A Schemata Calculus for Propositional Logic

Abstract

International audienc

Linear Temporal Logic and Propositional Schemata, Back and Forth

“…Logic [22] with the logic of propositional schemata introduced in [1]. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other.…”

“…It is widely used to reason on finite state transition systems. On the other hand, propositional schemata have been introduced in [1]. They extend the language of propositional logic with indexed propositions (such as p n , p 1 or p i+1 ) and iterated connectives of the form n i=0 φ or n i=0 φ.…”

“…Note that n denotes a parameter, interpreted as a natural number. If arbitrary expressions for indices and iterations are allowed in the schema, then the satisfiability problem is undecidable, but we have identified in [1], [2], [4] some subclasses for which this problem is decidable. The simplest of these classes is called regular: it is defined by restricting both the indices of the propositions, that must be of the form k or n+k where k ∈ Z and n is a variable, and iterations, that must be non-nested and of the form n+l i=k φ where n is a variable and k, l ∈ Z.…”

“…The simplest of these classes is called regular: it is defined by restricting both the indices of the propositions, that must be of the form k or n+k where k ∈ Z and n is a variable, and iterations, that must be non-nested and of the form n+l i=k φ where n is a variable and k, l ∈ Z. Decision procedures are designed in [1], [2] and an implementation is available [3].…”

Simplified handling of iterated term schemata

“…, n] (see Examples 2 and 3). This feature is particularly important in the context of formula schematisation [3]. For instance, it allows to express a clause of the form n i=1 P i or a set of clauses of the form {P i ∨ Q i+1 | i ∈ [1..n − 1]}, which is impossible using less expressive formalisms [2].…”