Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories

Abstract

In program analysis, the synthesis of models of logical theories representing the program semantics is often useful to prove program properties. We use Order-Sorted First-Order Logic as an appropriate framework to describe the semantics and properties of programs as given theories. Then we investigate the automatic synthesis of models for such theories. We use convex polytopic domains as a flexible approach to associate different domains to different sorts. We introduce a framework for the piecewise definition of functions and predicates. We develop its use with linear expressions (in a wide sense, including linear transformations represented as matrices) and inequalities to specify functions and predicates. In this way, algorithms and tools from linear algebra and arithmetic constraint solving (e.g., SMT) can be used as a backend for an efficient implementation.

Multi-Dimensional Interpretations for Termination of Term Rewriting

“…In order to prove the finiteness of a given DP problem, we need a weakly monotone F -algebra for the signature F indicated by this problem, rather than for a predefined signature like N *+max . We fill the gap by employing the notion of derivers [24,33] to derive an F -algebra from one of another signature G. Definition 4 (deriver). An F /G-deriver is a pair of a sort δ ∈ S and a map- Now we show that an F /G-deriver yields a weakly monotone F -algebra if the base G-algebra is known to be weakly monotone.…”

“…The first step to the main contribution is the use of derivers [24,33], which allow us to abstract away the mathematical details of polynomials or maxpolynomials. We will obtain a key soundness result that derivers derive monotone interpretations from monotone interpretations (Section 4).…”

Dietary Composition Perception Algorithm Using Social Robot Audition for Mandarin Chinese

“…According to the above definitions and analysis, using the first-order predicate logic theory [44], the predicate logic reasoning formula (1) can be used to determine whether the eating action occurs or not.…”

Use of logical models for proving infeasibility in term rewriting

“…In order to guarantee that A in Theorem 6 has a nonempty domain, we just add (∃x) x = x to R ∪ {¬ϕ} (this is not necessary if T (F) = ∅). Structures A can be automatically generated from R and sentence(s) ¬(∃ x)( n i=1 s i i t i ) by using tools like AGES [5], implementing the methods in [7]. Essentially, AGES interprets the sort, function and predicate symbols of an input (many-sorted) first-order theory as parametric domains, functions and predicates.…”

“…The research in this paper was first presented in [8] (a restricted, nonsystematic use in proofs of operational termination of CTRSs is sketched in [7,Section 11.1]) and then settled by the first author in a first-order logic framework in [6]. In this paper we have extended the treatment of [8] to more general properties of rewrite systems (e.g., reducibility or root-stability, see Section 4.3).…”