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Linear time computable problems and first-order descriptions

Published online by Cambridge University Press:  19 April 2018

Detlef Seese*
University Karlsruhe (TH), Germany Email:


It is well known that every algorithmic problem definable by a formula of first-order logic can be solved in polynomial time, since all these problems are in L (see Aho and Ullman (1979) and Immerman (1987)). Using an old technique of Hanf (Hanf 1965) and other techniques developed to prove the decidability of formal theories in mathematical logic, it is shown that an arbitrary FO-problem over relational structures of bounded degree can be solved in linear time.

Research Article
Copyright © Cambridge University Press 1996

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Abitoul, S., Vardi, M. and Vianu, V. (1992) Fixpoint logics, relational machines and computational complexity, Proceedings of the 33rd Annual Symposium on Foundations of Computer Science 156-192.Google Scholar
Aho, A., Hopcroft, J. and Ullman, J. D. (1974) The design and analysis of computer algorithms, Addison-Wesley.Google Scholar
Aho, A. and Ullman, J. D. (1979) Universality of Data Retrieval Languages, 6th Symposium on Principles of Programming Languages 110-117.CrossRefGoogle Scholar
Ajtai, M. and Fagin, R. (1990) Reachability is harder for directed than for undirected finite graphs. Journal of Symbolic Logic 55 (1) 113-150.Google Scholar
Arnborg, S., Courcelle, B., Proskurowski, A. and Seese, D. (1993) An algebraic theory of graph reduction. JACM 40 (5) 11341164.Google Scholar
Arnborg, S., Lagergren, J. and Seese, D. (1991) Easy problems for tree-decomposable graphs. Journal of Algorithms 12 308340.CrossRefGoogle Scholar
Baudisch, A., Seese, D., Tuschik, P. and Weese, M. (1982) Decidability and Quantifier-Elimination. In: Barwise, J. and Feferman, S. (eds.) Model-Theoretic Logics, Springer-Verlag 235268.Google Scholar
Behmann, H. (1922) Beitrage zur Algebra der Logik, insbesondere zum Entscheidungsproblem. Math. Ann. 86 163229.Google Scholar
Chandra, A. K. and Harel, D. (1980) Stucture and complexity of relational queries. J. Comput. Syst. Sci. 25 156178.Google Scholar
Church, A. and Quine, W. (1953) Some theorems on definability and decidability. Journ. Symb. Logic 17 179187.Google Scholar
Compton, K. and Henson, C. (1987) A uniform method for proving lower bounds on the computational complexity of logical theories, The University of Michigan, Computing Research Laboratory, CRL-TR-04-87.Google Scholar
Cormen, T., Leiserson, C. and Rivest, R. (1990) Introduction to Algorithms, The MIT Press.Google Scholar
Courcelle, B. (1994) Monadic second-order definable transductions: a survey. Theoretical Computer Science 126 5375.Google Scholar
Courcelle, B. and Mosbah, M. (1993) Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science 109 4982.CrossRefGoogle Scholar
Creignou, N. (1993) The class of problems that are linearly equivalent to Satisfiability or a uniform method for proving NP–completeness. In: Computer Science Logic 1992, San Miniato. Springer-Verlag Lecture Notes in Computer Science 702 115133.Google Scholar
Dahlhaus, E. (1983) Reduction to NP-complete problems by interpretations. In: Boerger, , Hasenjaeger, and Roedding, (eds.) Logic and Machines, Decision problems and Complexity. Springer-Verlag Lecture Notes in Computer Science 171 357365.Google Scholar
Eaves, B. and Rothblum, U. (1994) Linear Problems and linear algorithms, Rutcor Research Report, RRR28-94.Google Scholar
Ebbinghaus, H.-D., Flum, J. and Thomas, W. (1993) Mathematical Logic, Second Edition, Springer-Verlag.Google Scholar
Fagin, R. (1974) Generalized First-Order spectra and polynomial-time recognizable sets. In: Karp, R. (ed.) Complexity of Computation. SIAM-AMS Proc. 7 2741.Google Scholar
Fagin, R., Stockmeyer, L. and Vardi, M. (1993) On monadic NP vs. monadic co-NR In: The Proceedings of the 8th Annual IEEE Conference on Structure in Complexity Theory 19-30. (Full paper in Information and Computation (1995) 120 (1) 7892.)Google Scholar
