The Vampire and the FOOL

This paper presents new features recently implemented in the theorem prover Vampire, namely support for first-order logic with a first class boolean sort (FOOL) and polymorphic arrays. In addition to having a first class boolean sort, FOOL also contains if-then-else and let-in expressions. We argue that presented extensions facilitate reasoning-based program analysis, both by increasing the expressivity of first-order reasoners and by gains in efficiency

An End-to-End Pipeline from Law Text to Logical Formulas

We propose a pipeline for converting natural English law texts into logical formulas via a series of structural representations. Text texts are first parsed using a formal grammar derived from light-weight annotations. An intermediate representation called assembly logic is then used for logical interpretation and supports translations to different back-end logics and visualisations. The approach, while rule-based and explainable, is also robust: it can deliver useful results from day one, but allows subsequent refinements and variations

HOL(y)Hammer: Online ATP Service for HOL Light

HOL(y)Hammer is an online AI/ATP service for formal (computer-understandable) mathematics encoded in the HOL Light system. The service allows its users to upload and automatically process an arbitrary formal development (project) based on HOL Light, and to attack arbitrary conjectures that use the concepts defined in some of the uploaded projects. For that, the service uses several automated reasoning systems combined with several premise selection methods trained on all the project proofs. The projects that are readily available on the server for such query answering include the recent versions of the Flyspeck, Multivariate Analysis and Complex Analysis libraries. The service runs on a 48-CPU server, currently employing in parallel for each task 7 AI/ATP combinations and 4 decision procedures that contribute to its overall performance. The system is also available for local installation by interested users, who can customize it for their own proof development. An Emacs interface allowing parallel asynchronous queries to the service is also provided. The overall structure of the service is outlined, problems that arise and their solutions are discussed, and an initial account of using the system is given

Learning-Assisted Automated Reasoning with Flyspeck

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided

Going Polymorphic - TH1 Reasoning for Leo-III

While interactive proof assistants for higher-order logic (HOL) commonly admit reasoning within rich type systems, current theorem provers for HOL are mainly based on simply typed lambda-calculi and therefore do not allow such flexibility. In this paper, we present modifications to the higher-order automated theorem prover Leo-III for turning it into a reasoning system for rank-1 polymorphic HOL. To that end, a polymorphic version of HOL and a suitable paramodulation-based calculus are sketched. The implementation is evaluated using a set of polymorphic TPTP THF problems

Proof Support for Common Logic

We present an extension of the Heterogeneous Tool Set HETS that enables proof support for Common Logic. This is achieved via logic translations that relate Common Logic and some of its sublogics to already supported logics and automated theorem proving systems. We thus provide the first full theorem proving support for Common Logic, including the possibility of verifying meta-theoretical relationships between Common Logic theories

HaskHOL: A Haskell Hosted Domain Specific Language for Higher-Order Logic Theorem Proving

HaskHOL is an implementation of a HOL theorem proving capability in Haskell. Motivated by a need to integrate theorem proving capabilities into a Haskell-based tool suite, HaskHOL began as a simple port of HOL Light to Haskell. However, Haskell's laziness, immutable data, and monadic extensions both complicate an implementation and enable a new feature class. This thesis describes HaskHOL, its motivation and implementation. Its use to implement a primitive, interactive theorem prover is explored and its performance is evaluated using a collection of intuitionistically valid problems

Representation, Verification, and Visualization of Tarskian Interpretations for Typed First-order Logic

peer reviewedThis paper describes a new format for representing Tarskian-style interpretations for formulae in typed first-order logic, using the TPTP TF0 language. It further describes a technique and an implemented tool for verifying models using this representation, and a tool for visualizing interpretations. The research contributes to the advancement of automated reasoning technology for model finding, which has several applications, including verification