Decision procedures for linear arithmetic

In this thesis, we present new decision procedures for linear arithmetic in the context of SMT solvers and theorem provers: 1) CutSat++, a calculus for linear integer arithmetic that combines techniques from SAT solving and quantifier elimination in order to be sound, terminating, and complete. 2) The largest cube test and the unit cube test, two sound (although incomplete) tests that find integer and mixed solutions in polynomial time. The tests are especially efficient on absolutely unbounded constraint systems, which are difficult to handle for many other decision procedures. 3) Techniques for the investigation of equalities implied by a constraint system. Moreover, we present several applications for these techniques. 4) The Double-Bounded reduction and the Mixed-Echelon-Hermite transformation, two transformations that reduce any constraint system in polynomial time to an equisatisfiable constraint system that is bounded. The transformations are beneficial because they turn branch-and-bound into a complete and efficient decision procedure for unbounded constraint systems. We have implemented the above decision procedures (except for Cut- Sat++) as part of our linear arithmetic theory solver SPASS-IQ and as part of our CDCL(LA) solver SPASS-SATT. We also present various benchmark evaluations that confirm the practical efficiency of our new decision procedures.In dieser Arbeit präsentieren wir neue Entscheidungsprozeduren für lineare Arithmetik im Kontext von SMT-Solvern und Theorembeweisern: 1) CutSat++, ein korrekter und vollständiger Kalkül für ganzzahlige lineare Arithmetik, der Techniken zur Entscheidung von Aussagenlogik mit Techniken aus der Quantorenelimination vereint. 2) Der Größte-Würfeltest und der Einheitswürfeltest, zwei korrekte (wenn auch unvollständige) Tests, die in polynomieller Zeit (gemischt-)ganzzahlige Lösungen finden. Die Tests sind besonders effizient auf vollständig unbegrenzten Systemen, welche für viele andere Entscheidungsprozeduren schwer sind. 3) Techniken zur Ermittlung von Gleichungen, die von einem linearen Ungleichungssystem impliziert werden. Des Weiteren präsentieren wir mehrere Anwendungsmöglichkeiten für diese Techniken. 4) Die Beidseitig-Begrenzte-Reduktion und die Gemischte-Echelon-Hermitesche- Transformation, die ein Ungleichungssystem in polynomieller Zeit auf ein erfüllbarkeitsäquivalentes System reduzieren, das begrenzt ist. Vereint verwandeln die Transformationen Branch-and-Bound in eine vollständige und effiziente Entscheidungsprozedur für unbeschränkte Ungleichungssysteme. Wir haben diese Techniken (ausgenommen CutSat++) in SPASS-IQ (unserem theory solver für lineare Arithmetik) und in SPASS-SATT (unserem CDCL(LA) solver) implementiert. Basierend darauf präsentieren wir Benchmark-Evaluationen, die die Effizienz unserer Entscheidungsprozeduren bestätigen

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

A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems

International audienceWe present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded Reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two transformations turn any system of linear mixed constraints into a bounded system, i.e., a system for which termination can be achieved easily. Existing approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from proofs, only explore a finite search space after application of our two transformations. Instead of generating a priori bounds for the variables, e.g., as suggested by Papadimitriou, unbounded variables are eliminated through the two transformations. The transformations orient themselves on the structure of an input system instead of computing a priori (over- )approximations out of the available constants. Experiments provide further evidence to the efficiency of the transformations in practice. We also present a polynomial method for converting certificates of (un)satisfiability from the transformed to the original system

