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

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

An Efficient Subsumption Test Pipeline for {BS(LRA)} Clauses

International audienceThe importance of subsumption testing for redundancy elimination in first-order logic automatic reasoning is well-known. Although the problem is already NP-complete for first-order clauses, the meanwhile developed test pipelines efficiently decide subsumption in almost all practical cases. We consider subsumption between first-oder clauses of the Bernays-Schönfinkel fragment over linear real arithmetic constraints: BS(LRA). The bottleneck in this setup is deciding implication between the LRA constraints of two clauses. Our new sample point heuristic preempts expensive implication decisions in about 94% of all cases in benchmarks. Combined with filtering techniques for the first-order BS part of clauses, it results again in an efficient subsumption test pipeline for BS(LRA) clauses

Symbolic Model Construction for Saturated Constrained Horn Clauses

Clause sets saturated by hierarchic ordered resolution do not offer a model representation that can be effectively queried, in general. They only offer the guarantee of the existence of a model. We present an effective symbolic model construction for saturated constrained Horn clauses. Constraints are in linear arithmetic, the first-order part is restricted to a function-free language. The model is constructed in finite time, and non-ground clauses can be effectively evaluated with respect to the model. Furthermore, we prove that our model construction produces the least model

SCL with Theory Constraints

We lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution inferences by a partial model assumption instead of an a priori fixed order as done for instance in hierarchic superposition. The model representation consists of ground background theory literals and ground foreground first-order literals. One major advantage of the model guided approach is that clauses generated by SCL(T) enjoy a non-redundancy property that makes expensive testing for tautologies and forward subsumption completely obsolete. SCL(T) is a semi-decision procedure for pure clause sets that are clause sets without first-order function symbols ranging into the background theory sorts. Moreover, SCL(T) can be turned into a decision procedure if the considered combination of a first-order logic modulo a background theory enjoys an abstract finite model property.Comment: 22 page

{SCL} with Theory Constraints

We lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution inferences by a partial model assumption instead of an a priori fixed order as done for instance in hierarchic superposition. The model representation consists of ground background theory literals and ground foreground first-order literals. One major advantage of the model guided approach is that clauses generated by SCL(T) enjoy a non-redundancy property that makes expensive testing for tautologies and forward subsumption completely obsolete. SCL(T) is a semi-decision procedure for pure clause sets that are clause sets without first-order function symbols ranging into the background theory sorts. Moreover, SCL(T) can be turned into a decision procedure if the considered combination of a first-order logic modulo a background theory enjoys an abstract finite model property

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 Datalog Hammer for Supervisor Verification Conditions Modulo Simple Linear Arithmetic

The Bernays-Sch\"onfinkel first-order logic fragment over simple linear real arithmetic constraints BS(SLR) is known to be decidable. We prove that BS(SLR) clause sets with both universally and existentially quantified verification conditions (conjectures) can be translated into BS(SLR) clause sets over a finite set of first-order constants. For the Horn case, we provide a Datalog hammer preserving validity and satisfiability. A toolchain from the BS(LRA) prover SPASS-SPL to the Datalog reasoner VLog establishes an effective way of deciding verification conditions in the Horn fragment. This is exemplified by the verification of supervisor code for a lane change assistant in a car and of an electronic control unit for a supercharged combustion engine

A Datalog Hammer for Supervisor Verification Conditions Modulo Simple Linear Arithmetic

The Bernays-Sch\"onfinkel first-order logic fragment over simple linear real arithmetic constraints BS(SLR) is known to be decidable. We prove that BS(SLR) clause sets with both universally and existentially quantified verification conditions (conjectures) can be translated into BS(SLR) clause sets over a finite set of first-order constants. For the Horn case, we provide a Datalog hammer preserving validity and satisfiability. A toolchain from the BS(LRA) prover SPASS-SPL to the Datalog reasoner VLog establishes an effective way of deciding verification conditions in the Horn fragment. This is exemplified by the verification of supervisor code for a lane change assistant in a car and of an electronic control unit for a supercharged combustion engine.Comment: 26 page

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