NATURAL DEDUCTION AS HIGHER-ORDER RESOLUTION

An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. Resolution gives both forwards and backwards proof, supporting a large class of logics. Isabelle has been used to prove theorems in Martin-L\"of's Constructive Type Theory. Quantifiers pose several difficulties: substitution, bound variables, Skolemization. Isabelle's representation of logical syntax is the typed lambda-calculus, requiring higher- order unification. It may have potential for logic programming. Depth-first subgoaling along inference rules constitutes a higher-order Prolog

Deduction modulo theory

This paper is a survey on Deduction modulo theor

Higher-order Linear Logic Programming of Categorial Deduction

We show how categorial deduction can be implemented in higher-order (linear) logic programming, thereby realising parsing as deduction for the associative and non-associative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions.Comment: 8 pages LaTeX, uses eaclap.sty, to appear EACL9

A Vernacular for Coherent Logic

We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a frag- ment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many ev- eryday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle/Isar and Coq, as well as proofs expressed in a natural language form (formatted in LATEX or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014

NaDeA: A Natural Deduction Assistant with a Formalization in Isabelle

We present a new software tool for teaching logic based on natural deduction. Its proof system is formalized in the proof assistant Isabelle such that its definition is very precise. Soundness of the formalization has been proved in Isabelle. The tool is open source software developed in TypeScript / JavaScript and can thus be used directly in a browser without any further installation. Although developed for undergraduate computer science students who are used to study and program concrete computer code in a programming language we consider the approach relevant for a broader audience and for other proof systems as well.Comment: Proceedings of the Fourth International Conference on Tools for Teaching Logic (TTL2015), Rennes, France, June 9-12, 2015. Editors: M. Antonia Huertas, Jo\~ao Marcos, Mar\'ia Manzano, Sophie Pinchinat, Fran\c{c}ois Schwarzentrube