Compilability of Domain Descriptions in the Language

In this note we analyze the possibility of reducing the complexity of entailment in A by a compilation of the domain description Since a single domain description D must in general be queried many times with respect to many di erent queries V it makes sense to reduce it in a form that allows the solving problem of entailment in polynomial time Using results from the eld of language compilation we prove that such a compilation is impossible if we impose the result of compilation to be a polynomial data structure The language A is one of the most popular languages for representing the e ect of actions Since the seminal paper by Gelfond and Lifshitz many other researchers have been worked on extending the language to incor porate the indirect e ect of actions the possibility of concurrent actions etc Very recently two papers appeared on the computational complexity of reasoning about actions and The latter regards the complexity of deciding whether a domain description in A is consistent and what it implies These problems have been found to be NP and coNP complete respectively for the basic de nition of the classes NP and coNP we refer to the work of Johnson Thus these problem are intractable that is no polynomial algorithm exists for solving them In order to solve intractable problems a compilation of part of the data may be useful A general theory of compilation of intractable problems has been de ned in In this paper we apply the results given there to the problems of A The problem of language compilation has been deeply analyzed in AI in the last few years The general pattern is the following There is an intractable problem The input of this problem is composed by two parts The typical situation is that of a knowledge base KB We extract infor mation from this knowledge base by checking which facts are true that is whether KB j Q for a fact Q Even for the propositional case this problem is coNP complete thus intractable In many cases a single knowl edge base KB must be queried many times w r t many di erent Q s In such cases it makes sense to compile KB into a new data structure H that allows the solution of the problem KB j Q in polynomial time What is required is that the data structure H has size polynomial in the size of the initial knowledge base KB If this compilation is possible we say that the problem is compilable A graphical representation of compilation is given in Figure