should be send to Iris van Rooij, Donders Institute for Brain, Cognition, and Behaviour, Radboud University Nijmegen, Montessorilaan 3, 6525 HR Nijmegen, The Netherlands. E-mail: i.vanrooij@donders.ru.nl

Abstract

Four articles in this issue of topiCS (volume 4, issue 1) argue against a computational approach in cognitive science in favor of a dynamical approach. I concur that the computational approach faces some considerable explanatory challenges. Yet the dynamicists’ proposal that cognition is self-organized seems to only go so far in addressing these challenges. Take, for instance, the hypothesis that cognitive behavior emerges when brain and body (re-)configure to satisfy task and environmental constraints. It is known that for certain systems of constraints, no procedure can exist (whether modular, local, centralized, or self-organized) that reliably finds the right configuration in a realistic amount of time. Hence, the dynamical approach still faces the challenge of explaining how self-organized constraint satisfaction can be achieved by human brains and bodies in real time. In this commentary, I propose a methodology that dynamicists can use to try to address this challenge.

Dixon, Holden, Mirman, and Stephen (2012), Gibbs and Van Orden (2012), Riley, Shockley, and Van Orden (2012), and Silberstein and Chemero (2012) are critical of computational approaches to cognition and aim to promote a dynamical perspective of cognitive behavior. The authors argue that certain phenomena are difficult or impossible to explain from a computational perspective, yet they are naturally explained from a dynamical perspective. The phenomena range from pink noise and context sensitivity to ultrafast cognition and even consciousness. Whether or not computational approaches can explain these phenomena I will leave for others to discuss. In this commentary, I would like to instead focus on the explanatory power and limits of the dynamicists’ proposal that cognitive behavior emerges when brain and body (re-)configure to satisfy task and environmental constraints. Does this proposal really explain the speed of cognitive behavioral change and adaptation as the dynamicists seem to suggest? I will show that we need to answer this question in the negative. The reason is that there exist systems of constraints for which constraint satisfaction cannot be realized quickly by any method, be it modular, local, centralized, or self-organized. This means that a dynamical account will need more ingredients than self-organization alone if it is to explain the speed of cognitive behavioral change and adaptation. I will present ideas on what these extra ingredients could be and how dynamicists can use complexity-theoretic proof techniques to test their candidate dynamical explanations for validity.

2. A formalism for constraint satisfaction

A hypothesis shared between the target articles is that cognitive behavior is self-organized, emerging from the physical interaction among a large number of elements (Dixon et al., 2012) that together comprise an extended and non-linearly coupled brain–body–environment system (Silberstein & Chemero, 2012). Gibbs and Van Orden (2012) and Riley et al. (2012) furthermore propose that this hypothesis can explain the context-sensitivity, flexibility, and adaptability of cognitive behavior, because self-organized systems can efficiently converge on a configuration that satisfies task and situational constraints (cf. Dixon et al., 2012; Silberstein & Chemero, 2012). As the authors do not present a particular formalism of constraint satisfaction, and I think the discussion can benefit from having one, I propose to consider an existing formalism, called the Constraint Satisfaction Problem (CSP).

Constraint satisfaction problem

Input: A set of variables V, a domain D, and a set of constraints C.

Output: A value assignment A: V→D, assigning to variables in V possible values from domain D, such that all constraints in C are satisfied.

The CSP is a widely studied problem in both applied and theoretical computer science (Marx, 2005; Tsang, 1996). Given its relevance for the field of computer science, it is important to forestall the possible misunderstanding that the CSP formalism is only relevant for, or its use commits one to, a computational theory of mind. The formalism merely specifies a function, with a particular input domain (possible instantiations of V, D, and C) and output domain (possible values assignments A), and a mapping between them. In the context of a computational theory, where CSP may be postulated as a function describing the computation performed by a mental module or cognitive subsystem, “input” and “output” can be read as meaning “representation” (see, e.g., Thagard & Verbeurgt, 1998). In the context of a dynamical theory that postulates CSP as a characterization of the transition of a brain–body–environment configuration from an unstable to a stable state, “input” and “output” may be read as meaning “state.” In the latter case, possibly a dynamicist may want to add an initial value assignment to the input part of the problem definition, reflecting the “unstable state,” for example, as per the following generalization of CSP:

Constraint satisfaction problem-G1

Input: A set of variables V, a domain D, an initial value assignment A_{i}: V→D, and a set of constraints C (where A_{i} need not yet satisfy all constraints in C).

