Combination of Uniform Interpolants via Beth Definability

Abstract

Uniform interpolants were largely studied in non-classical propositional logics since the nineties, and their connection to model completeness was pointed out in the literature. A successive parallel research line inside the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. In this paper, we investigate cover transfer to theory combinations in the disjoint signatures case. We prove that, for convex theories, cover algorithms can be transferred to theory combinations under the same hypothesis needed to transfer quantifier-free interpolation (i.e., the equality interpolating property, aka strong amalgamation property). The key feature of our algorithm relies on the extensive usage of the Beth definability property for primitive fragments to convert implicitly defined variables into their explicitly defining terms. In the non-convex case, we show by a counterexample that covers may not exist in the combined theories, even in case combined quantifier-free interpolants do exist. However, we exhibit a cover transfer algorithm operating also in the non-convex case for special kinds of theory combinations; these combinations (called ‘tame combinations’) concern multi-sorted theories arising in many model-checking applications (in particular, the ones oriented to verification of data-aware processes).

Linear-Time Verification of Data-Aware Processes Modulo Theories via Covers and Automata

“…The tool interfaces CVC5 (Deters et al 2014) for SMT checks, and QE in LIRA. On top of that, we implemented the cover computation procedure for EUF with unary functions and arbitrary relations from (Calvanese et al 2021, p. 961), and the TameCombCover routine (Calvanese et al 2022) to compute covers in tame combinations of EUF and LIRA. LINDMT is a command-line tool written in Python, but it is also accessible via a web interface (https://lindmt.unibz.it); sources and benchmarks are available as well.…”

“…We will sometimes refer to common SMT theories (Barrett et al 2021): the theory of equality and uninterpreted functions (EUF) for a given Σ, and linear arithmetic over rationals (LRA), integers (LIA), or both (LIRA). While the arithmetic theories have QE (Presburger 1929), EUF admits model completion and so do certain tame combinations of LIRA and EUF (Calvanese et al 2020a(Calvanese et al , 2022. We denote the sorts of rationals and integers by rat and int.…”

“…While in purely numeric settings this idea has been explored by relying on quantifier elimination (Felli, Montali, and Winkler 2022), the same approach cannot be lifted to theories formalizing data structures and relational databases, as quantifiers cannot be eliminated therein. We deal with this essential technical difficulty by combining automata-theoretic and cover computation (Calvanese et al 2021(Calvanese et al , 2022Gulwani and Musuvathi 2008) techniques to respectively deal with the temporal and datatype dimensions. More specifically, this paper makes the following four contributions: (1) We devise a model checking procedure for DMTs w.r.t.…”

Linear Temporal Logic Modulo Theories over Finite Traces

“…Constrained Horn clauses (Gurfinkel and Bjørner 2019) are also promising for quantifier-free first-order automata, while for quantified ones, quantifier elimination (Calvanese et al 2022;Calvanese et al 2021), also of second-order predicates (Gabbay, Schmidt, and Szalas 2008), has to be used.…”

Safety Verification and Universal Invariants for Relational Action Bases

“…This data-awareness is essential to capture relevant systems in AI [Baral and De Giacomo, 2015] and business process management [Vianu, 2009;. Notable examples are: (i) Situation Calculus [De Giacomo et al, 2016] and knowledge-based [Bagheri Hariri et al, 2013b] action theories, (ii) relational MDPs [Yang et al, 2022], (iii) data-/artifact-centric business processes [Bagheri Hariri et al, 2013a;Calvanese et al, 2019], and (iv) Petri nets with identifiers [Polyvyanyy et al, 2019;Ghilardi et al, 2022a] or with advanced interaction mechanisms [Fahland, 2019]. The subtle interplay between, data and processes calls for dedicated techniques for verifying at design-time whether such relational dynamic systems behave as expected [Baral and De Giacomo, 2015;.…”