We present a pragmatic approach to extending a Boolean-free higher-order superposition calculus to support Boolean reasoning. Our approach extends inference rules that have been used only in a firstorder setting, uses some well-known rules previously implemented in higher-order provers, as well as new rules. We have implemented the approach in the Zipperposition theorem prover. The evaluation shows highly competitive performance of our approach and clear improvement over previous techniques.
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such questions have not been formalised. We present a thesis that a new algorithm for ascertaining satisfiability of formulae over the reals via Cylindrical Algebraic Coverings [Ábrahám, Davenport, England, Kremer, \emph{Deciding the Consistency of Non-Linear Real Arithmetic Constraints with a Conflict Driver Search Using Cylindrical Algebraic Coverings}, 2020] might provide trace and outputs that allow the results to be more susceptible to machine verification than those of competing algorithms.
The TPTP world is a well established infrastructure that supports research, development, and deployment of Automated Theorem Proving (ATP) systems for classical logics. The TPTP language is one of the keys to the success of the TPTP world. Originally the TPTP world supported only first-order clause normal form (CNF). Over the years support for full first-order form (FOF), monomorphic typed first-order form (TF0), rank-1 polymorphic typed first-order form (TF1), and monomorphic typed higher-order form (TH0) have been added. The TF0, TF1, and TH0 languages also include constructs for arithmetic. This paper introduces the TH1 form, an extension of TH0 with TF1-style rank-1 polymorphism. TH1 is designed to be easy to process by existing reasoning tools that support ML-style polymorphism. The hope is that TH1 will be implemented in many popular ATP systems for typed higher-order logic.
This paper describes the translation of proofs in the Thousands of Solutions from Theorem Provers (TSTP) solution library to the Proof Markup Language (PML), and the subsequent use of Inference Web (IW) tools to provide new presentations of the proofs. The translation enriches the TSTP proofs with proof provenance meta-data, and provides new possibilities for proof processing.
The ESHOL sessions of the PAAR workshop focussed on the use of higher-order reasoning systems. A particular focus was on means to evaluate higher-order reasoning systems. The notion of higher-order included, but was not limited to, ramified type theory, simple type theory, intuitionistic and constructive type theory, and logical frameworks. The notion of reasoning systems included automated and semi-automated provers, model generators, as well as proof and model checkers. There were two parts to the ESHOL sessions: (i) higher-order system demonstrations, and (ii) a panel discussion. Additionally, one of the PAAR invited speakers, Rob Arthan, gave a talk in the ESHOL topic area.
