Lifting congruence closure with free variables to λ-free higher-order logic via SAT encoding

International audienceRecent work in extending SMT solvers to higher-order logic (HOL) has not explored lifting quantifier instantiation algorithms to perform higher-order unification. As a consequence, widely used instantiation techniques, such as trigger-and particularly conflictbased, can only be applied in a limited manner. Congruence closure with free variables (CCFV) is a decision procedure for the E-ground (dis-)unification problem, which is at the heart of these instantiation techniques. Here, as a first step towards fully supporting trigger-and conflict-based instantiation in HOL, we define the E-ground (dis-)unification problem in λ-free higher-order logic (λfHOL), an extension of first-order logic where function symbols may be partially applied and functional variables may occur, and extend CCFV to solve it. To improve scalability in the context of handling higher-order variables, we rely on an encoding of the CCFV search as a propositional formula. We present a solution reconstruction procedure so that models for the propositional formula lead to solutions for the E-ground (dis-)unification problem. This is instrumental to port triggerand conflict-based instantiation to be fully applied in λfHOL. * The order of authors is inverse alphabetic

Automatic instantiation of abstract tests on specific configurations for large critical control systems

Computer-based control systems have grown in size, complexity, distribution and criticality. In this paper a methodology is presented to perform an abstract testing of such large control systems in an efficient way: an abstract test is specified directly from system functional requirements and has to be instantiated in more test runs to cover a specific configuration, comprising any number of control entities (sensors, actuators and logic processes). Such a process is usually performed by hand for each installation of the control system, requiring a considerable time effort and being an error prone verification activity. To automate a safe passage from abstract tests, related to the so called generic software application, to any specific installation, an algorithm is provided, starting from a reference architecture and a state-based behavioural model of the control software. The presented approach has been applied to a railway interlocking system, demonstrating its feasibility and effectiveness in several years of testing experience

Equivalence of two Fixed-Point Semantics for Definitional Higher-Order Logic Programs

Two distinct research approaches have been proposed for assigning a purely extensional semantics to higher-order logic programming. The former approach uses classical domain theoretic tools while the latter builds on a fixed-point construction defined on a syntactic instantiation of the source program. The relationships between these two approaches had not been investigated until now. In this paper we demonstrate that for a very broad class of programs, namely the class of definitional programs introduced by W. W. Wadge, the two approaches coincide (with respect to ground atoms that involve symbols of the program). On the other hand, we argue that if existential higher-order variables are allowed to appear in the bodies of program rules, the two approaches are in general different. The results of the paper contribute to a better understanding of the semantics of higher-order logic programming.Comment: In Proceedings FICS 2015, arXiv:1509.0282

Second-Order Functions and Theorems in ACL2

SOFT ('Second-Order Functions and Theorems') is a tool to mimic second-order functions and theorems in the first-order logic of ACL2. Second-order functions are mimicked by first-order functions that reference explicitly designated uninterpreted functions that mimic function variables. First-order theorems over these second-order functions mimic second-order theorems universally quantified over function variables. Instances of second-order functions and theorems are systematically generated by replacing function variables with functions. SOFT can be used to carry out program refinement inside ACL2, by constructing a sequence of increasingly stronger second-order predicates over one or more target functions: the sequence starts with a predicate that specifies requirements for the target functions, and ends with a predicate that provides executable definitions for the target functions.Comment: In Proceedings ACL2 2015, arXiv:1509.0552

Language and Proofs for Higher-Order SMT (Work in Progress)

Satisfiability modulo theories (SMT) solvers have throughout the years been able to cope with increasingly expressive formulas, from ground logics to full first-order logic modulo theories. Nevertheless, higher-order logic within SMT is still little explored. One main goal of the Matryoshka project, which started in March 2017, is to extend the reasoning capabilities of SMT solvers and other automatic provers beyond first-order logic. In this preliminary report, we report on an extension of the SMT-LIB language, the standard input format of SMT solvers, to handle higher-order constructs. We also discuss how to augment the proof format of the SMT solver veriT to accommodate these new constructs and the solving techniques they require.Comment: In Proceedings PxTP 2017, arXiv:1712.0089