Extending SMT Solvers to Higher-Order Logic

International audienceSMT solvers have throughout the years been able to cope with increasingly expressive formulas, from ground logics to full first-order logic (FOL). In contrast, the extension of SMT solvers to higher-order logic (HOL) is mostly un-explored. We propose a pragmatic extension for SMT solvers to support HOL reasoning natively without compromising performance on FOL reasoning, thus leveraging the extensive research and implementation efforts dedicated to efficient SMT solving. We show how to generalize data structures and the ground decision procedure to support partial applications and extensionality, as well as how to reconcile quantifier instantiation techniques with higher-order variables. We also discuss a separate approach for redesigning an HOL SMT solver from the ground up via new data structures and algorithms. We apply our pragmatic extension to the CVC4 SMT solver and discuss a redesign of the veriT SMT solver. Our evaluation shows they are competitive with state-of-the-art HOL provers and often outperform the traditional encoding into FOL

Deletions of the RUNX2 Gene Are Present in About 10% of Individuals with Cleidocranial Dysplasia

Cleidocranial Dysplasia (CCD) is an autosomal dominant skeletal disorder characterized by hypoplastic or absent clavicles, increased head circumference, large fontanels, dental anomalies, and short stature. Hand malformations are also common. Mutations in RUNX2 cause CCD, but are not identified in all CCD patients. In this study we screened 135 unrelated patients with the clinical diagnosis of CCD for RUNX2 mutations by sequencing analysis and demonstrated 82 mutations 48 of which were novel. By quantitative PCR we screened the remaining 53 unrelated patients for copy number variations in the RUNX2 gene. Heterozygous deletions of different size were identified in 13 patients, and a duplication of the exons 1 to 4 of the RUNX2 gene in one patient. Thus, heterozygous deletions or duplications affecting the RUNX2 gene may be present in about 10% of all patients with a clinical diagnosis of CCD which corresponds to 26% of individuals with normal results on sequencing analysis. We therefore suggest that screening for intragenic deletions and duplications by qPCR or MLPA should be considered for patients with CCD phenotype in whom DNA sequencing does not reveal a causative RUNX2 mutation. (C) 2010 Wiley-Liss, Inc

Superposition with Lambdas

International audienceWe designed a superposition calculus for a clausal fragment of extensional polymorphic higher-order logic that includes anonymous functions but excludes Booleans. The inference rules work on βη-equivalence classes of λ-terms and rely on higher-order unification to achieve refuta-tional completeness. We implemented the calculus in the Zipperposition prover and evaluated it on TPTP and Isabelle benchmarks. The results suggest that superposition is a suitable basis for higher-order reasoning