Output: A new value assignment A: V→D, assigning to variables in V possible values from domain D, such that all constraints in C are satisfied.

Motivated by additional intuitions about what characterizes “constraint satisfaction” for self-organized cognitive systems, dynamicists may wish to add more features to the formalism, increasing its expressive power further. For instance, one could assume some variables in V have values that are fixed by task or environmental constraints, different constraints in C may differ in how important they are to satisfy, and in cases where satisfaction of all constraints in C is impossible, the system will converge on a configuration A that maximally satisfies constraints (analogous to “harmony maximization” or, conversely, “energy minimization”). These are just a few options that combined would yield a generalization of the CSP formalism as given below:

Constraint satisfaction problem-G2

Input: A set of variables V = V_{fixed}∪V_{change}, a domain D, an initial value assignment A_{i}: V→D, and a set of constraints C with for each constraint c∈C a numerical weight w(c).

Output: A new value assignment A: V_{change}→D, such that the sum of weights of satisfied constraints is maximized (here SAT_{C,A} denotes the set of constraints in C that are satisfied by value assignment A).

Evidently, any dynamical account of self-organized constraint satisfaction can adapt the formalism to its own needs and intuitions. By doing so, it may be possible to give metaphorical or otherwise underspecified dynamical accounts of cognitive phenomena more substance and provide bases for new experimental tests of such accounts.

Explication of the nature of CSPs solved by self-organized cognitive systems may also be necessary to support the claim that the dynamical approach can explain how cognitive behavior can change and adapt so quickly in real time. Currently, it seems that dynamicists suggest the speed of cognitive change and adaptation is explained by self-organization itself. This is, however, to attribute to self-organization more magical powers that it is likely to possess. Why this is so, I explain next.

3. Time-complexity of constraint satisfaction

Constraint satisfaction problem is one of the many problems in mathematics and computer science that are infamous for being so-called computationally intractable (Arora & Barak, 2009; Garey & Johnson, 1979). Informally, this means that there does not exist any method for reliably solving all instances of CSP in a realistic amount of time. Formally, it means CSP belongs to the class of NP-hard problems. NP-hard problems owe their name to the fact that they are as “hard” to solve as any problem in the class NP, and possibly even harder.^{1} Here, the class NP consists of all those problems for which a solution can be checked for its correctness in polynomial time (i.e., a time on the order of n^{c}, denoted O(n^{c}), where n is some measure of the input size and c is some (usually small) constant). It is known that the class P, consisting of problems for which correct solutions can also be found in polynomial time, is a subset of NP (i.e., P⊆NP) and it is strongly^{2} conjectured that the inclusion is proper (i.e., P ≠ NP). As the P ≠ NP conjecture is widely accepted among mathematicians, in the remainder of this commentary I will work under the assumption that it is true.^{3} This implies, then, that CSP, being an NP-hard problem, does not afford any polynomial-time method for solving it, and that all methods (be they modular, local, centralized, or self-organized) require exponential time or worse to complete the transformation from input state to output state, if not for all inputs, then at least for a non-empty set of inputs. Note that the same applies for CSP-G1 and CPS-G2 as these problems include CSP as a special case and hence are NP-hard as well.

To see that an exponential time, that is, O(c^{n}), takes unrealistically long even for medium sized inputs, consider that 2^{n} is already more than the number of milliseconds in 30 years for n =40. To put it differently, if we assume that a self-organized system can find the right configuration by following a trajectory through an exponential number of intermediate configurations (defined by intermediate value assignments to the variables in the system) while transitioning at the rate of one complete configuration per millisecond, it would take a system of 40 variables decades to converge on the right configuration. Given that the systems of constraints hypothesized by the authors of the target articles are expected to be quite large, possibly ranging in the hundreds or even thousands variables, the postulation that cognitive behavior can be understood as self-organized constraint satisfaction raises the question how this hypothesis explains it speed. If anything, it seems to predict that behavioral change and adaptation should regularly proceed horribly slow. Evidently, behavioral change and adaptation typically does proceed quickly. The dynamical approach may be able to explain this, but this seems to require that dynamicists formulate explicit hypotheses about how brain, body, and environment are mutually constrained in exactly such a way that states that are impossible to re-configure quickly are effectively ruled out.

Before turning to a discussion of a methodology for modeling and testing such hypotheses, I would like to fence off two possible objections to the use of time-complexity measures for assessing the quality of cognitive explanations. A first possible objection reads as follows: The intractability of NP-hard problems, such as CSP or its generalizations, is a consequence of the computational limitations of classical computing machines, such as Turing machines; under alternative formalisms of “computation,” such as (analog) neural networks, cellular automata, or quantum computers, it is possible to compute NP-hard problems efficiently. This objection builds on a misrepresentation of the state of the art: Currently there does not exist any (physically plausible) formalism of “computation” under which NP-hard problems are, or can be shown to be, solvable in polynomial time (Arora & Barak, 2009; Parberry, 1994; Siegelmann & Sontag, 1994; see also van Rooij, 2008, pp. 963–970). A second possible and closely related objection is as follows: Cognitive systems are physical systems operating by the laws of physics, not the rules of computation, and hence can quickly solve systems of constraints that are not quickly solvable by any computational procedure, such as NP-hard problems. As an empirical hypothesis this claim has no support that I know of and as a theoretical claim based on classical physics it seems to be demonstrably false (Aaronson, 2005).^{4}

These two objections seem to me the most relevant in the context of this commentary. I am aware that many other objections have been raised—including the popular but unsubstantiated claims that “approximate” or “good enough” solutions to NP-hard problems can be efficiently found—but discussing all of them in this commentary would detract from its message. I refer the reader to Kwisthout, Wareham, and van Rooij (2011), van Rooij (2008), and van Rooij, Wright, and Wareham (2010) for replies to these other possible objections.

4. Identifying conditions for tractability: A methodology

I will now sketch a methodology dynamicists could adopt for building models of self-organized constraint satisfaction with low time-complexity. Let me restate what is at stake. Dynamical accounts wish to use the notion of self-organized constraint satisfaction to explain that, under normal conditions, cognitive behavior quickly adjusts to task and environmental constraints (cf. Riley et al., 2012). We have seen that, if no specific commitments are made to characterize “normal conditions,” constraint satisfaction is NP-hard. Now, it is well known that even if a function F: I→O is NP-hard, that function restricted to a smaller input domain, that is, F′: I′→O, with I′⊂I, can be efficiently solvable (Garey & Johnson, 1979). Therefore, if it is psychologically plausible and empirically supported that the inputs I* = I − I′ are irrelevant, abnormal, or otherwise negligible, then a cognitive scientist can replace the hypothesis that F explains cognitive behavior by the hypothesis F′. In such case, no explanatory power is lost, yet what is won is that the newly hypothesized function can be realized in real time by some (possibly self-organized) process.

This strategy for building tractable models is probably best seen as an entire research program. Here I can merely explain its rationale. I will use the following function as an illustration:

Constraint satisfaction problem-GR

Input: A set of variables V = V_{fixed}∪V_{change}, a domain D = {0,1}, an initial value assignment A_{i}: V→D, and a set of binary constraints C⊆V × V with for each constraint (u, v)∈C a numerical weight w(u, v), where C is partitioned into positive constraints C^{+} and negative constraints C^{−} (i.e., C = C^{+}∪C^{−} and C^{+}∩C^{−} = ∅). A positive constraint (u, v)∈C^{+} is satisfied if u and v are assigned the same value (both “0” or both “1”) and a negative constraint (u, v)∈C^{−} is satisfied if u and v are assigned a different value (one a “1” and the other “0”).

Output: A new value assignment A: V_{change}→D, such that the sum of weights of satisfied constraints is maximized (here SAT_{C,A} denotes the set of constraints in C that are satisfied by value assignment A).

Note how this function is the same as CSP-G2 but with a severely restricted domain of inputs: The domain of values is restricted to be binary (allow only two values), the constraints are binary (apply to only pairs of variables), and the constraints come only in two types (positive and negative constraints with special conditions of satisfaction). Is CSP-GR tractable? No, it is not. Notwithstanding the strong restrictions on its input domain, the problem CSP-GR remains NP-hard.

The NP-hardness of CSP-GR follows from the following observation: If the initial assignment A_{i} were to be deleted from the input and V_{fixed} were to be empty, then the problem CSP-GR is identical to the CSP Coherence—equivalent to harmony maximization in Hopfield networks—proven NP-hard by Thagard and Verbeurgt (1998). To see that the availability of an initial value assignment, by itself, does not make solving CSP-GR easier than Coherence, note that no conditions on the initial assignment have been specified, so it is computationally trivial to make one. Further note that nowhere is it defined that V_{fixed} cannot be empty.^{5} It follows that CSP-GR is at least as hard as Coherence and thus NP-hard.