Gaifman, H. (1982) On local and nonlocal properties. In: Stern, J. (ed.) Logic Colloquium ’81, North-Holland 105135.Google Scholar
Giakoumakis, V. (1995) On imperfect P 4 - sparse graphs. In: Faigle, U. and Hoede, C. (eds.) Proceedings 4th Twente workshop on graphs and combinatorial optimization 110-113.Google Scholar
Giammarresi, D., Restivo, A., Seibert, S. and Thomas, W. (1993) Monadic second order-logic over rectangular pictures and recognizability by tiling systems, Report No. 9318, Christian-Albrechts-Universität Kiel (also to appear in Information and Computation).Google Scholar
Grädel, E. (1990) On the notion of linear time computability. Internat. J. Foundations Comput. Sci. 1 295307.CrossRefGoogle Scholar
Grandjaen, E. (1993) Linear time algorithms and NP-complete problems. In: Proceedings Computer Science Logic 1992, San Miniato. Springer-Verlag Lecture Notes in Computer Science 702 248273 (also to appear in SIAM J. on Computing).Google Scholar
Gurevich, V. (1984) Toward logic tailored for computational complexity. In: Börger, E. et al. (eds.) Computation and Proof Theory. Springer-Verlag Lecture Notes in Mathematics 1104 175216.Google Scholar
Gurevich, Y. (1985) Monadic second-order theories. In: Barwise, J. and Feferman, S. (eds.) Model- Theoretic Logics Chap. XIII, Springer-Verlag 479506.Google Scholar
Hanf, W. (1965) Model-theoretic methods in the study of elementary logic. In: Addison, J., Henkin, L. and Tarski, A. (eds.) The Theory of Models, North-Holland 132145.Google Scholar
Hauschild, K. and Rautenberg, W. (1971) Interpretierbarkeit in der Gruppentheorie. Algebra universalis 1 fasc. 2 136151.CrossRefGoogle Scholar
Immerman, N. (1982) Upper and Lower Bounds for First Order Expressability. JCSS 25 (1).Google Scholar
Immerman, N. (1986) Relational queries computable in polynomial time. Information and Control 68 86104.Google Scholar
Immerman, N. (1987) Languages that capture complexity classes. SIAM J. Comput. 16 (4) 760778.Google Scholar
Maltsev, A. I. (1959) Regular products of models, Izv. Acad. Nauk. SSSR, Ser. Mat., 23 489-502.Google Scholar
Mehlhorn, K. (1984) Graph Algorithms and NP-Completeness, Springer-Verlag.Google Scholar
Minsky, M. L. (1967) Computation: Finite and Infinite Machines, Prentice-Hall.Google Scholar
Papadimitriou, C. (1994) Computational Complexity, Addison-Wesley.Google Scholar
Rabin, M. (1965) A simple method for undecidability proofs and some applications. In: Bar-Hillel, (ed.) Logic, Methodology and Philosophy of Science II, North-Holland 5868.Google Scholar
Rabin, M. (1977) Decidable Theories. In: Barwise, J. (ed.) Handbook of Mathematical Logic, North-Holland 595629.Google Scholar
Robertson, N. and Seymour, P. (1985) Graph minors — A survey. In: Anderson, I. (ed.) Surveys in Combinatorics, Cambridge University Press 153171.Google Scholar
de Rougemont, M. (1987) Second-order and inductive definability on finite structures. Zeitschrift f. math. Logik und Grundlagen d. Math. 33 4763.Google Scholar
Schwentick, T. (1994) Graph connectivity and monadic NP, Informatik-Bericht Nr. 2/94, Johannes Gutenberg - Universität Mainz, Institut für Informatik, FB 17 1-33.Google Scholar
Seese, D. (1992) Interpretability and tree automata: a simple way to solve algorithmic problems on graphs closely related to trees. In: Nivat, M. and Podelski, V (eds.) Tree Automata and Languages, Elsevier Science Publishers 83114.Google Scholar
Stolboushkin, A. P. and Taitslin, M. A. (1994) Is First Order Contained in an Initial Segment of PTIME? In: Pacholski, L. and Tiuryn, J. (eds.) Proceedings of CSL'94. Springer-Verlag Lecture Notes in Computer Science 933 242248.Google Scholar
Thomas, W. (1991) On logics, tilings and automata. In Proc. 18th ICALP. Springer-Verlag Lecture Notes in Computer Science 510 441454.Google Scholar
Wagner, K. and Wechsung, G. (1986) Computational Complexity, VEB Deutscher Verlag der Wissenschaften, Berlin.Google Scholar