Schematic Refutations of Formula Schemata

Abstract

Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.

A Special Case of Schematic Syntactic Unification

“…Also left to future work is extending our results to full loop unification and developing an algorithm based on our sufficient conditions. This algorithm can then be integrated into the computational proof analysis method introduced in [7].…”

“…The automated reasoning methods for induction used in [5] are quite powerful in comparison to other existing methods [6], however, the overall method is quite weak proof analytically. This observation lead to the recent generalization of the proof analysis method introduced in [7]. This new framework introduces the formal definition of the schematic unification which motivated this work.…”

“…In this paper, we introduce a variation of the schematic unification problem discussed in [7] and an important special case when only one side of the problem is schematic, what we refer to as semiloop unification. This seems like a significant simplification, though the problem remains difficult as complex cyclic structures may occur during unification.…”

Higher-order pattern generalization modulo equational theories

“…Discovering such a class is essential for automated reasoning methods exploiting this type of unification. Integration with a resolution calculus, such as the one described in [15] would be step towards more expressive inductive theorem proving based on schematic formula representation. One can imagine this work playing a foundational role for a cyclic resolution calculus for inexpressive theories of arithmetic [10,25].…”

“…The required unification procedure should unify terms containing infinitely many variables that each occur finitely many times. This type of unification was first discussed in [14,15] and partially solved in [12]; a sufficient condition for unifiability was given. In this work we provide both a necessary and sufficient condition for unifiability of simple linear loops with a finite depth term checkable in finite time.…”