Abstract
We develop a variant of Least Fixed Point logic based on First Orderlogic with a relaxed version of guarded quantification. We develop aGame Theoretic Semantics of this logic, and find that under reasonableconditions, guarding quantification does not reduce the expressibilityof Least Fixed Point logic. But we also find that the guarded version ofa least fixed point algorithm may have a greater time complexity thanthe unguarded version, by a linear factor.
Access this article
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Similar content being viewed by others
References
Abadi, M., Lamport, L., and Wolper, P., 1989, “Realizable and unrealizable specifications of reactive systems, ” pp. 1-17 in Proceedings of 16th ICALP, Stresa, Italy, G. Ausiello et al., eds., Berlin: Springer-Verlag.
Abiteboul, S., Hull, R., and Vianu, V., 1995, Foundations of Databases, Reading, MA: Addison-Wesley.
Aczel, P., 1975, “Quantifiers, games and inductive definitions, ” pp. 1-14 in Proceedings of the 3rd Scandinavian Logic Symposium, S. Kanger, ed., Amsterdam: North-Holland.
Aho, A. and Ullman, J., 1979, “Universality of data retrieval languages, ” pp. 110-117 in Proceedings of the 6th ACM Symposium on Principles of Programming Languages, Association for Computing Machinery.
Andréka, H., van Benthem, J., and Németi, I., 1996, “Modal Languages and Bounded Fragments of Predicate Logic, ” ILLC Research Repport and Technical Notes Series.
Back, R.-J. and von Wright, J., 1998, Refinement Calculus: A Systematics Introduction, Berlin: Springer-Verlag.
Barwise, J., 1977, “On Moschovakis Closure Ordinals, ” Journal of Symbolic Logic 42, 292-296.
Barwise, J. and Moss, L., 1996, Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena, Stanford, CA: CSLI.
Berlekamp, E.R., Conway, J.H., and Guy, R.K., 2001, Winning Ways for your Mathematical Plays, London, New York: A.K. Peters.
Blass, A. and Gurevich, Y., 1987, “Existential fixed-point logic, ” pp. 20-36 in Computation Theory and Logic, E. Börger, ed., Berlin: Springer-Verlag.
Chandra, A. and Harel, D., 1982, “Structure and complexity of relational queries, ” Journal of Computer and System Sciences 25, 99-128.
Clark, K., 1978, “Negation as Failure, ” pp. 293-322 in Logic and Databases, H. Gallaire and J. Minke, eds., New York: Plenum Press.
Codd, E. F., 1970, “A relational model of data for larged shared data banks, ” Communications of the ACM 13, 377-387.
Compton, K., 1983, “Some useful preservation theorems, ” Journal of Symbolic Logic 48, 427-440.
Conway, J.H., 2001, On Numbers and Games, London, New York: A.K. Peters.
Dahlhaus, E., 1987, “Skolem normal forms concerning the least fixed point, ” pp. 101-106 in Computation Theory and Logic, E. Börger, ed., Berlin: Springer-Verlag.
Date, C.J., 1990, An Introduction to Database Systems, 5th edition, Reading, MA: Addison-Wesley.
Ebbinghaus, H.-D. and Flum, J., 1995, Finite Model Theory, Berlin: Springer-Verlag.
Ehrenfeucht, A., 1961, “An application of games to the completeness problem for formalized theories, ” Fundamenta Mathematicae 49, 129-141.
Fagin, R., 1974, “Generalized first-order spectra and polynomial time recognizable sets, ” pp. 43-73 in Complexity of Computations, R. Karp, ed., SIAM-AMS Proceedings, Vol. 7, Philadelphia, PA: SIAM.
Fagin, R., Halpern, J., Moses, Y., and Vardi, M., 1995, Reasoning about Knowledge, Cambridge, MA: MIT Press.
Gale, D. and Stewart, F.M., 1953, “Infinite games with perfect information, ” Annals of Mathematics Studies 28, 245-266.
Grädel, E., 1992, “On transitive closure logic, ” pp. 149-163 in '91), Berlin: Springer-Verlag.
Grädel, E. and McColm, G., 1996, “Hierarchies in transitive closure logic, stratified datalog and infinitary logic, ” Annals of Pure and Applied Logic 77, 169-199.
Grädel, E. and Wakiewicz, I., 1999, “Guarded fixed point logic, ” pp. 45-54 in Proceedings 14th IEEE Symposium on Logic in Computer Science, New York: IEEE.
Grohe, M., 1996, “Arity hierarchies, ” Annals of Pure and Applied Logic 82, 103-163.
Guy, R.K., ed., 1991, Combinatorial Games, Providence, RI: American Mathematical Society.
Harel, D. and Kozen, D., 1984, “A programming language for the inductive sets, and applications, ” Information and Control 63, 118-139.
