Analysis of Algorithms for Monotonic and Non-Monotonic Reasoning

i Preface Logic has always provided a rigorous foundation for mathematics and philosophy, and more recently it has been called upon to support formal concepts and problem-solving mechanisms for computer science, operations research, and artificial intelligence. Recent work emphasizes that, in addition to traditional concerns such as soundness and completeness of proof systems, it is important to study computational requirements. Notably, a system must produce relatively short proofs or else it will take too much time to be useful. What are the most powerful systems that will admit only short proofs? Even if short proofs are guaranteed by a system, short proofs obtained by a particular implementation of the system may not be guaranteed. Is there a method for finding short proofs when they exist? For a given proof system, a valid formula, and a function f , what is the probability that a system will produce a proof of length bounded by f (n), where n is the size of the input statement? If f is a low degree polynomial and the probability in question is close to 1, then the system may be suitable for practical use. To answer questions such as those above, considerable research has focused on algorithm development and complexity analysis for proof systems. In classical logic, enlarging a set of axioms results in an enlarged set of conclusions that can be drawn from the axioms. In other words, existing assertions or conclusions cannot be retracted if the set of axioms is added to. For example , a proposition can be expressed in Conjunctive Normal Form (CNF) as follows: (a ∨ b ∨ ¯ c) ∧ (¯ a ∨ ¯ c ∨ d) ∧ (¯ b ∨ c ∨ ¯ d). Each disjunction, called a clause, is a constraint. Adding constraints (that is, enlarging the set of axioms) shrinks the set of satisfying interpretations (models) and enlarges the set of clauses that are implied (conclusions or im-plicants). Many systems for dealing with this more traditional, monotonic form of logic have been proposed and algorithms implemented. Most useful algorithms rely on monotonicity for determining whether satisfying interpretations exist. Since the question whether a propositional formula has a satisfying interpretaion is N P-complete, none of these algorithms has worst-case, polynomially bounded time complexity. However, if the structure of a given formula is restricted in certain ways, for example, by restricting the number of literals in a clause …