Abstract
In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes.
Highlights
Decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them
How can we solve the problem presented at the end of the previous section? How can computationally intractable systems of logic work as foundations of mathematics if they are considered to be unfit for modelling human cognitive capacities? I will argue that this is a pseudo-problem that is the result of unwarranted application of results from the study of computational complexity in the domain of computational modelling of cognitive processes
Problems become prohibitively complex because the only method for solving them is through brute force by going through all possible combinations. Quite clearly this is not how mathematical problems are solved by human agents, aside from particular cases involving small domains. This is consistent with the way I have argued the P-cognition thesis to be misguided: limits on relevant input sizes and considerations on human problem solving algorithms require thinking of mathematical problem solving in a different context, one in which computational complexity classes are not applied directly
Summary
Decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. The entire range of relevant cognitive phenomena is enormous and a thorough analysis is not possible in a single paper For this reason, I focus on mathematics and mathematical problem solving to show that the computational complexity measures are too coarse to be used as general principles concerning the modelling of human cognitive capacities in these domains. While there have been important critical assessments of how computational complexity should be applied in cognitive science (e.g., van Rooij 2008; Isaac et al 2014; Szymanik 2016; Szymanik and Verbrugge 2018; Van Rooij et al 2019; Pantsar 2019b; Fabry & Pantsar 2019), limitations like the above P-cognition thesis are still too often misunderstood and given excessive importance These problems can extend to applications of descriptive complexity through the connections between computational complexity measures and the complexity of logical systems. I will conclude that with well-considered specifications with regard to relevant input sizes and problem solving algorithms, tractability principles like the P-cognition thesis can be given their proper place as potentially useful guidelines, but not as anything resembling strict limits
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