Typed Answer Set Programming and Inverse Lambda Algorithms

TL;DR

This paper introduces algorithms for constructing ASP-lambda formulas from lambda calculus expressions, advancing automatic translation of English sentences into Answer Set Programming for better question understanding.

Contribution

It presents two inverse algorithms for building ASP-lambda formulas from lambda expressions, with correctness and complexity analyses, enhancing scalable language translation methods.

Findings

Algorithms are correct and have proven complexity bounds.

Development of typed ASP-lambda calculus theories.

Improved scalability in translating English to ASP.

Abstract

Our broader goal is to automatically translate English sentences into formulas in appropriate knowledge representation languages as a step towards understanding and thus answering questions with respect to English text. Our focus in this paper is on the language of Answer Set Programming (ASP). Our approach to translate sentences to ASP rules is inspired by Montague's use of lambda calculus formulas as meaning of words and phrases. With ASP as the target language the meaning of words and phrases are ASP-lambda formulas. In an earlier work we illustrated our approach by manually developing a dictionary of words and their ASP-lambda formulas. However such an approach is not scalable. In this paper our focus is on two algorithms that allow one to construct ASP-lambda formulas in an inverse manner. In particular the two algorithms take as input two lambda-calculus expressions G and H and compute a lambda-calculus expression F such that F with input as G, denoted by F@G, is equal to H; and similarly G@F = H. We present correctness and complexity results about these algorithms. To do that we develop the notion of typed ASP-lambda calculus theories and their orders and use it in developing the completeness results. (To appear in Theory and Practice of Logic Programming.)