Skip to main content
×
Home
    • Aa
    • Aa

Some Aspects of Model Theory and Finite Structures

  • Eric Rosen (a1)
Abstract

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory (see [26, 46]). In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.

It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems (see [35, 26]). The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.

From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] M. Ajtai and Y. Gurevich , Monotone versus positive, Journal of the ACM, vol. 34 (1987), pp. 10041015.

[2] M. Ajtai and Y. Gurevich , Datalog vs first-order logic, Journal of Computer and System Sciences, vol. 49 (1994), pp. 562–58.

[3] N. Alechina and Y. Gurevich , Syntax vs. semantics on finite structures, Structures in logic and computer science. A selection of essays in honor of A. Ehrenfeucht ( J. Mycielski , G. Rozenberg , and A. Salomaa , editors), LNCS, vol. 1261, Springer-Verlag, 1997, pp. 1433.

[6] J. T. Baldwin and O. Lessmann , Amalgamation properties and finite models in Ln-theories, Archive for Mathematical Logic, vol. 41 (2002), pp. 155167.

[7] J. T. Baldwin and S. Shelah , Randomness and semigenericity, Transactions of the American Mathematical Society, vol. 349 (1997), pp. 13591376.

[8] J. T. Baldwin and N. Shi , Stable generic structures, Annals of Pure and Applied Logic, vol. 79 (1996), pp. 135.

[12] E. Börger , E. Grädel , and Y. Gurevich , The classical decision problem, Springer-Verlag, 1997.

[13] C. C. Chang and H. J. Keisler , Model theory, 3rd ed., North-Holland, 1990.

[14] Z. Chatzidakis , Model theory of finite fields and pseudo-finite fields, Annals of Pure and Applied Logic, vol. 88 (1997), pp. 95108.

[17] G. Cherlin , Large finite structures with few types, Algebraic model theory ( B. T. Hart , A. H. Lachlan , and M. A. Valeriote , editors), Kluwer Academic Publishers, 1997, pp. 53105.

[19] G. Cherlin and U. Felgner , Homogeneous finite groups, Journal of the London Mathematical Society (2), vol. 62 (2000), pp. 784794.

[20] G. Cherlin , L. Harrington , and A. H. Lachian , 0-categorical, ℵ0-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 103135.

[23] A. Dawar , S. Lindell , and S. Weinstein , Infnitary logic and inductive defnability over fnite structures, Information and Computation, vol. 119 (1994), pp. 160175.

[26] H.-D. Ebbinghaus and J. Flum , Finite model theory, Springer-Verlag, 1995.

[27] R. Fagin , Probabilities on fnite models, The Journal of Symbolic Logic, vol. 41 (1976), pp. 5058.

[29] A. Gardiner , Homogeneous graphs, Journal of Combinatorial Theory. Series B., vol. 20 (1976), pp. 94102.

[31] E. Grädel and G. McColm , Hierarchies in transitive closure logic, stratifed Datalog and infnitary logic, Annals of Pure and Applied Logic, vol. 77 (1996), pp. 166199.

[32] E. Grädel and E. Rosen , On preservation theorems for two-variable logic, Mathematical Logic Quarterly, vol. 45 (1999), pp. 315325.

[33] M. Grohe , Existential least fxed-point logic and its relatives, Journal of Logic and Computation, vol. 7 (1997), pp. 205228.

[35] Y. Gurevich , Toward logic tailored for computational complexity, Computation and proof theory ( M. M. Richter et al., editors), Lecture Notes in Mathematics, vol. 1104, Springer-Verlag, 1984, pp. 175216.

[37] B. Herwig , Extending partial isomorphisms for the small index property of many co-categorical structures, Israel Journal of Mathematics, vol. 107 (1997), pp. 93123.

[38] B. Herwig and D. Lascar , Extending partial isomorphisms and the profnite topology on free groups, Transactions of the American Mathematical Society, vol. 352 (2000), pp. 19852021.

[39] W. Hodges , Finite extensions and fnite groups, Models and sets (Aachen, 1983) ( G. H. Müller and M. M. Richter , editors), Lecture Notes in Mathematics, vol. 1103, Springer-Verlag, 1984, pp. 193206.

[40] W. Hodges , Model theory, Cambridge University Press, 1993.

[42] E. Hrushovski , Extending partial isomorphisms of graphs, Combinatorica, vol. 12 (1992), pp. 411416.

[45] T. Hyttinen , On stability in finite models, Archive for Mathematical Logic, vol. 39 (2000), pp. 89102.

[47] N. Immerman and D. Kozen , Definability with bounded number of bound variables, Information and Computation, vol. 83 (1989), pp. 121139.

[52] A. H. Lachlan , Stable finitely homogeneous structures a survey, Algebraic model theory ( B. T. Hart , A. H. Lachlan , and M. A. Valeriote , editors), Kluwer Academic Publishers, 1997, pp. 145159.

[53] A. H. Lachlan and A. Tripp , Finite homogeneous 3-graphs, Mathematical Logic Quarterly, vol. 41 (1995), pp. 287306.

[55] R. C. Lyndon , An interpolation theorem in the predicate calculus, Pacific Journal of Mathematics, vol. 9 (1959), pp. 129142.

[56] R. C. Lyndon , Properties preserved under homomorphism, Pacific Journal of Mathematics, vol. 9 (1959), pp. 143154.

[57] D. Macpherson , Finite axiomatizability and theories with trivial algebraic closure, Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 188192.

[59] A. Robinson , Introduction to model theory and to the metamathematics of algebra, North-Holland, 1963.

[61] E. Rosen , Modal logic over finite structures, Journal of Logic, Language and Information, vol. 6 (1997), pp. 427439.

[64] E. Rosen and S. Weinstein , Preservation theorems in finite model theory, Logic and computational complexity ( D. Leivant , editor), LNCS, vol. 960, Springer-Verlag, 1995, pp. 480502.

[65] D. Saracino and C. Wood , Homogeneous finite rings in characteristic 2n, Annals of Pure and Applied Logic, vol. 40 (1988), pp. 1128.

[66] S. Shelah and J. Spencer , Zero-one laws for sparse random graphs, Journal of the American Mathematical Society, vol. 1 (1988), pp. 97115.

[68] A. Stolboushkin , Finite monotone properties, Proceedings of 10th IEEE Symposium on Logic in Computer Science, 1995, pp. 324330.

[70] S. Thomas , Theories with finitely many models, The Journal of Symbolic Logic, vol. 51 (1986), pp. 374376.

[72] J. S. Wilson , On simple pseudofinite groups, Journal of the London Mathematical Society (2), vol. 51 (1995), pp. 471490.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 51 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd September 2017. This data will be updated every 24 hours.