Strict General Setting for Building Decision Procedures into Theorem Provers

The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking etc. In this paper we present a variant of one general setting for building decision procedures into theorem provers (gs framework [18]). That setting is based on macro inference rules motivated by techniques used in different approaches. The general setting enables a simple describing of different combination/augmentation schemes. In this paper, we further develop and extend this setting by an imposed ordering on the macro inference rules. That ordering leads to a ”strict setting”. It makes implementing and using variants of well-known or new schemes within this framework a very easy task even for a non-expert user. Also, this setting enables easy comparison of different combination/augmentation schemes and combination of their ideas

Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation

AC-completion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground AC-completion for deciding formulas in the combination of the theory of equality with user-defined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground AC-completion with the canonizer and solver present for the theory X. This integration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for publication by LMCS (Logical Methods in Computer Science

New results on rewrite-based satisfiability procedures

Program analysis and verification require decision procedures to reason on theories of data structures. Many problems can be reduced to the satisfiability of sets of ground literals in theory T. If a sound and complete inference system for first-order logic is guaranteed to terminate on T-satisfiability problems, any theorem-proving strategy with that system and a fair search plan is a T-satisfiability procedure. We prove termination of a rewrite-based first-order engine on the theories of records, integer offsets, integer offsets modulo and lists. We give a modularity theorem stating sufficient conditions for termination on a combinations of theories, given termination on each. The above theories, as well as others, satisfy these conditions. We introduce several sets of benchmarks on these theories and their combinations, including both parametric synthetic benchmarks to test scalability, and real-world problems to test performances on huge sets of literals. We compare the rewrite-based theorem prover E with the validity checkers CVC and CVC Lite. Contrary to the folklore that a general-purpose prover cannot compete with reasoners with built-in theories, the experiments are overall favorable to the theorem prover, showing that not only the rewriting approach is elegant and conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page

Combination of convex theories: Modularity, deduction completeness, and explanation

AbstractDecision procedures are key components of theorem provers and constraint satisfaction systems. Their modular combination is of prime interest for building efficient systems, but their effective use is often limited by poor interface capabilities, when such procedures only provide a simple “sat/unsat” answer. In this paper, we develop a framework to design cooperation schemas between such procedures while maintaining modularity of their interfaces. First, we use the framework to specify and prove the correctness of classic combination schemas by Nelson–Oppen and Shostak. Second, we introduce the concept of deduction complete satisfiability procedures, we show how to build them for large classes of theories, then we provide a schema to modularly combine them. Third, we consider the problem of modularly constructing explanations for combinations by re-using available proof-producing procedures for the component theories

Quantifier-Free Interpolation of a Theory of Arrays

The use of interpolants in model checking is becoming an enabling technology to allow fast and robust verification of hardware and software. The application of encodings based on the theory of arrays, however, is limited by the impossibility of deriving quantifier- free interpolants in general. In this paper, we show that it is possible to obtain quantifier-free interpolants for a Skolemized version of the extensional theory of arrays. We prove this in two ways: (1) non-constructively, by using the model theoretic notion of amalgamation, which is known to be equivalent to admit quantifier-free interpolation for universal theories; and (2) constructively, by designing an interpolating procedure, based on solving equations between array updates. (Interestingly, rewriting techniques are used in the key steps of the solver and its proof of correctness.) To the best of our knowledge, this is the first successful attempt of computing quantifier- free interpolants for a variant of the theory of arrays with extensionality

Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds

Given a formula in quantifier-free Presburger arithmetic, if it has a satisfying solution, there is one whose size, measured in bits, is polynomially bounded in the size of the formula. In this paper, we consider a special class of quantifier-free Presburger formulas in which most linear constraints are difference (separation) constraints, and the non-difference constraints are sparse. This class has been observed to commonly occur in software verification. We derive a new solution bound in terms of parameters characterizing the sparseness of linear constraints and the number of non-difference constraints, in addition to traditional measures of formula size. In particular, we show that the number of bits needed per integer variable is linear in the number of non-difference constraints and logarithmic in the number and size of non-zero coefficients in them, but is otherwise independent of the total number of linear constraints in the formula. The derived bound can be used in a decision procedure based on instantiating integer variables over a finite domain and translating the input quantifier-free Presburger formula to an equi-satisfiable Boolean formula, which is then checked using a Boolean satisfiability solver. In addition to our main theoretical result, we discuss several optimizations for deriving tighter bounds in practice. Empirical evidence indicates that our decision procedure can greatly outperform other decision procedures.Comment: 26 page

Galaxy Disks

The formation and evolution of galactic disks is particularly important for understanding how galaxies form and evolve, and the cause of the variety in which they appear to us. Ongoing large surveys, made possible by new instrumentation at wavelengths from the ultraviolet (GALEX), via optical (HST and large groundbased telescopes) and infrared (Spitzer) to the radio are providing much new information about disk galaxies over a wide range of redshift. Although progress has been made, the dynamics and structure of stellar disks, including their truncations, are still not well understood. We do now have plausible estimates of disk mass-to-light ratios, and estimates of Toomre's $Q$ parameter show that they are just locally stable. Disks are mostly very flat and sometimes very thin, and have a range in surface brightness from canonical disks with a central surface brightness of about 21.5 $B$-mag arcsec$^{-2}$ down to very low surface brightnesses. It appears that galaxy disks are not maximal, except possibly in the largest systems. Their HI layers display warps whenever HI can be detected beyond the stellar disk, with low-level star formation going on out to large radii. Stellar disks display abundance gradients which flatten at larger radii and sometimes even reverse. The existence of a well-defined baryonic Tully-Fisher relation hints at an approximately uniform baryonic to dark matter ratio. Thick disks are common in disk galaxies and their existence appears unrelated to the presence of a bulge component; they are old, but their formation is not yet understood. Disk formation was already advanced at redshifts of $\sim 2$, but at that epoch disks were not yet quiescent and in full rotational equilibrium. Downsizing is now well-established. The formation and history of star formation in S0s is still not fully understood.Comment: This review has been submitted for Annual Reviews of Astronomy & Astrophysics, vol. 49 (2011); the final printed version will have fewer figures and a somewhat shortened text. A pdf-version of this preprint with high-resolution figures is available from http://www.astro.rug.nl/~vdkruit/jea3/homepage/disks-ph.pdf. (table of contents added; 71 pages, 24 figures, 529 references