Abstract
Whereas first-order logic was developed to confront the infinite it is often used in computer science in such a way that infinite models are meaningless. We discuss the first-order theory of finite structures and alternatives to first-order logic, especially polynomial time logic.
Supported in part by NSF grant MCS83-01022
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.V. Aho and J.D. Ullman, "Universality of Data Retrieval Languages", Proc. of 6th ACM Symposium on Principles of Programming Languages, 1979, 110–117.
J. Bennet, "On Spectra", Doctoral Dissertation, Princeton University, N.J., 1962.
E. Börger, "Spektralproblem and Completeness of Logical Decision Problems", in "Logic and Machines: Decision Problems and Complexity (ed. E. Börger, G. Hasenjaeger, D. Rödding), Springer Lecture Notes in Computer Science, to appear.
A.K. Chandra and D. Harel, "Computable Queries for Relational Databases", Journal of Computer and System Sciences 21 (1980), 156–178.
A.K. Chandra and D. Harel, "Structure and Complexity of Relational Queries", Journal of Computer and System Sciences 25 (1982), 99–128.
C.C. Chang and H.J. Keisler, "Model Theory", North-Holland, 1977.
A.K. Chandra, D.C. Kozen and L.J. Stockmeyer, "Alternation", Journal of the ACM 28 (1981), 114–133.
K. Compton, A private communication.
W. Craig, "Three Uses of the Herbrand-Gentzen Theorem in Relating Model Theory and Proof Theory", J. Symbolic Logic 22 (1957), 250–268.
A. Ehrenfeucht, "An Application of Games to the Completeness Problem for Formalized Theories", Fund. Math. 49 (1961), 129–141.
R. Fagin, "Generalized First-Order Spectra and Polynomial-Time Recognizable Sets", in "Complexity of Computation" (ed. R. Karp), SIAM-AMS Proc. 7 (1974), 43–73.
R. Fagin, "Monadic generalized spectra", Zeitschr. f. math. Logik und Grundlagen d. Math. 21, 1975, 89–96.
H. Friedman, "The Complexity of Explicit Definitions", Advances in Mathematics 20 (1976), 18–29.
S. Feferman and R.L. Vaught, "The First-Order Properties of Algebraic Systems", Fund. Math. 47 (1959), 57–103.
M.R. Garey and D.S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-completeness", Freeman, 1979.
Yu. V. Glebskii, D.I. Kogan, M.I. Liogon'kii and V.A. Talanov, "Range and Degree of Relizability of Formulas in the Restricted Predicate Calculus", Cybernetics 5:2 (1969), 17–28; translation (1972), 142–154.
Y. Gurevich and H.R. Lewis, "A Logic for Constant-Depth Circults", Manuscript, 1983.
H. Gaifman and M.Y. Vardi, "A Simple Proof that Connectivity is not First-Order", Manuscript, 1983.
W. Goldfarb, "The Gödel Class with Identity is Unsolvable", Journal of Symbolic Logic, to appear.
Y. Gurevich, "Existential Interpretation", Algebra and Logic, 4:4 (1965), 71–85 (or "Existential Interpretation II", Archiv Math. Logik 22 (1982), 103–120).
Y. Gurevich, "Recognizing Satisfiability of Predicate Formulas", Algebra and Logic 5:2 (1966), 25–55; "The Decision Problem for Logic of Predicates and Operations", Algebra and Logic 8 (1969), pages 160–174 of English translation; "The Decision Problem for Standard Classes", Journal of Symbolic Logic 41 (1976), 460–464.
Y. Gurevich, "Algebras of Feasible Functions", Proc. of 24th IEEE Symposium on Foundations of Computer Science, 1983, 210–214.
J.E. Hopcroft and J.D. Ullman, "Introduction to Automata Theory, Languages and Computation", Addison-Wesley, 1979.
N. Immerman, "Relational Queries Computable in Polynomial Time", Proc. of 14th ACM Symposium on Theory of Computing, 1982, 147–152.
N. Immerman, "Languages which Capture Complexity Classes", Proc. of 15th ACM Symposium on Theory of Computing, 1983, 347–354.
N.G. Jones and A.L. Selman, "Turing Machines and the Spectra of First-Order Formulas", Journal of Symbolic Logic 39 (1974), 139–150.
V.F. Kostyrko, "To the Decision Problem for Predicate Logic", Ph.D. Thesis, Kiev, 1965 (Russian).
G. Kreisel, Technical Report No. 3, Applied Mathematics and Statistics Labs., Stanford University, January 1961.
A.B. Livchak, "The Relational Model for Systems of Automatic Testing", Automatic Documentation and Mathematical Linguistics 4 (1982), pages 17–19 of Russian original.
A.B. Livchak, "The Relational Model for Process Control", Automatic Documentation and Mathematical Linguistics 4 (1983), pages 27–29 of Russian original.
P. Lindström, "On Extensions of Elementary Logic", Theoria 35 (1969), 1–11.
D. Mundici, "Complexity of Craig's Interpolation", Annales Societ. Math. Polonae, Series IV: Fundamenta Informaticae, vol. 3–4 (1982); "NP and Craig's Interpolation Theorem", Proc. of Florence Logic Colloquium 1982, North Holland, to appear.
V. Sazonov, "Polynomial Computability and Recursivity in Finite Domains", Elektronische Informationsverarbeitung and Kybernetik 16 (1980), 319–323.
L.J. Stockmeyer, "The Polynomial — Time Hierarchy", Theoretical Computer Science 3 (1977), 1–22.
W.W. Tait, "A Counterexample to a Conjecture of Scott and Suppes", Journal of Symbolic Logic 24 (1959), 15–16.
A. Tarski, "A Lattice-Theoretical Fixpoint Theorem and its Application", Pasific Journal of Mathematics 5 (1955), 285–309.
B.A. Trakhtenbrot, "Impossibility of an Algorithm for the Decision Problem on Finite Classes", Doklady 70 (1950), 569–572.
J.D. Ullman, "Principles of Database Systems", Computer Science Press, 1982.
L.G. Valiant, "Relative Complexity of Checking and Evaluating", Information Processing 5 (1976), 20–23.
M.Y. Vardi, "Complexity of Relational Query Languages", Proc. of 14th ACM Symposium on Theory of Computing, 1982, 137–146.
M.M. Zloof, "Query-by-Example: a Database Languages`, IBM Syst. Journal 16 (1977), 324–343.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Gurevich, Y. (1984). Toward logic tailored for computational complexity. In: Börger, E., Oberschelp, W., Richter, M.M., Schinzel, B., Thomas, W. (eds) Computation and Proof Theory. Lecture Notes in Mathematics, vol 1104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099486
Download citation
DOI: https://doi.org/10.1007/BFb0099486
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13901-0
Online ISBN: 978-3-540-39119-7
eBook Packages: Springer Book Archive