The NP-hardness of CSP-GR may be sobering, but it is important to note that under further restrictions the problem can be made tractable. For instance, if the number of negative constraints is small or if almost all constraints can be satisfied, then CSP-GR is efficiently solvable (van Rooij, 2003, 2008). The important lesson is two-fold. First, finding conditions on the input domain of a constraint satisfaction model that guarantee tractability is non-trivial; after all, even highly restricted constraint satisfaction can be NP-hard. Second, there do exist restricted versions of constraint satisfaction that are tractable, and for all we know there exist ones that can characterize the conditions that afford—and thereby explain—speedy self-organization of brain–body–environment systems.

It is unlikely that CSP-GR (plus the conditions that make it tractable) has sufficient descriptive power to characterize all forms of self-organized constraint satisfaction that subserve human cognitive behavior. The challenge for dynamicists remains to develop models of constraint satisfaction that are descriptively powerful enough. Yet these models cannot be made so descriptively powerful that they become computationally intractable if they are to provide plausible accounts of how cognitive behavior evolves in real time. The modeling task ahead is thus a careful balancing act between descriptive power and tractability. I hope to have illustrated how such a balancing act may be implemented; namely, by making functional-level models of constraint satisfaction with explicit conditions on the input domain and then testing these models for tractability. For details on proof techniques that can be used for testing the tractability of such models see, for instance, van Rooij, Stege, and Kadlec (2005) and van Rooij and Wareham (2008), and for existing tractability results relevant for constraint satisfaction, see, for instance, Gottlob and Szeider (2008).

5. Conclusion

I agree with the authors of the target articles that computational approaches to cognition have real problems in explaining the context sensitivity and speed of human cognition (see, e.g., Haselager, van Dijk, & van Rooij, 2008; Kwisthout et al., 2011). I am also sympathetic with the ideas that the human brain need not be organized as a massively modular system, behavior need not be planned by a central executive, and behavior may be emergent from the complex interplay between brain, body and world (see, e.g., Haselager et al., 2008; van Dijk, Kerkhofs, van Rooij, & Haselager, 2008; van Rooij, Bongers, & Haselager, 2002). The proposal that such an interplay can be characterized as a process of self-organized constraint satisfaction seems to me quite elegant. But, with this proposal, the dynamical approach is not yet home free. In order to really explain why and how a brain–body–world coupled system can quickly converge on constraint-satisfying configurations, dynamical theorists need to build explicit models of self-organized constrained satisfaction that have the property that they afford such quick convergence. I sincerely hope that dynamical modelers will take on this challenge, as I believe that the approach holds promise of solving one of the key problems in cognitive science, that is, the inexplicable speed of cognition and behavior.

Footnotes

^{1}

NP-hard problems that are in NP are also known as NP-complete problems.

^{2}

The famous P ≠ NP conjecture is mathematically unproven yet widely believed by mathematicians on both theoretical and empirical grounds (Fortnow, 2009; Garey & Johnson, 1979).

^{3}

Some dynamicists may wish to object at this point, in that they believe the P ≠ NP conjecture is false. Such theorists indeed may be able to parry my charge of having to explain how self-organized constraint satisfaction can proceed quickly but at the cost of having to explain why the intuitions of so many mathematicians are false. The latter, I suspect, will be much harder than building a research program aimed at explaining the tractability of self-organized constraint satisfaction within the constraints imposed by the P ≠ NP conjecture.

^{4}

Aaronson (2005) considers, for instance, the popular claim that soap bubbles can solve an NP-hard problem called the Steiner Tree problem. Interestingly, he observes bubbles settle on suboptimal as well as invalid solutions even for small inputs (physically realized as pegs on a board; for details, see Aaronson, 2005, pp. 32–33) and that although re-configurations where the bubbles escape a local optimum do happen, these re-configurations seem to take considerable time.

^{5}

Even if the set V_{fixed} were forced to be non-empty, tractability would not follow: Coherence always has two optimal solutions which are each other’s complement (because of the symmetry in the constraints); therefore, one could always arbitrarily pick at least one variable to fix. It is conceivable that with further restrictions on the nature of the set V_{fixed}, tractability may be achieved. To exploit this option, though, dynamicists will have to explicate what those restrictions are and then establish—for example, by mathematical proof—that the restrictions indeed make constraint satisfaction tractable.