A Framework for the Flexible Integration of a Class of Decision Procedures into Theorem Provers

The role of decision procedures is often essential in theorem proving. Decision procedures can reduce the search space of heuristic components of a prover and increase its abilities. However, in some applications only a small number of conjectures fall within the scope of the available decision procedures. Some of these conjectures could in an informal sense fall ‘just outside’ that scope. In these situations a problem arises because lemmas have to be invoked or the decision procedure has to communicate with the heuristic component of a theorem prover. This problem is also related to the general problem of how to exibly integrate decision procedures into heuristic theorem provers. In this paper we address such problems and describe a framework for the exible integration of decision procedures into other proof methods. The proposed framework can be used in different theorem provers, for different theories and for different decision procedures. New decision procedures can be simply ‘plugged-in’ to the system. As an illustration, we describe an instantiation of this framework within the Clam proof-planning system, to which it is well suited. We report on some results using this implementation

Existential and Positive Theories of Equations in Graph Products

We prove that the existential theory of equations with normalized rational constraints in a fixed graph product of finite monoids, free monoids, and free groups is PSPACE-complete. Under certain restrictions this result also holds if the graph product is part of the input. As the second main result we prove that the positive theory of equations with recognizable constraints in graph products of finite and free groups is decidable

Appendix B: Sato–Tate theorem for families and low-lying zeros of automorphic L-functions

International audienc