −2) and τ₀(x, R) = min(x + 1, 2). For the other agent, let τ₁((x, b), T) = (x, −b) and τ₁((x, b), F) = (min(max(x + b, −2), 2), b). Now we have labeled automata X0=(X0,M0,τ0,h,S) and X1=(X1,M1,τ1,h,S).

It is not hard to see that the one-to-one map h₀:X₀ → {−2, −1, 0, 1, 2} with h₀(x) = x is a sufficient refinement of h which is minimal (see Figure 7A). Thus, every state needs to be distinguished by the agent for it to be possible to determine the following sensation from the current one. The derived information space automaton X0/h0 isomorphic to X0 (Proposition 2.36).

For the second automaton, consider the labeling h₁:X₁ → {−2, −1, 0, 1, 2} defined by h₁(x, b) = b·x (see Figure 7C).

Claim. h₁ is a minimal sufficient refinement of h in X1.

Proof. See Appendix B               □

Both minimal sufficient labelings, h₀ and h₁ have five values; thus, they categorize the environment into five distinct state-types. However, the resulting derived information spaces are different in the sense that the quotients X0/h0 and X1/h1 are not isomorphic; compare Figure 7B with Figure 7D.

Example 4.26. Figure 8A shows a filtering example from Tovar et al. (2014). More complex versions have been studied more recently in O'Kane and Shell (2017), and are found through automaton minimization algorithms and some extensions. It can be shown that this example's four-state derived information space depicted on Figure 8B corresponds to the unique minimal sufficient refinement of the labeling that only distinguishes between “are in the same region” and “are not in the same region.” To see this, first note that this labeling is sufficient (since it can be represented as an automaton, this follows from Theorem 4.5). It follows from Theorem 4.19 that if this labeling is not minimal, then there is a minimal one which is strictly coarser, and so can be obtained by merging the states in the automaton of Figure 8B. This is impossible: the state T cannot be merged with anything because it violates the base-labeling; if, say D_a and D_c, are merged, then transition a will lead to inconsistency as it can lead either to D_b (from D_c) or to T (from D_a). This proves that this derived information space is indeed minimal sufficient, and by Corollary 4.19 there are no others up to isomorphism.

This section sketches some computational problems and presents computed examples. The problem of computing the minimal sufficient refinement in some cases reduces to classical deterministic finite automaton (DFA) minimization, and in other cases it becomes NP-hard (O'Kane and Shell, 2017).

Consider an automaton (X, M, τ) and a labeling function h₀, and the corresponding labeled automaton described using the quintuple (X, M, τ, h₀, L). Suppose that the automaton (X, M, τ) corresponds to that of an body-environment system. Hence, X corresponds to the states of this coupled system. Suppose h₀ is not sufficient and consider the problem of computing a (minimal) sufficient refinement of h₀, that is, the coarsest refinement of h₀ that is sufficient.

Despite the uniqueness of the minimal sufficient refinement of h₀ (by Corollary 4.19), we can argue that the formulation of the problem, in particular, the input, can differ based on the level at which we are addressing the problem (for example, global perspective, agent perspective or something in between). Since the labeled automaton corresponding to an agent-environment coupling is described from a global perspective, the input to an algorithm that addresses the problem from this perspective is the labeled automaton A=(X,τ,M,h0,L) itself. Then, the problem is defined as given A compute A′=(X,M,τ,h,L) such that h is the minimal sufficient refinement of h₀.

A special case of this problem from the global perspective occurs if the preimages of h₀ partition X in two classes which can be interpreted as the “accept” and “reject” states, for example, goal states at which the agent accomplishes a task and others. Furthermore, suppose that the initial state of the agent is known to be some x₀ ∈ X. Then, computing a minimal sufficient refinement becomes identical to minimization of a finite automaton, that is, given a DFA (X, M, τ, x₀, F) in which x₀ is the initial state and F is the set of accept states find (X′,M,τ′,x0′,F′) such that no DFA with fewer states recognizes the same language. Existing algorithms, for example Hopcroft (1971), can be used to compute a minimal automaton.

