Proof Assistants for Natural Language Semantics

AbstractIn this paper we discuss the use of interactive theorem provers (also called proof assistants) in the study of natural language semantics. It is shown that these provide useful platforms for NL semantics and reasoning on the one hand, and allow experiments to be performed on various frameworks and new theories, on the other. In particular, we show how to use Coq, a prominent type theory based proof assistant, to encode type theoretical semantics of various NL phenomena. In this respect, we can encode the NL semantics based on type theory for quantifiers, adjectives, common nouns, and tense, among others, and it is shown that Coq is a powerful engine for checking the formal validity of these accounts as well as a powerful reasoner about the implemented semantics. We further show some toy semantic grammars for formal semantic systems, like the Montagovian Generative Lexicon, Type Theory with Records and neo-Davidsonian semantics. It is also explained that experiments on new theories can be done as well, testing their validity and usefulness. Our aim is to show the importance of using proof assistants as useful tools in natural language reasoning and verification and argue for their wider application in the field.</jats:p

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

Gentzen-Prawitz Natural Deduction as a Teaching Tool

We report a four-years experiment in teaching reasoning to undergraduate students, ranging from weak to gifted, using Gentzen-Prawitz's style natural deduction. We argue that this pedagogical approach is a good alternative to the use of Boolean algebra for teaching reasoning, especially for computer scientists and formal methods practionners

Mechanized semantics

The goal of this lecture is to show how modern theorem provers---in this case, the Coq proof assistant---can be used to mechanize the specification of programming languages and their semantics, and to reason over individual programs and over generic program transformations, as typically found in compilers. The topics covered include: operational semantics (small-step, big-step, definitional interpreters); a simple form of denotational semantics; axiomatic semantics and Hoare logic; generation of verification conditions, with application to program proof; compilation to virtual machine code and its proof of correctness; an example of an optimizing program transformation (dead code elimination) and its proof of correctness