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Using Program Schemes to Capture Polynomial-Time Logically on Certain Classes of Structures

Published online by Cambridge University Press:  01 February 2010

Iain A. Stewart
Iain A. Stewart
Affiliation:
Department of Computer Science, University of Durham, Durham DH1 3LEi.a.stewart@durham.ac.uk, http://www.dur.ac.uk/i.a.stewart

Abstract

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In this paper, the study of the expressive power of certain classes of program schemes on finite structures is continued, in relation to more mainstream logics studied in finite model theory and to computational complexity. The author shows that there exists a program scheme – whose constructs are assignments and while-loops with quantifier-free tests and which has access to a stack – that can accept a P-complete problem, the deterministic path system problem, even in the absence of non-determinism, so long as problem instances are presented in a functional style. (The proof given here leans heavily on Cook's proof that the classes of formal languages accepted by deterministic and non-deterministic logspace auxiliary pushdown machines coincide.) However, whilst this result is of independent interest, in that it leads to a deterministic model of computation capturing P, whose non-deterministic variant also captures P, the program scheme can also be used to build a successor relation in certain classes of structures (namely: the class of strongly connected locally ordered digraphs, the class of connected planar embeddings, and the class of triangulations), with the consequence that on these classes of graphs, (a fragment of) path system logic (with no built-in relations) captures exactly the polynomial-time solvable problems.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 40 - 67
Copyright
Copyright © London Mathematical Society 2003

References

1Arratia-Quesada, A.A., Chauhan, S. R. and Stewart, I. A., ‘Hierarchies in classes of program schemes’, J. Logic Comput. 9 (1999) 915–957.Google Scholar
2Cook, S. A., ‘Characterization of pushdown machines in terms of time-bounded computers’, J. ACM 18 (1971) 4–18.Google Scholar
3Diestel, R., Graph theory (Springer, 1997).Google Scholar
4Ebbinghais, H.D. and Flum, J., Finite model theory (Springer, 1995).Google Scholar
5Etessami, K. and Immerman, N., ‘Reachability and the power of local ordering’, Theoret. Comput. Sci. 148 (1995) 261–279.Google Scholar
6Grohe, M., ‘Fixed-point logics on planar graphsy’, Proc. 13th Annual IEEE Symposium on Logic in Computer Science (IEEE Computer Society, 1998) 6–15.CrossRefGoogle Scholar
7Grohe, M. and Mariño, J., ‘Definability and descriptive complexity on databases of bounded tree-width’, Proc 7th International Conference on Database Theory, Lecture Notes in Comput. Sci. 1540 (ed. Beeri, C. and Buneman, P., Springer, Berlin, 1999) 70–82.Google Scholar
8Immerman, N. and , E. Lander, ‘Describing graphs: a first-order approach to graph canonization’, Complexity theory retrospective (ed. Selman, A., Springer, 1990) 59–81.Google Scholar
9Otto, M., Bounded variable logics and counting, Lecture Notes in Logic 9 (Springer, 1997).Google Scholar
10Saaty, T.L. and Kainen, P.C., The four-colour problem (McGraw-Hill International, 1977).Google Scholar
11, I.A. Stewart, ‘Logical and schematic characterization of complexity classes’, Acta Informatica 30 (1993) 61–87.CrossRefGoogle Scholar
12, I. A. Stewart, ‘Logical description of monotone NP problems’, J. Logic Comput. 4 (1994) 337–357.Google Scholar
13, I. A. Stewart, ‘Program schemes, queues, the recursive spectrum and zero-one laws’, Proc.7th Annual International Computing and Combinatorics Conference, (COCOON 2001), Lecture Notes in Comput. Sci. 2108 (ed. Wang, J., Springer, Berlin, 2001) 39–8.Google Scholar
14, I. A. Stewart, ‘Program schemes, arrays, Lindström quantifiers and zero-one laws’, Theoret.Comput.Sci. 275 (2002) 283–310.CrossRefGoogle Scholar