{SCL(EQ)}: {SCL} for First-Order Logic with Equality

International audienceAbstract We propose a new calculus SCL(EQ) for first-order logic with equality that only learns non-redundant clauses. Following the idea of CDCL (Conflict Driven Clause Learning) and SCL (Clause Learning from Simple Models) a ground literal model assumption is used to guide inferences that are then guaranteed to be non-redundant. Redundancy is defined with respect to a dynamically changing ordering derived from the ground literal model assumption. We prove SCL(EQ) sound and complete and provide examples where our calculus improves on superposition

SCL(EQ): SCL for First-Order Logic with Equality

We propose a new calculus SCL(EQ) for first-order logic with equality thatonly learns non-redundant clauses. Following the idea of CDCL (Conflict DrivenClause Learning) and SCL (Clause Learning from Simple Models) a ground literalmodel assumption is used to guide inferences that are then guaranteed to benon-redundant. Redundancy is defined with respect to a dynamically changingordering derived from the ground literal model assumption. We prove SCL(EQ)sound and complete and provide examples where our calculus improves onsuperposition.<br

Decidability of the Monadic Shallow Linear First-Order Fragment with Straight Dismatching Constraints

The monadic shallow linear Horn fragment is well-known to be decidable and has many application, e.g., in security protocol analysis, tree automata, or abstraction refinement. It was a long standing open problem how to extend the fragment to the non-Horn case, preserving decidability, that would, e.g., enable to express non-determinism in protocols. We prove decidability of the non-Horn monadic shallow linear fragment via ordered resolution further extended with dismatching constraints and discuss some applications of the new decidable fragment.Comment: 29 pages, long version of CADE-26 pape

Automating Security Analysis: Symbolic Equivalence of Constraint Systems

We consider security properties of cryptographic protocols, that are either trace properties (such as confidentiality or authenticity) or equivalence properties (such as anonymity or strong secrecy). Infinite sets of possible traces are symbolically represented using deducibility constraints. We give a new algorithm that decides the trace equivalence for the traces that are represented using such constraints, in the case of signatures, symmetric and asymmetric encryptions. Our algorithm is implemented and performs well on typical benchmarks. This is the first implemented algorithm, deciding symbolic trace equivalence

On the Expressivity and Applicability of Model Representation Formalisms

A number of first-order calculi employ an explicit model representation formalism for automated reasoning and for detecting satisfiability. Many of these formalisms can represent infinite Herbrand models. The first-order fragment of monadic, shallow, linear, Horn (MSLH) clauses, is such a formalism used in the approximation refinement calculus. Our first result is a finite model property for MSLH clause sets. Therefore, MSLH clause sets cannot represent models of clause sets with inherently infinite models. Through a translation to tree automata, we further show that this limitation also applies to the linear fragments of implicit generalizations, which is the formalism used in the model-evolution calculus, to atoms with disequality constraints, the formalisms used in the non-redundant clause learning calculus (NRCL), and to atoms with membership constraints, a formalism used for example in decision procedures for algebraic data types. Although these formalisms cannot represent models of clause sets with inherently infinite models, through an additional approximation step they can. This is our second main result. For clause sets including the definition of an equivalence relation with the help of an additional, novel approximation, called reflexive relation splitting, the approximation refinement calculus can automatically show satisfiability through the MSLH clause set formalism.Comment: 15 page

A Sorted Datalog Hammer for Supervisor Verification Conditions Modulo Simple Linear Arithmetic

International audienceAbstract In a previous paper, we have shown that clause sets belonging to the Horn Bernays-Schönfinkel fragment over simple linear real arithmetic (HBS(SLR)) can be translated into HBS clause sets over a finite set of first-order constants. The translation preserves validity and satisfiability and it is still applicable if we extend our input with positive universally or existentially quantified verification conditions (conjectures). We call this translation a Datalog hammer. The combination of its implementation in SPASS-SPL with the Datalog reasoner VLog establishes an effective way of deciding verification conditions in the Horn fragment. We verify supervisor code for two examples: a lane change assistant in a car and an electronic control unit of a supercharged combustion engine. In this paper, we improve our Datalog hammer in several ways: we generalize it to mixed real-integer arithmetic and finite first-order sorts; we extend the class of acceptable inequalities beyond variable bounds and positively grounded inequalities; and we significantly reduce the size of the hammer output by a soft typing discipline. We call the result the sorted Datalog hammer. It not only allows us to handle more complex supervisor code and to model already considered supervisor code more concisely, but it also improves our performance on real world benchmark examples. Finally, we replace the before file-based interface between SPASS-SPL and VLog by a close coupling resulting in a single executable binary

A Sorted Datalog Hammer for Supervisor Verification Conditions Modulo Simple Linear Arithmetic

In a previous paper, we have shown that clause sets belonging to the HornBernays-Sch\"onfinkel fragment over simple linear real arithmetic (HBS(SLR))can be translated into HBS clause sets over a finite set of first-orderconstants. The translation preserves validity and satisfiability and it isstill applicable if we extend our input with positive universally orexistentially quantified verification conditions (conjectures). We call thistranslation a Datalog hammer. The combination of its implementation inSPASS-SPL with the Datalog reasoner VLog establishes an effective way ofdeciding verification conditions in the Horn fragment. We verify supervisorcode for two examples: a lane change assistant in a car and an electroniccontrol unit of a supercharged combustion engine. In this paper, we improve ourDatalog hammer in several ways: we generalize it to mixed real-integerarithmetic and finite first-order sorts; we extend the class of acceptableinequalities beyond variable bounds and positively grounded inequalities; andwe significantly reduce the size of the hammer output by a soft typingdiscipline. We call the result the sorted Datalog hammer. It not only allows usto handle more complex supervisor code and to model already consideredsupervisor code more concisely, but it also improves our performance on realworld benchmark examples. Finally, we replace the before file-based interfacebetween SPASS-SPL and VLog by a close coupling resulting in a single executablebinary.<br

Hierarchic Superposition Revisited

Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory