In this chapter, we consider a computational-level theory of communication as Bayesian inference. We again illustrate the use of classical complexity analysis to assess the theory’s intractability. In addition, we show that parameterized complexity analysis can converge with theoretically informed intuitions about possible sources of this intractability, in this case based on the Gricean Maxims in pragmatic theories of communication.

This chapter discusses a set of possible objections to the tractability constraint on computational-level theories. Any of these objections may naturally arise when reading Chapter 1. It is an option to read this chapter directly after Chapter 1. However we believe that the responses to the listed objections can be best appreciated after having mastered the formal concepts and techniques in Chapters 2–7 and familiarized oneself with the conceptual arguments in Chapter 8.

In this chapter we introduce the classical complexity classes P and NP, where P consists of decision problems that are solvable in polynomial time and NP consists of decision problems for which yes-instances are verifiable in polynomial time. We show that the latter problem class contains many problems for which it is conjectured that no polynomial-time algorithm exists. We introduce the formal notions of NP-hardness and NP-completeness to show that a problem is as least as hard as the hardest problems in NP, and we describe why NP-hard problems are considered to be intractable (i.e., not computable in polynomial time). We explain how one can prove that a problem is a member of either P or NP, and how one can use the technique of polynomial-time reduction, introduced and practiced in Chapter 3, to prove NP-hardness and NP-completeness.

In this chapter we review different "coping strategies" that cognitive scientists have been adopting for dealing with the intractability\sind{intractability} of computational-level theories\sind{computational-level theories} and reflect on their validity and usefulness. We next turn to the question how the concepts and techniques from Chapters 2–7 can be useful tools for cognitive scientists to deal with intractability of computational-level theories of cognition in a constructive and conceptually coherent way.

In this chapter, we consider a computational-level theory of analogy derivation as structure mapping. We again illustrate the use of classical complexity analysis to assess the theory's intractability. In addition, we show how parameterized complexity analysis can be used to formally assess intuitive conjectures about possible sources of this intractability. This illustration demonstrates that such intuitions can often be wrong, underscoring the importance of formal analyses.

In this chapter we introduce the parameterized analogues of \pt\ and \np, namely, \fpt and other complexity classes such as \wone, \wtwo, and \xp\ comprising with \fpt\ the \whi-hierarchy, and the formal notions of \whi-hardness and \whi-completeness. We describe why \whi-hard parameterized problems are considered to be fixed-parameter intractable (i.e., not computable in fixed-parameter tractable time). We explain how one can prove that a problem is a member of a class in the \whi-hierarchy, and how one can use the technique of parameterized reduction, introduced and practiced in Chapter 6, to prove \whi-hardness and \whi-completeness.

In this chapter, we consider a computational-level theory of coherence as constraint satisfaction. We illustrate the use of classical and parameterized complexity analysis to assess the theory's (in)tractability and identify some of its sources of intractability.

In this chapter, we introduce the concepts of parameterized problem and fixed-parameter tractability. Using these concepts, one can show that problems that are polynomial-time intractable in general (i.e., \np-hard) may yet be practically solvable provided only that certain input parameters are constrained in terms of their values. This conception of tractability underlies the FPT-Cognition Thesis introduced in Chapter 1. We illustrate three techniques for showing that a parameterized problem is fixed-parameter tractable, namely, brute-force combinatorics, bounded search trees, and reduction to a problem kernel.We also include several exercises for practicing these techniques.

In this chapter we introduce the notion of polynomial-time reductions. We explain how this technique can be used to transform an input for problem A to an input for problem B, mapping yes-instances for A to yes-instances for B and vice versa. If this transformation can be done in polynomial time, this implies that if B is polynomial-time computable, then so is A; also, it implies that if A has an exponential-time lower bound, then so must B. These polynomial-time reductions are thus a powerful technique to relate problems to each other. We will demonstrate several reduction strategies, namely reduction by restriction, by local replacement, and by component design. We include several exercises for practicing this technique.

In this chapter we introduce the notion of parameterized reductions. We explain how this technique can be used to transform an input for a parameterized problem $K$-$A$ into an input or parameterized problem $K$-$B$, mapping yes-instances for $K$-$A$ to yes-instances for $K$-$B$ and vice versa. If this transformation can be done in fixed-parameter tractable time, this implies that if $K$-$B$ is fixed-parameter-tractable, then so is $K$-$A$; conversely, if $K$-$A$ is not fixed-parameter tractable, then neither is $K$-$B$. Like the polynomial-time reductions introduced in Chapter 3, parameterized reductions are a powerful technique for relating problems to each other. We will demonstrate parameterized analogues of each of the reduction strategies described in Chapter 3. We also include several exercises for practicing this technique.