Orthologic with Axioms

We study the proof theory and algorithms for orthologic, a logical system based on ortholattices, which have shown practical relevance in simplification and normalization of verification conditions. Ortholattices weaken Boolean algebras while having polynomial-time equivalence checking that is sound with respect to Boolean algebra semantics. We generalize ortholattice reasoning and obtain an algorithm for proving a larger class of classically valid formulas. As the key result, we analyze a proof system for orthologic augmented with axioms. An important feature of the system is that it limits the number of formulas in a sequent to at most two, which makes the extension with axioms non-trivial. We show a generalized form of cut elimination for this system, which implies a sub-formula property. From there we derive a cubic-time algorithm for provability from axioms, or equivalently, for validity in finitely presented ortholattices. We further show that propositional resolution of width 5 proves all formulas provable in orthologic with axioms. We show that orthologic system subsumes resolution of width 2 and arbitrarily wide unit resolution and is complete for reasoning about generalizations of propositional Horn clauses. Moving beyond ground axioms, we introduce effectively propositional orthologic, presenting its semantics as well as a sound and complete proof system. Our proof system implies the decidability of effectively propositional orthologic, as well as its fixed-parameter tractability for a bounded maximal number of variables in each axiom. As a special case, we obtain a generalization of Datalog with negation and disjunction

Equivalence Checking for Orthocomplemented Bisemilattices in Log-Linear Time

We present a quasilinear time algorithm to decide the word problem on a natural algebraic structures we call orthocomplemented bisemilattices, a subtheory of boolean algebra. We use as a base a variation of Hopcroft, Ullman and Aho algorithm for tree isomorphism which we combine with a term rewriting system to decide equivalence of two terms. We prove that the rewriting system is terminating and confluent and hence the existence of a normal form, and that our algorithm is computing it. We also discuss applications and present an effective implementation in Scala.LAR

Formula Normalizations in Verification

We propose a new approach for normalization and simplification of logical formulas. Our approach is based on algorithms for lattice-like structures. Specifically, we present two efficient algorithms for computing a normal form and deciding the word problem for two subtheories of Boolean algebra, giving a sound procedure for propositional logical equivalence that is incomplete in general but complete with respect to a subset of Boolean algebra axioms. We first show a new algorithm to produce a normal form for expressions in the theory of ortholattices (OL) in time O(n^2). We also consider an algorithm, recently presented but never evaluated in practice, producing a normal form for a slightly weaker theory, orthocomplemented bisemilattices (OCBSL), in time O(n log(n)^2). For both algorithms, we present an implementation and show efficiency in two domains. First, we evaluate the algorithms on large propositional expressions, specifically combinatorial circuits from a benchmark suite, as well as on large random formulas. Second, we implement and evaluate the algorithms in the Stainless verifier, a tool for verifying the correctness of Scala programs. We used these algorithms as a basis for a new formula simplifier, which is applied before valid verification conditions are saved into a persistent cache. The results show that normalization substantially increases cache hit ratio in large benchmarks.LAR

Confirmation of Psoriasis Susceptibility Loci on Chromosome 6p21 and 20p13 in French Families

Plaque psoriasis is a chronic inflammatory disorder of the skin. It is inherited as a multifactorial trait, with a strong genetic component. Linkage studies have identified a large number of disease loci, but very few could be replicated in independent family sets. In this study, we present the results of a genome-wide scan carried out in 14 French extended families. Candidate regions were then tested in a second set of 32 families. Analysis of the pooled samples confirmed linkage to chromosomes 6p21 (ZMLB score ¼ 3.5, P ¼ 0.0002) and 20p13 (ZMLB score ¼ 2.9, P ¼ 0.002), although there was little contribution of the second family set to the 20p13 linkage signal. Moreover, we identified four additional loci potentially linked to psoriasis. The major histocompatibility complex region on 6p21 is a major susceptibility locus, referred to as PSORS1, which has been found in most of the studies published to date. The 20p13 locus segregates independently of PSORS1 in psoriasis families. It has previously been thought to be involved in the predisposition to psoriasis and other inflammatory disorders such as atopic dermatitis (AD) and asthma. Although psoriasis and AD rarely occur together, this reinforces the hypothesis that psoriasis is influenced by genes with general effects on inflammation and immunity