The B-book: assigning programs to meanings
The B-book: assigning programs to meanings
Simplification by Cooperating Decision Procedures
ACM Transactions on Programming Languages and Systems (TOPLAS)
Unions of non-disjoint theories and combinations of satisfiability procedures
Theoretical Computer Science
IJCAR '01 Proceedings of the First International Joint Conference on Automated Reasoning
Spectra of Monadic Second-Order Formulas with One Unary Function
LICS '03 Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science
AI Communications
Combining Nonstably Infinite Theories
Journal of Automated Reasoning
The design and implementation of VAMPIRE
AI Communications - CASC
Combining Non-stably Infinite, Non-first Order Theories
Electronic Notes in Theoretical Computer Science (ENTCS)
Decidability of invariant validation for paramaterized systems
TACAS'03 Proceedings of the 9th international conference on Tools and algorithms for the construction and analysis of systems
Combining theories with shared set operations
FroCoS'09 Proceedings of the 7th international conference on Frontiers of combining systems
Combining theories with shared set operations
FroCoS'09 Proceedings of the 7th international conference on Frontiers of combining systems
On theorem proving for program checking: historical perspective and recent developments
Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming
On Deciding Satisfiability by Theorem Proving with Speculative Inferences
Journal of Automated Reasoning
Combining theories: the Ackerman and guarded fragments
FroCoS'11 Proceedings of the 8th international conference on Frontiers of combining systems
Automatic verification for a class of proof obligations with SMT-solvers
ABZ'10 Proceedings of the Second international conference on Abstract State Machines, Alloy, B and Z
Combination of disjoint theories: beyond decidability
IJCAR'12 Proceedings of the 6th international joint conference on Automated Reasoning
Integration of SMT-solvers in B and Event-B development environments
Science of Computer Programming
Instantiation Schemes for Nested Theories
ACM Transactions on Computational Logic (TOCL)
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The design of decision procedures for first-order theories and their combinations has been a very active research subject for thirty years; it has gained practical importance through the development of SMT (satisfiability modulo theories) solvers. Most results concentrate on combining decision procedures for data structures such as theories for arrays, bitvectors, fragments of arithmetic, and uninterpreted functions. In particular, the well-known Nelson-Oppen scheme for the combination of decision procedures requires the signatures to be disjoint and each theory to be stably infinite; every satisfiable set of literals in a stably infinite theory has an infinite model. In this paper we consider some of the best-known decidable fragments of first-order logic with equality, including the Löwenheim class (monadic FOL with equality, but without functions), Bernays-Schönfinkel-Ramsey theories (finite sets of formulas of the form ∃*¬* φ, where φ is a function-free and quantifier-free FOL formula), and the two-variable fragment of FOL. In general, these are not stably infinite, and the Nelson-Oppen scheme cannot be used to integrate them into SMT solvers. Noticing some elementary results about the cardinalities of the models of these theories, we show that they can nevertheless be combined with almost any other decidable theory.