Hesselink, W.H., 1994, “Nondeterminism and recursion via games and stacks, ” Theoretical Computer Science 124, 273-295.
Hilpenen, R., 1982, “On C. S. Peirce's theory of the proposition: Peirce as a precursor of gametheoretic semantics, ” The Monist 62, 182-189.
Hintikka, J., 1972, Language Games and Information, Oxford: Clarendon.
Hintikka, J. and Kulas, J., 1983, The Game of Language: Studies in Game-Theoretical Semantics and its Applications, Dordrecht: Reidel.
Hintikka, J. and Sandu, G., 1997, “Game-theoretic semantics, ” pp. 361-410 in Handbook of Logic and Language, J. van Benthem and A. ter Meulen, eds., Cambridge, MA: MIT Press and Amsterdam: North-Holland.
Hirsch, R. and Hodkinson, I., 2002, Relation Algebras by Games, Amsterdam: Elsevier.
Immerman, N., 1981, “Number of quantifiers is better than number of tape cells, ” Journal of Computer and System Sciences 22, 384-406.
Immerman, N., 1982, “Upper and lower bounds for first order expressibility, ” Journal of Computer and System Sciences 25, 76-98.
Immerman, N., 1986, “Relational queries computable in polynomial time, ” Information and Control 68, 86-104.
Immerman, N., 1987, “Languages that capture complexity classes, ” SIAM Journal of Computing 16, 760-778.
Immerman, N., 1999, Descriptive Complexity, Berlin: Springer-Verlag.
Kanellakis, P., 1991, “Elements of relational database theory, ” pp. 1074-1156 in Handbook of Theoretical Computer Science, J. van Leeuwen, ed., Amsterdam: Elsevier.
Kolaitis, P., 1985, “Game quantification, ” pp. 365-421 in Model-Theoretic Logics, J. Barwise and S. Feferman, eds., Berlin: Springer-Verlag.
Kolaitis, P., 1991, “The expressive power of stratified logic programs, ” Information and Computation 90, 50-66.
McColm, G., 1989, “Some restrictions on simple fixed points of the integers, ” Journal of Symbolic Logic 54, 1324-1345.
McColm, G., 1990a, “Parametrization over inductive relations of a bounded number of variables, ” Annals of Pure and Applied Logic 48, 103-134.
McColm, G., 1990b, “When is arithmetic possible?, ” Annals of Pure and Applied Logic 50, 29-51.
McColm, G., 1995a, “Pebble games and subroutines in least fixed point logic, ” Information and Computation 122, 201-220.
McColm, G., 1995b, “Dimension versus number of variables, and connectivity, too, ” Mathematical Logic Quarterly 41, 111-134.
Milner, R., 1999, Communicating and Mobile Systems: The π-Calculus, Cambridge: Cambridge University Press.
Moschovakis, Y., 1972, “The game quantifier, ” Proceedings of the American Mathematical Society 31, 245-250.
Moschovakis, Y., 1974, Elementary Induction on Abstract Structures, Amsterdam: North-Holland.
Moschovakis, Y., 1980, Descriptive Set Theory, Amsterdam: North-Holland.
Moschovakis, Y., 1983, “Abstract recursion as a foundation for the theory of algorithms, ” pp. 289-364 in Computation and Proof Theory, M.M. Richter et al., eds., Berlin: Springer-Verlag.
Moschovakis, Y., 1991, “A model of concurrency with fair merge and full recursion, ” Information and Computation 93, 114-171.
Nowakowski, R.J., ed., 2002, More Games of No Chance, Cambridge: Cambridge University Press.
Parikh, R., 1985, “The logic of games and its applications, ” Annals of Discrete Mathematics 24, 111-140.
Plotkin, G., 1981, A Structural Approach to Operational Semantics, Aarhus: Computer Science Department, Aarhus University.
Rogers, Jr., H., 1967, Theory of Recursive Functions and Effective Computability, New York: McGraw-Hill.
Ullman, J.D., 1988, 1989, Principles of Database Systems and Knowledge Base Systems I and II, Rockville, MD: Computer Science Press.
van Benthem, J.F.A.K., 1996, Exploring Logical Dynamics, Stanford, CA: CSLI.
van Benthem, J.F.A.K., 2002, “Extensive games as process models, ” Journal of Logic, Language and Information 11, 289-313.
Vardi, M., 1982, “Complexity of relational database systems, ” pp. 137-146 in Proceedings 14th ACM Symposium on the Theory of Computing, Association for Computing Machinery (ACM).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
McColm, G. Guarded Quantification in Least Fixed Point Logic. Journal of Logic, Language and Information 13, 61–110 (2004). https://doi.org/10.1023/A:1026107209351
Issue Date:
DOI: https://doi.org/10.1023/A:1026107209351