  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.)
