The Parameterized Complexity of Some Problems in Logic and Linguistics

The theory of parameterized computational complexity introduced in [DF1-3] appears to be of wide applicability in the study of the complexity of concrete problems [ADF,BDFW,BFH,DEF,FHW,FK]. We believe the theory may be of particular importance to practical applications of logic formalisms in programming language design and in system specification. The reason for this relevance is that while many computational problems in logic are extremely intractable generally, realistic applications often involve a “hidden parameter” according to which the computational problem may be feasible according to the more sensitive criteria of fixed-parameter tractability that is the central issue in parameterized computational complexity. We illustrate how this theory may apply to problems in logic, programming languages and linguistics by describing some examples of both tractability and intractability results in these areas. It is our strong expectation that these results are just the tip of the iceberg of interesting applications of parameterized complexity theory to logic and linguistics. Our main results are as follows. (1) The problem of determining whether a word x can be derived in k steps in a context-sensitive grammar G (Short CSL Derivation) is complete for the parameterized complexity class W[1]. (2) The problem Minimum Axiom Set [GJ] is complete for the parameterized complexity class W[P]. The latter result has the added significance of providing a starting point for a proof of the important structural result W[P]=monotone W[P]. We also report on the parameterized complexity of reachability in Petri nets (equivalently, vector addition systems) and of some problems concerning phonological derivation in linguistic theory.

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