Automatic Generation of Invariants for Circular Derivations in SUP(LA)

Abstract

The hierarchic combination of linear arithmetic and first-order logic with free function symbols, FOL(LA), results in a strictly more expressive logic than its two parts. The SUP(LA) calculus can be turned into a decision procedure for interesting fragments of FOL(LA). For example, reachability problems for timed automata can be decided by SUP(LA) using an appropriate translation into FOL(LA). In this paper, we extend the SUP(LA) calculus with an additional inference rule, automatically generating inductive invariants from partial SUP(LA) derivations. The rule enables decidability of more expressive fragments, including reachability for timed automata with unbounded integer variables. We have implemented the rule in the SPASS(LA) theorem prover with promising results, showing that it can considerably speed up proof search and enable termination of saturation for practically relevant problems.

Symbol Elimination for Parametric Second-Order Entailment Problems (with Applications to Problems in Wireless Network Theory)

“…for fragments of theories of arrays with read and write in the presence of iterators and selectors in [4]. Similar ideas are used in the superposition calculus in [16,27], and in approaches which combine superposition and induction [31] or use solutions for recurrences in loop invariant generation [33,32]. We plan to analyze such aspects in future work.…”

“…The main challenge when using saturation approaches for symbol elimination is the fact that the saturated sets might be infinite. Sometimes finite representations of possibly infinite sets of clauses exist: for this, Horbach and Weidenbach introduced a melting calculus [27], later used in [25,26] and [16]. Similar aspects were explored in the study of acceleration for program verification modulo Presburger arithmetic by Boigelot, Finkel and Leroux [14,17], in relationship with array systems by [4], or in the study of constrained Horn clauses (cf.…”