Here, we also consider this problem from the agent's perspective for which the information about the environment states is obtained through its sensors, more generally, through a labeling function. Note that by agent's perspective we do not necessarily imply that the agent is the one making the computation (or any computation) but it means that no further information can be gathered regarding the environment other than the actions taken and what is sensed by the agent. At this level we address the following problem; given a set M of actions, a domain X, and a labeling function h₀ defined on X, compute the minimal sufficient refinement of h₀. The crux of the problem is that unlike the global perspective described above, the labeled automaton A is not given, in particular, the state transitions are not known a priory. Instead, the information regarding the state transitions can only be obtained locally by means of applying actions and observing the outcomes, that is, through sensorimotor interactions. Hence, the current body-environment state is also not observable. To show that an algorithm exists to compute a sufficient refinement of h₀ at this level, we propose an iterative algorithm (Algorithm 1) that explores X through agent's actions and sensations by keeping the history information state, that is, the history of actions and sensations (labels). We then show, by empirical results, that the sufficient refinement computed by Algorithm 1 is minimal for the selected problem.

The functioning of Algorithm 1 is as follows. Starting from an initial sensation s₀ = h(x₀), the agent moves by taking an action⁶ given by the mapping policy:L → M. Particularly, we used a fixed policy which samples an action m from a uniform distribution over M for each s ∈ S. In principle, any policy that ensures all states that are reachable from x₀ will be visited infinitely often should be enough. The history information state is implemented as a list, denoted by H, of triples (s, m, s′) such that s = h(x) and s′ = h(x′) in which x′ = τ(x, m). At each step, it is checked whether the current sensation is consistent with the history (Line 7). Current sensation is inconsistent with the history if there exists a triple (s, m, s″) in the history such that s′ ≠ s″. If it is not consistent then the label is split, which means that h⁻¹(s) is partitioned into two parts P and Q. In particular, we apply a balanced random partitioning, that is, we select P and Q randomly from a uniform distribution over the partitions of h⁻¹(s) that have two elements with balanced cardinalities. The labeling function is updated by a split operation as

Recall that labels or subscripts do not carry any meaning from the agent's perspective.

Even a trivial strategy that splits the preimage of the label seen at each step would succeed computing a sufficient refinement. However, this would result in h being a one-to-one mapping. Hence, the finest possible refinement. Splitting only at the instances when an inconsistency is detected might reach a coarser refinement that is sufficient but there might be more equivalence classes than the ones induced by the minimal sufficient refinement of h₀. Therefore, a merge operation is introduced (Line 10). Let s and s′ be two distinct labels for which ∃s″∈h0[X] such that h-1(s)⊆h0-1(s″) and h-1(s′)⊆h0-1(s″). Let t denote a triple in H and let t_k, k = 1, 2, 3, denote the k^th element of that triple. Suppose s′ = s, if there are at least N number of triples in H such that for each triple t, (t₁, t₂) = (s, m) and ∀m ∈ M and ∀t, t′ ∈ H such that (t1,t2)=(t1′,t2′)=(s,m) it is true that t3=t3′ then labels s and s′ are merged. The merge procedure goes through all labels and updates h as

for each pair of labels s and s′ that satisfies the aforementioned condition. Note that in principle, one can merge two labels regardless of the number of occurrences in the history. However, we noticed that this can result in oscillatory behaviour between split and merge operations especially for states that are reached less frequently. At present, we considered N as a tunable parameter and we know that it depends on the cardinality of the state space X such that larger the number of states, larger N should be. The problem of defining N as a function of the problem description remains open.

In the following, we present an illustrative example to show the practical implications of the previously introduced concepts in Section 4.2. In particular, we show how a simple algorithm like Algorithm 1 can be used by a computing unit which relies only on the sensorimotor interactions of an agent to further categorize the environment such that there are no inconsistencies in terms of the actions taken by the agent and the resulting sensations with respect to an initial categorization induced by h₀ (Figure 9C).

