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. 2016 Jul 29:7:1139.
doi: 10.3389/fpsyg.2016.01139. eCollection 2016.

Systematicity and a Categorical Theory of Cognitive Architecture: Universal Construction in Context

Steven Phillips  1 , William H Wilson  2
Affiliations

Systematicity and a Categorical Theory of Cognitive Architecture: Universal Construction in Context

Steven Phillips et al. Front Psychol. .

Abstract

Why does the capacity to think certain thoughts imply the capacity to think certain other, structurally related, thoughts? Despite decades of intensive debate, cognitive scientists have yet to reach a consensus on an explanation for this property of cognitive architecture-the basic processes and modes of composition that together afford cognitive capacity-called systematicity. Systematicity is generally considered to involve a capacity to represent/process common structural relations among the equivalently cognizable entities. However, the predominant theoretical approaches to the systematicity problem, i.e., classical (symbolic) and connectionist (subsymbolic), require arbitrary (ad hoc) assumptions to derive systematicity. That is, their core principles and assumptions do not provide the necessary and sufficient conditions from which systematicity follows, as required of a causal theory. Hence, these approaches fail to fully explain why systematicity is a (near) universal property of human cognition, albeit in restricted contexts. We review an alternative, category theory approach to the systematicity problem. As a mathematical theory of structure, category theory provides necessary and sufficient conditions for systematicity in the form of universal construction: each systematically related cognitive capacity is composed of a common component and a unique component. Moreover, every universal construction can be viewed as the optimal construction in the given context (category). From this view, universal constructions are derived from learning, as an optimization. The ultimate challenge, then, is to explain the determination of context. If context is a category, then a natural extension toward addressing this question is higher-order category theory, where categories themselves are the objects of construction.

Keywords: (co)algebra; (co)recursion; category theory; classicism; compositionality; connectionism; systematicity; universal construction.

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Figures

Figure 1
Figure 1
Systematicity with regard to (A) the pair (square, triangle) and (B) the pair (triangle, square). The arrow 〈s, t〉 corresponds to the capacity to generate the representation for (square, triangle), indicated as the symbol pair (□, △). This arrow is composed of the arrow 〈1, 1〉, which is common to all shape pairs, and the arrow s × t. Arrow s × t is constructed by applying a functor (not shown) to the pair of arrows (s, t), which correspond to the basic capacity to generate a representation for square and triangle, respectively. The objects in these categories are sets and the arrows are functions, where ↦ indicates the action of the function on set elements, and dashed arrows indicate uniqueness, i.e., the only arrow that satisfies the equation 〈s, t〉 = s × t ○ 〈1, 1〉. The symbol * indicates an element whose name is unimportant. Note that {(*, *)} is a one-element set containing the (ordered) pair with the (unnamed) element * at both positions, and ({*}, {*}) is a pair with the one-element set {*} at both positions. Systematicity with regard to (C) the capacity to infer that the first shape of the pair (square, triangle) is square and (D) the capacity to infer that the first shape of the pair (triangle, square) is triangle, where the common arrow π1 is the projection that returns the first component of each pair. The arrows that are the sources of 〈s, t〉 and 〈t, s〉 in (C,D) are not shown.
Figure 2
Figure 2
Universal construction and comma category. The diagram in (A) shows the general form of a universal arrow, which is the pair (A, ϕ), from an object X in a category C to a functor F: D → C from a category D to C. To be a universal arrow, as such, for every object Y in D and every arrow f: X → F(Y) in C there must exist a unique arrow u: A → Y, in D, such that f = F(u) ○ ϕ. The diagram in (B) shows the corresponding comma category, denoted (X ↓ F), whose objects are the pairs (Y, f) and arrows are the arrows u that uniquely satsify the “triangle” equation, f = F(u) ○ ϕ. The collection of objects in the comma category includes the universal arrow, (A, ϕ), because for Y set to A and f set to ϕ the triangle equation is uniquely satisfied by setting u to the identity arrow 1A. The universal arrow is the initial object in the comma category, which is straightforward to prove. Dashed lines indicate that the arrows are unique. An example universal construction (C) and corresponding comma category (D) is the universal arrow from the object (real number) 2.9 to the inclusion functor (function) from the integers to the real numbers (regarded as posets with the usual order ≤, hence categories) is the pair (3, ≤). The number 3 corresponds to the smallest integer greater than or equal to 2.9. In general, for a real number x ∈ ℝ, the object component of the universal arrow (a, ≤) from x to the inclusion function/functor is obtained by rounding up to the nearest integer a ∈ ℤ greater than or equal to x, i.e., obtained by the ceiling function, a = ⌈x⌉, and the corresponding comma category consists of all the integers y greater than or equal to a, i.e., the set {y ∈ ℤ|⌈x⌉ ≤ y}.
Figure 3
Figure 3
Function minimization as a universal construction. Within each row, the left panel shows the function and the right panel shows the corresponding universal construction. (A,B) a quadratic function, (C,D) an arbitrary function, (E,F) a fitness/error function over network connection weights, and (G,H) a generalized fitness/error function over a collection of algebras.
Figure 4
Figure 4
An example cue-target prediction task for a schema induction paradigm showing (A) six cue-target pairs and (B) their geometric interpretation, where trigrams correspond to vertices of a triangle and shapes correspond to clockwise and anticlockwise rotation.
Figure 5
Figure 5
(A) The comma category corresponding to the universal construction shown in Figure 3B as a directed graph. (B) The comma category corresponding to the universal construction shown in Figure 3D as a directed graph. Edges corresponding to identity arrows are not shown. (C,D) The iterative process for finding the initial nodes in corresponding directed graphs.
Figure 6
Figure 6
Universal construction and comma category. (A) Universal arrow from object X to bifunctor F(−, −) and (B) corresponding comma category (bifunctor). (C) The universal arrow from the object (point) 2.9 to the composition of the addition function/functor with the inclusion function/functor and (D) corresponding comma category.
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