The Inverse Lambda Calculus Algorithm for Typed First Order Logic Lambda Calculus and Its Application to Translating English to FOL

Abstract

In order to answer questions and solve problems that require deeper reasoning with respect to a given text, it is necessary to automatically translate English sentences to formulas in an appropriate knowledge representation language. This paper focuses on a method to translate sentences to First-Order Logic (FOL). Our approach is inspired by Montague’s use of lambda calculus formulas to represent the meanings of words and phrases. Since our target language is FOL, the meanings of words and phrases are represented as FOL-lambda formulas. In this paper we present algorithms that allow one to construct FOL-lambda formulas in an inverse manner. Given a sentence and its meaning and knowing the meaning of several words in the sentence our algorithm can be used to obtain the meaning of the other words in that sentence. In particular the two algorithms take as input two FOL-lambda formulas G and H and compute a FOL-lambda formula F such that F with input G, denoted by F@G, is H; respectively, G@F = H. We then illustrate our algorithm and present soundness, completeness and complexity results, and briefly mention the use of our algorithm in a NL Semantics system that translates sentences from English to formulas in formal languages.