The TPTP Typed First-Order Form with Arithmetic

Abstract

Abstract. The TPTP World is a well established infrastructure supporting research, development, and deployment of Automated Theorem Proving systems. Recently, the TPTP World has been extended to include a typed first-order logic, which in turn has enabled the integration of arithmetic. This paper describes these developments. Motivation and HistoryThe TPTP World [32] is a well established infrastructure that supports research, development, and deployment of Automated Theorem Proving (ATP) systems. The TPTP World is based on the Thousands of Problems for Theorem Provers (TPTP) problem library [30], and includes the TPTP language, the SZS ontologies, the Thousands of Solutions from Theorem Provers (TSTP) solution library, various tools associated with the libraries, and the CADE ATP System Competition (CASC). This infrastructure has been central to the progress that has been made in the development of high performance first-order ATP systems -most state of the art systems natively read the TPTP language, many produce proofs or models in the TSTP format, much testing and development is done using the TPTP problem library, and CASC is an annual focal point where developers meet to discuss new ideas and advances in ATP techniques.Originally the TPTP supported only first-order problems in clause normal form (CNF). Over the years support for the full first-order form (FOF) and typed higher-order form (THF) have been added. Recently the simply typed first-order form (TFF) has been added. TFF has in turn been used as the basis for supporting arithmetic. Problems that use these new features have been added to the TPTP problem library, and ATP systems that can solve these problems have been developed. This paper describes the key steps of these developments.While the development of the TPTP World for typed first-order logic is new, several similar logics have been described previously, e.g., [36,26,13]. However, there are no contemporary ATP systems that implement those logics. There is active related research in the SMT community, which started in 2003 [24]. There are high performance systems for the various logics of the SMT-LIB 1 , e.g., those1 See the SMT-LIB web page

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E 2.0 User Manual

“…As such, it implements an instance of the superposition calculus described in [BG94]. We have added some stronger contraction rules and a more general approach to literal selection, and have also extended the calculus to simple, monomorphic many-sorted logic (in the sense of the TPTP-3 TFF format [SSCB12]). …”

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“…The TFF format described in [6] is just the core TFF0 of a polymorphic typed first-order format TFF1 [2]. The paper [6] also extends TFF by predefined types and operators for integer, rational and real arithmetics, which is the target language of SMTtoTPTP .…”

“…The paper [6] also extends TFF by predefined types and operators for integer, rational and real arithmetics, which is the target language of SMTtoTPTP .…”

“…In this paper I focus on the more difficult direction, from the SMT-LIB format to the appropriate TPTP format, the typed first-order TPTP format with arithmetics, TFF [6]. This converter, SMTtoTPTP , is meant to be…”