Example 4.27. Consider an agent (a mouse) that is placed in a maze where certain paths lead to cheese and others do not (see Figure 9A). At each intersection the agent can go either left or right and it can not go back. Hence, at each step the agent takes one of the two actions; go right or go left. Figure 9B shows the corresponding automaton with 15 states describing the agent-environment system together with the initial labeling h₀ that partitions the state space into states in which the agent has reached a cheese (light blue) and others (dark blue). The initial state x₀ is when the agent is at the entrance of the maze. Once the end of the maze is reached (a leaf node) the state does not change regardless of which action is taken. After a predetermined number of steps the system reverts back to the initial state, similar to an episode in the reinforcement learning terminology (see, for example, Sutton and Barto, 2018). However, despite the system going back to the initial state the history information state still includes the prior actions and sensations. Figure 10 reports the updates of h, initialized at h₀, by Algorithm 1 being run for 1,000 steps. It converged to a final labeling h (Figure 10R), that is the minimal sufficient refinement of h₀, in 435 steps. For 20 initializations of Algorithm 1 for the same problem, on average, it took 364 steps to converge to a minimal sufficient refinement of h₀ (Figure 9C).

We have also applied the same algorithm to variations of this example with different depths of maze and different number of cheese and cheese placements (varying h₀). Empirical evidence shows that the same algorithm was capable of consistently finding the minimal sufficient refinement of the initial labeling. However, it is likely that it might fail for more complicated problems, for example, when the number of actions are significantly larger. It remains an open problem finding a provably correct algorithm for computing the minimal sufficient refinement of h₀ from the agent's perspective.

Section 2 introduced SM-systems, including the special class of quasifilters. We showed that quasifilters can be thought of as labeled transition systems, and we worked with such systems in Sections 4, 4.4. Let us see how do the concepts introduced in those sections work for SM-systems. We also defined coupling of SM-systems (Definition 2.22), but we have not defined what it means for a coupling to be “good.” We will use sufficiency to approach this subject.

Let E=(E,(S×M),T) and I=(I,S×M,B) be SM-systems. We think intuitively of E as “the environment” and I as the “agent,” even though they share the set of sensorimotor parameters S × M. When is the coupling E*I “successful”? Given another I′=(I′,S×M,B′), how can we compare I and I′ in the context of E? The coupled system E*I is not labeled; therefore, we cannot apply the definition of sufficiency. However, as soon as we apply some labeling to it, we can. There are many different ways to do it, intuitively corresponding to the “agent's perspective,” the “environment's perspective” and a “god's perspective” (or “global perpsective”).

The first one is the labeling h:E × I → I, which is the projection to the right coordinate, h_I(e, i) = i. The second one is the projection to the left coordinate h_E(e, i) = i, and the third one is the labeling of states by themselves, h_G(e, i) = (e, i). Clearly, h_G is a refinement of both h_E and h_I. Yet another option is to use the sensory data as labelings, which is a coarser labeling than h_I. Or perhaps there was already a labeling h:E → S to begin with, so then we can ask about the property of ĥ:E × I → S defined by ĥ(e, i) = h(e). We focus on what we called the agent's perspective, h_I, for the rest of this section.

Recall Definition 4.11 of the degree of insufficiency. Given SM-systems E (environment) and I (agent), we can ask what is the degree of insufficiency of h_I in E*I? The smaller the degree, the better the agent is attuned to the environment. This says something about the way in which the agent is adapted or attuned to the environment without attributing contentful states or representations to the agent in alignment with (EA2) and (EA4).

Let E, I, and I′ be SM-systems. When is degins(E*I,hI)<degins(E*I′,hI′)? Of course, if I is fully constrained (Definition 2.31), then degins(E*I)=∞. This corresponds to the agent never engaging in any sensorimotor interaction with the environment. No wonder that it can always “predict” the result of such passive existence. Assume, however, that there some constraints on the coupling. For example, we may demand that the agent must regularly visit states of some particular type to survive. Subject to such constrains, what can we say about degins(E*I)? This seems to be a good preliminary notion⁷ of attunement.