Proving theorems by program transformation

In this paper we present an overview of the unfold/fold proof method, a method for proving theorems about programs, based on program transformation. As a metalanguage for specifying programs and program properties we adopt constraint logic programming (CLP), and we present a set of transformation rules (including the familiar unfolding and folding rules) which preserve the semantics of CLP programs. Then, we show how program transformation strategies can be used, similarly to theorem proving tactics, for guiding the application of the transformation rules and inferring the properties to be proved. We work out three examples: (i) the proof of predicate equivalences, applied to the verification of equality between CCS processes, (ii) the proof of first order formulas via an extension of the quantifier elimination method, and (iii) the proof of temporal properties of infinite state concurrent systems, by using a transformation strategy that performs program specialization

The Structure of Differential Invariants and Differential Cut Elimination

The biggest challenge in hybrid systems verification is the handling of differential equations. Because computable closed-form solutions only exist for very simple differential equations, proof certificates have been proposed for more scalable verification. Search procedures for these proof certificates are still rather ad-hoc, though, because the problem structure is only understood poorly. We investigate differential invariants, which define an induction principle for differential equations and which can be checked for invariance along a differential equation just by using their differential structure, without having to solve them. We study the structural properties of differential invariants. To analyze trade-offs for proof search complexity, we identify more than a dozen relations between several classes of differential invariants and compare their deductive power. As our main results, we analyze the deductive power of differential cuts and the deductive power of differential invariants with auxiliary differential variables. We refute the differential cut elimination hypothesis and show that, unlike standard cuts, differential cuts are fundamental proof principles that strictly increase the deductive power. We also prove that the deductive power increases further when adding auxiliary differential variables to the dynamics

Transformation Rules for Locally Stratified Constraint Logic Programs

We propose a set of transformation rules for constraint logic programs with negation. We assume that every program is locally stratified and, thus, it has a unique perfect model. We give sufficient conditions which ensure that the proposed set of transformation rules preserves the perfect model of the programs. Our rules extend in some respects the rules for logic programs and constraint logic programs already considered in the literature and, in particular, they include a rule for unfolding a clause with respect to a negative literal.Comment: To appear in: M. Bruynooghe, K.-K. Lau (Eds.) Program Development in Computational Logic, Lecture Notes in Computer Science, Springe

Applying Constraint Databases in the Determination of Potential Minimal Conflicts to Polynomial Model-Based Diagnosis

Model-based Diagnosis allows the identification of the parts which fail in a system. The models are based on the knowledge of the system to diagnose, and may be represented by constraints associated to the components. The variables of these constraints can be observable or non-observable, depending on the situation of the sensors. In order to obtain the potential minimal diagnosis in a system, an important issue is related to finding out the potential minimal conflicts in an efficient way. We consider that Constraint Databases represent an excellent option in order to solve this problem in complex systems. In this work we have used a novel logical architecture of Constraint Databases which has allowed obtaining these potential conflicts by means of the corresponding queries. Moreover, we have considered Gröbner Bases as a projection operator to obtain the potential minimal conflicts of a system. The first results obtained on this work, which are shown in a heat exchangers example, have been very promising.Ministerio de Ciencia y Tecnología DPI2003-07146-C02-0

Dynamic Clock Elimination in Parametric Timed Automata

The formalism of parametric timed automata provides designers with a formal way to specify and verify real-time concurrent systems where iming requirements are unknown (or parameters). Such models are usually subject to the state space explosion. A popular way to partially reduce the size of the state space is to reduce the number of clock variables. In this work, we present a technique for dynamically eliminating clocks. Experiments using IMITATOR show a diminution of the number of states and of the computation time, and in some cases allow termination of the analysis of models that could not terminate otherwise. More surprisingly, even when the number of clocks remains constant, there is little noticeable overhead in applying the proposed clock elimination

Parameterized Construction of Program Representations for Sparse Dataflow Analyses

Data-flow analyses usually associate information with control flow regions. Informally, if these regions are too small, like a point between two consecutive statements, we call the analysis dense. On the other hand, if these regions include many such points, then we call it sparse. This paper presents a systematic method to build program representations that support sparse analyses. To pave the way to this framework we clarify the bibliography about well-known intermediate program representations. We show that our approach, up to parameter choice, subsumes many of these representations, such as the SSA, SSI and e-SSA forms. In particular, our algorithms are faster, simpler and more frugal than the previous techniques used to construct SSI - Static Single Information - form programs. We produce intermediate representations isomorphic to Choi et al.'s Sparse Evaluation Graphs (SEG) for the family of data-flow problems that can be partitioned per variables. However, contrary to SEGs, we can handle - sparsely - problems that are not in this family

Reducing the size and number of linear programs in a dynamic Gr\"obner basis algorithm

The dynamic algorithm to compute a Gr\"obner basis is nearly twenty years old, yet it seems to have arrived stillborn; aside from two initial publications, there have been no published followups. One reason for this may be that, at first glance, the added overhead seems to outweigh the benefit; the algorithm must solve many linear programs with many linear constraints. This paper describes two methods of reducing the cost substantially, answering the problem effectively.Comment: 11 figures, of which half are algorithms; submitted to journal for refereeing, December 201