Journal of Automated Reasoning

The Intuitionistic Logic Theorem Proving (ILTP) library provides a platform for testing and benchmarking automated theorem proving (ATP) systems for intuitionistic propositional and first-order logic. It includes about 2,800 problems in a standardized syntax from 24 problem domains. For each problem an intuitionistic status and difficulty rating were obtained by running comprehensive tests of currently available intuitionistic ATP systems on all problems in the library. Thus, for the first time, the testing and evaluation of ATP systems for intuitionistic logic have been put on a firm basis.

This system description paper introduces the OWL 2 reasoner HermiT. The reasoner is fully compliant with the OWL 2 Direct Semantics as standardised by the World Wide Web Consortium (W3C). HermiT is based on the hypertableau calculus, and it supports a wide range of standard and novel optimisations that improve the performance of reasoning on real-world ontologies. Apart from the standard OWL 2 reasoning task of entailment checking, HermiT supports several specialised reasoning services such as class and property classification, as well as a range of features outside the OWL 2 standard such as DL-safe rules, SPARQL queries, and description graphs. We discuss the system’s architecture, and we present an overview of the techniques used to support the mentioned reasoning tasks. We further compare the performance of reasoning in HermiT with that of FaCT++ and Pellet—two other popular and widely used OWL 2 reasoners.

Several methods to compute the prime implicants and the prime implicates of a negation normal form (NNF) formula are developed and implemented. An algorithm PI is introduced that is an extension to negation normal form of an algorithm given by Jackson and Pais. A correctness proof of the PI algorithm is given. The PI algorithm alone is sufficient in a computational sense. However, it can be combined with path dissolution, and it is shown empirically that this is often an advantage. None of these variations rely on conjunctive normal form or on disjunctive normal form. A class of formulas is described for which reliance on CNF or on DNF results in an exponential increase in the time required to compute prime implicants/implicates. The possibility of avoiding this problem with efficient structure preserving clause form translations is examined briefly and appears unfavorable.

HeerHugo is a propositional formula checker that determines whether a given formula is satisfiable or not. The underlying algorithm is based on a specific breadth-first search procedure, with several enhancements including unit resolution and 2-satisfiability tests. Its main ingredient is the branch/merge rule inspired by an algorithm proposed by Stållmarck, which is protected by a software patent. In this paper, the main elements of the algorithm are discussed, and its remarkable effectiveness is illustrated with some examples and computational results.

Extended Static Checking (ESC) is a fully automated formal verification technique. Verification in ESC is achieved by translating programs and their specifications into verification conditions (VCs). Proof of a VC establishes the correctness of the program. The implementations of many seemingly simple algorithms are beyond the ability of traditional Extended Static Checking (ESC) tools to verify. Not being able to verify toy examples is often enough to turn users off of the idea of using formal methods. ESC4, the ESC component of the JML4 project, is able to verify many more kinds of methods in part because of its use of novel techniques which apply multiple theorem provers. In particular, we present Offline User-Assisted ESC (OUA-ESC), a new form of verification that lies between ESC and Full Static Program Verification (FSPV). ESC is generally quite efficient, as far as verification tools go, but it is still orders of magnitude slower than simple compilation. As can be imagined, proving VCs is computationally expensive: While small classes can be verified in seconds, verifying larger programs of 50 KLOC can take hours. To help address the added cost of using multiple provers and this lack of scalability, we present the multi-threaded version of ESC4 and its distributed prover back-end.

The interoperability of proof assistants and the integration of their libraries is a highly valued but elusive goal in the field of theorem proving. As a preparatory step, in previous work, we translated the libraries of multiple proof assistants, specifically the ones of Coq, HOL Light, IMPS, Isabelle, Mizar, and PVS into a universal format: OMDoc/MMT. Each translation presented great theoretical, technical, and social challenges, some universal and some system-specific, some solvable and some still open. In this paper, we survey these challenges and compare and evaluate the solutions we chose. We believe similar library translations will be an essential part of any future system interoperability solution, and our experiences will prove valuable to others undertaking such efforts.

Some basic theorems about composition and other key constructs of set theory were proved using McCune’s computer program OTTER, building on Quaife’s modification of Gödel’s class theory. Our proofs use equational definitions in terms of Gödel’s flip and rotate functors. A new way to prove the composition of homomorphisms theorem is also presented.

It is shown, that there exist a unification problem〈s=t〉 AI , for which the set of solutions under associativity and idempotence is not empty. But $$\mu U\sum _{Al} \left( {s,t} \right)$$ , the complete and minimal subset of this set of solutions does not exist, i.e.A+I is of type nullary. This is the first known standard first order theory with this unpleasant feature.