New Techniques for Linear Arithmetic: Cubes and Equalities

International audienceWe present several new techniques for linear arithmetic constraint solving. They are all based on the linear cube transformation, a method presented here, which allows us to efficiently determine whether a system of linear arithmetic constraints contains a hypercube of a given edge length. Our first findings based on this transformation are two sound tests that find integer solutions for linear arithmetic constraints. While many complete methods search along the problem surface for a solution, these tests use cubes to explore the interior of the problems. The tests are especially efficient for constraints with a large number of integer solutions, e.g., those with infinite lattice width. Inside the SMT-LIB benchmarks, we have found almost one thousand problem instances with infinite lattice width. Experimental results confirm that our tests are superior on these instances compared to several state-of-the-art SMT solvers. We also discovered that the linear cube transformation can be used to investigate the equalities implied by a system of linear arithmetic constraints. For this purpose, we developed a method that computes a basis for all implied equalities, i.e., a finite representation of all equalities implied by the linear arithmetic constraints. The equality basis has several applications. For instance, it allows us to verify whether a system of linear arithmetic constraints implies a given equality. This is valuable in the context of Nelson-Oppen style combinations of theories

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

SCL with Theory Constraints

22 pagesWe 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

Phonon-pump XUV-photoemission-probe in graphene: evidence for non-adiabatic heating of Dirac carriers by lattice deformation

We modulate the atomic structure of bilayer graphene by driving its lattice at resonance with the in-plane E1u lattice vibration at 6.3um. Using time- and angle-resolved photoemission spectroscopy (tr-ARPES) with extreme ultra-violet (XUV) pulses, we measure the response of the Dirac electrons near the K-point. We observe that lattice modulation causes anomalous carrier dynamics, with the Dirac electrons reaching lower peak temperatures and relaxing at faster rate compared to when the excitation is applied away from the phonon resonance or in monolayer samples. Frozen phonon calculations predict dramatic band structure changes when the E1u vibration is driven, which we use to explain the anomalous dynamics observed in the experiment.Comment: 16 pages, 8 figure

Treatment of depressed mothers with disruptive children: A controlled evaluation of cognitive behavioral family intervention

This study compared the effects of two forms of behavioral family intervention in reducing mothers' depression and disruptive behavior problems in families with a clinically depressed parent and a child with significant conduct problems. Fortyseven parents were randomly assigned to either a Behavioral Family Intervention (BFI) or to Cognitive Behavioral Family Intervention (CBFI) which integrated cognitive therapy strategies to treat depression and teaching of parenting skills. Both treatments were equally effective in reducing mothers' depression and child disruptive behavior on observational and self-report measures at postintervention. However, at 6-month follow-up more families in CBFI (53%) compared to BFI (13%) experienced concurrent clinically reliable reductions in maternal depression and child disruptive behavior. These findings support the value of CBFI in reducing depression in mothers of children with disruptive behavior problems

The relationship between fertility and lifespan in humans

Evolutionary theories of aging predict a trade-off between fertility and lifespan, where increased lifespan comes at the cost of reduced fertility. Support for this prediction has been obtained from various sources. However, which genes underlie this relationship is unknown. To assess it, we first analyzed the association of fertility with age at menarche and menopause, and with mortality in 3,575 married female participants of the Rotterdam Study. In addition, we conducted a candidate gene study where 1,664 single nucleotide polymorphisms (SNPs) in 25 candidate genes were analyzed in relation to number of children as a measure of fertility. SNPs that associated with fertility were analyzed for association with mortality. We observed no associations between fertility and age at menarche (p = 0.38) and menopause (p = 0.07). In contrast, fertility was associated with mortality. Women with two to three children had significantly lower mortality (hazard ratio (HR), 0.82; 95% confidence interval (95% CI), 0.69–0.97) compared to women with no children. No such benefit was observed for women with four or more children, who had a similar mortality risk (HR, 0.93; 95% CI, 0.76–1.13) as women with no children. The analysis of candidate genes revealed four genes that influence fertility after correction for multiple testing: CGB/LHB gene cluster (p = 0.0036), FSHR (p = 0.023), FST (p = 0.023), and INHBA (p = 0.021). However, none of the independent SNPs in these genes predicted mortality. In conclusion, women who bear two to three children live longer than those who bear none or many children, but this relationship was not mediated by the candidate genes analyzed in this study