On the Finite Model Property in Order-Sorted Logic

The Schoenfinkel-Bernays-Ramsey class is a fragment of first-order logic with the Finite Model Property: a sentence in this class is satisfiable if and only if it is satisfied in a finite model. Since an upper bound on the size of such a model is computable from the sentence, the satisfiability problem for this family is decidable. Sentences in this form arise naturally in a variety of application areas, and several popular reasoning tools explicitly target this class. Others have observed that the class of sentences for which such a finite model theorem holds is richer in a many-sorted framework than in the one-sorted case. This paper makes a systematic study of this phenomenon in the general setting of order-sorted logic supporting overloading and empty sorts. We establish a syntactic condition generalizing the Schoenfinkel-Bernays-Ramsey form that ensures the Finite Model Property. We give a linear-time algorithm for deciding this condition and a polynomial-time algorithm for computing the bound on model sizes. As a consequence, model-finding is a complete decision procedure for sentences in this class. Our algorithms have been incorporated into Margrave, a tool for analysis of access-control and firewall policies, and are available in a standalone application suitable for analyzing input to the Alloy model finder.

[1]  A. Wilkie.  THE CLASSICAL DECISION PROBLEM (Perspectives in Mathematical Logic) By Egon Börger, Erich Grädel and Yuri Gurevich: 482 pp., DM.158.–, ISBN 3 540 57073 X (Springer, 1997). , 1998 .

[2]  John N. Hooker,et al.  Partial Instantiation Methods for Inference in First-Order Logic , 2002, Journal of Automated Reasoning.

[3]  Anna Philippou,et al.  Tools and Algorithms for the Construction and Analysis of Systems , 2018, Lecture Notes in Computer Science.

[4]  Nikolaj Bjørner,et al.  Z3: An Efficient SMT Solver , 2008, TACAS.

[5]  Harry R. Lewis,et al.  Complexity Results for Classes of Quantificational Formulas , 1980, J. Comput. Syst. Sci..

[6]  Jacques Herbrand.  Recherches sur la théorie de la démonstration , 1930 .

[7]  Michael Carl Tschantz,et al.  Verification and change-impact analysis of access-control policies , 2005, Proceedings. 27th International Conference on Software Engineering, 2005. ICSE 2005..

[8]  P. Bernays,et al.  Zum Entscheidungsproblem der mathematischen Logik , 1928 .

[9]  Alexander Moshe Rabinovich,et al.  Decidable fragments of many-sorted logic , 2010, J. Symb. Comput..

[10]  Herbert B. Enderton,et al.  A mathematical introduction to logic , 1972 .

[11]  K. Claessen,et al.  New Techniques that Improve MACE-style Finite Model Finding , 2007 .

[12]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[13]  Daniel Jackson,et al.  Software Abstractions - Logic, Language, and Analysis , 2006 .

[14]  Arnold Oberschelp,et al.  Order Sorted Predicate Logic , 1990, Sorts and Types in Artificial Intelligence.

[15]  Joseph A. Goguen,et al.  An Oxford survey of order sorted algebra , 1994, Mathematical Structures in Computer Science.

[16]  Ruzica Piskac,et al.  Deciding Effectively Propositional Logic Using DPLL and Substitution Sets , 2010, Journal of Automated Reasoning.

[17]  Pascal Fontaine,et al.  Decidability of Invariant Validation for Paramaterized Systems , 2003, TACAS.

[18]  Christoph Walther,et al.  Many-sorted unification , 1988, JACM.

[19]  Sanjit A. Seshia,et al.  The UCLID Decision Procedure , 2004, CAV.

[20]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[21]  L. D. Moura,et al.  Deciding Effectively Propositional Logic with Equality , 2008 .

[22]  Maurice Nivat,et al.  Resolution of Equations in Algebraic Structures , 1989 .

[23]  J. Harrison.  Exploiting sorts in expansion-based proof procedures , 2007 .

[24]  Trudy Weibel,et al.  An Order-Sorted Resolution in Theory and Practice , 1997, Theor. Comput. Sci..

[25]  Robert G. Jeroslow,et al.  Computation-oriented reductions of predicate to propositional logic , 1988, Decis. Support Syst..

[26]  Lee Momtahan.  Towards a Small Model Theorem for Data Independent Systems in Alloy , 2005, Electron. Notes Theor. Comput. Sci..

[27]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[28]  J. Goguen,et al.  Order-Sorted Equational Computation , 1989 .

[29]  José Meseguer,et al.  Order-Sorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations , 1992, Theor. Comput. Sci..

[30]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[31]  Leonardo Mendonça de Moura,et al.  Complete Instantiation for Quantified Formulas in Satisfiabiliby Modulo Theories , 2009, CAV.