On Deciding Local Theory Extensions via E-matching

Satisfiability Modulo Theories (SMT) solvers incorporate decision procedures for theories of data types that commonly occur in software. This makes them important tools for automating verification problems. A limitation frequently encountered is that verification problems are often not fully expressible in the theories supported natively by the solvers. Many solvers allow the specification of application-specific theories as quantified axioms, but their handling is incomplete outside of narrow special cases. In this work, we show how SMT solvers can be used to obtain complete decision procedures for local theory extensions, an important class of theories that are decidable using finite instantiation of axioms. We present an algorithm that uses E-matching to generate instances incrementally during the search, significantly reducing the number of generated instances compared to eager instantiation strategies. We have used two SMT solvers to implement this algorithm and conducted an extensive experimental evaluation on benchmarks derived from verification conditions for heap-manipulating programs. We believe that our results are of interest to both the users of SMT solvers as well as their developers

Efficient Interpolation for the Theory of Arrays

Existing techniques for Craig interpolation for the quantifier-free fragment of the theory of arrays are inefficient for computing sequence and tree interpolants: the solver needs to run for every partitioning $(A, B)$ of the interpolation problem to avoid creating $AB$-mixed terms. We present a new approach using Proof Tree Preserving Interpolation and an array solver based on Weak Equivalence on Arrays. We give an interpolation algorithm for the lemmas produced by the array solver. The computed interpolants have worst-case exponential size for extensionality lemmas and worst-case quadratic size otherwise. We show that these bounds are strict in the sense that there are lemmas with no smaller interpolants. We implemented the algorithm and show that the produced interpolants are useful to prove memory safety for C programs.Comment: long version of the paper at IJCAR 201

Ordered Sets in the Calculus of Data Structures

Our goal is to identify families of relations that are useful for reasoning about software. We describe such families using decidable quantifier-free classes of logical constraints with a rich set of operations. A key challenge is to define such classes of constraints in a modular way, by combining multiple decidable classes. Working with quantifierfree combinations of constraints makes the combination agenda more realistic and the resulting logics more likely to be tractable than in the presence of quantifiers. Our approach to combination is based on reducing decidable fragments to a common class, Boolean Algebra with Presburger Arithmetic (BAPA). This logic was introduced by Feferman and Vaught in 1959 and can express properties of uninterpreted sets of elements, with set algebra operations and equicardinality relation (consequently, it can also express Presburger arithmetic constraints on cardinalities of sets). Combination by reduction to BAPA allows us to obtain decidable quantifierfree combinations of decidable logics that share BAPA operations. We use the term Calculus of Data Structures to denote a family of decidable constraints that reduce to BAPA. This class includes, for example, combinations of formulas in BAPA, weak monadic second-order logic of k-successors, two-variable logic with counting, and term algebras with certain homomorphisms. The approach of reduction to BAPA generalizes the Nelson-Oppen combination that forms the foundation of constraint solvers used in software verification. BAPA is convenient as a target for reductions because it admits quantifier elimination and its quantifier-free fragment is NP-complete. We describe a new member of the Calculus of Data Structures: a quantifier-free fragment that supports 1) boolean algebra of finite and infinite sets of real numbers, 2) linear arithmetic over real numbers, 3) formulas that can restrict chosen set or element variables to range over integers (providing, among others, the power of mixed integer arithmetic and sets of integers), 4) the cardinality operators, stating whether a given set has a given finite cardinality or is infinite, 5) infimum and supremum operators on sets. Among the applications of this logic are reasoning about the externally observable behavior of data structures such as sorted lists and priority queues, and specifying witness functions for the BAPA synthesis problem. We describe an abstract reduction to BAPA for our logic, proving that the satisfiability of the logic is in NP and that it can be combined with the other fragments of the Calculus of Data Structures

On uniform word problems involving bridging operators on distributive lattices

Abstract. In this paper we analyze some fragments of the universal theory of distributive lattices with many sorted bridging operators. Our interest in such algebras is motivated by the fact that, in description logics, numerical features are often expressed by using maps that associate numerical values to sets (more generally, to lattice elements). We first establish a link between satisfiability of universal sentences with respect to algebraic models and satisfiability with respect to certain classes of relational structures. We use these results for giving a method for translation to clause form of universal sentences, and provide some decidability results based on the use of resolution or hyperresolution. Links between hyperresolution and tableau methods are also discussed, and a tableau procedure for checking satisfiability of formulae of type t1 ≤ t2 is obtained by using a hyperresolution calculus.

Locality results for certain extensions of theories with bridging functions

Abstract. We study possibilities of reasoning about extensions of base theories with functions which satisfy certain recursion (or homomorphism) properties. Our focus is on emphasizing possibilities of hierarchical and modular reasoning in such extensions and combinations thereof. We present practical applications in verification and cryptography.

STUDY REGARDING DELINEATION OF FLOOD HAZARD ZONES IN THE HYDROGRAPHIC BASIN OF THE SOMEŞ RIVER, BORDER AREA

The hydrological studies will provide the characteristic parameters for the floods occurred for the calculus discharges with overflow probabilities of 0,1%; 1%, 5%, 10%. The hydrologic and hydraulic models will be made by using the hydro-meteorological data base and the topographical measurements on site; them calibration will be done according to the records of the historical floods. The studies on the hydrologic and hydraulic models will be necessary for the establishment of the carrying capacity of the riverbeds, for the delimitation of the flood plains and for the detection of the transit discharges at the hydro-technical installations, but also for the establishment of the parameters needed for the structural measures’ projects. These will be based on the 1D and 2D unstable hydro-dynamic models. Therefore, the users would be able to assess the proposed measures and the impact over the river’s system; of course with the potential combination of the 1D and 2D. The main objectives followed by the project are: • identification of the river basins or river sub-basins with flood risks; • regionalization of the flood hazard; • presentation of the main flash floods occurred during the last 30 years, which induced floods; • assessment of the consequences of eventual flood over the population, properties and environment; • the establishment of the protection degree, accepted for the human settlements, for the economic and social objectives, for the farm areas, etc.

Representation Theorems and the Semantics of Non-Classical Logics , and Applications to Automated Theorem Proving

We give a uniform presentation of representation and decidability results related to the Kripke-style semantics of several nonclassical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, distributive lattices and semilattices) extends in a natural way to several classes of operators and allows to establish a relationship between algebraic and Kripke-style models. We illustrate the ideas on several examples. We conclude by showing how the Kripkestyle models thus obtained can be used (if rst-order axiomatizable) for automated theorem proving by resolution for some non-classical logics