Superposition Modulo Linear Arithmetic SUP(LA)

Abstract

Abstract. The hierarchical superposition based theorem proving calculus of Bachmair, Ganzinger, and Waldmann enables the hierarchic combination of a theory with full first-order logic. If a clause set of the combination enjoys a sufficient completeness criterion, the calculus is even complete. We instantiate and refine the calculus for the theory of linear arithmetic. In particular, we develop new effective versions for the standard superposition redundancy criteria taking the linear arithmetic theory into account. The resulting calculus is implemented in SPASS(LA) and extends the state of the art in proving properties of first-order formulas over linear arithmetic.

Model Evolution with Equality Modulo Built-in Theories

“…We assume, again with no loss of generality, that |B| is at most countably infinite and all of its elements are included in Σ as B-constant symbols. 1 Our running example for T will be the theory of linear integer arithmetic (LIA). For that example, Σ B 's operators are ≤, + and all the integer constants, all with the expected arities, T is the structure of the integer numbers with those operators, and |B| = {0, ±1, ±2, .…”

“…If L is a negative equation t 1 [s] ≈ t 2 , the Para rule could be improved by requiring that t 2 σ t 1 σ. That is, paramodulation into smaller sides of negative equations is not necessary.…”

“…Older theory reasoning techniques developed within first-order theorem proving are often impractical as they require the enumeration of complete sets of theory unifiers (in particular those in the tradition of Stickel's Theory Resolution [14]) or feature only weak or no redundancy criteria (e.g., Bürckert's Constraint Resolution [7]). Developing sophisticated automated reasoning systems that are also able to deal with quantified formulas has recently been an active area of research [8,10,13,4,1]. We contribute to this line of research and propose a novel instantiation-based method for a large fragment of first-order logic with equality modulo a given complete background theory, such as linear integer arithmetic.…”

“…Yet, we think ME E (T) is relevant through its combination of powerful techniques for first-order equational logic with equality, based on an adaptation of the Bachmair-Ganzinger theory of superposition, with a black-box theory reasoner, which greatly enhances its versatility. ME E (T) is thus similarly positioned as the hierarchic superposition calculus [1,3].…”

Rewriting modulo SMT and open system analysis

“…A key open research issue of current symbolic techniques is extensibility. Techniques that combine different methods have been proposed, e.g., decision procedures [50,51], unifications algorithms [7,11], theorem provers with decision procedures [1,10,53], and SMT solvers in model checkers [3,30,49,62,66]. However, there is still a lack of general extensibility techniques for symbolic analysis that simultaneously combine the power of SMT solving, rewriting-and narrowingbased analysis, and model checking.…”