Computationally hard instances of combinatorial problems arise at a certain critical ratio of constraints to variables. At the critical ratio, problem distributions undergo dramatic changes. I will discuss how an analogous phenomenon occurs in phase transitions studied in physics, and how experiments with critically constrained problems have led to surprising new insights into average-case complexity and stochastic search methods in AI.
Computational efficiency is a central concern in the design of knowledge representation systems. In order to obtain efficient systems, it has been suggested that one should limit the form of the statements in the knowledge base or use an incomplete inference mechanism. The former approach is often too restrictive for practical applications, whereas the latter leads to uncertainty about exactly what can and cannot be inferred from the knowledge base. We present a third alternative, in which knowledge given in a general representation language is translated (compiled) into a tractable form --- allowing for efficient subsequent query answering. We show how propositional logical theories can be compiled into Horn theories that approximate the original information. The approximations bound the original theory from below and above in terms of logical strength. The procedures are extended to other tractable languages (for example, binary clauses) and to the first-order case. Finally, we demonstrate the generality of our approach by compiling concept descriptions in a general frame-based language into a tractable form.
Critical Behavior in the Satisfiability of Random Boolean Expressions
(See also p. 1249 for news article entitled ``Pinning Down a Treacherous Border in Logical Statements'' by Barry CipraEntire paper
The satisfiability of randomly generated Boolean formulae with $k$ variables per clause is a popular testbed for the performance of search algorithms in artificial intelligence and computer science. For $k = 2$, formulae are almost aways satisfiable when the ratio of clauses to variables is less than 1; for ratios larger than 1, the formulae are almost never satisfiable. We present data showing a similar threshold behavior for higher values of $k$. We also show how finite-size scaling, a method from statistical physics, can be used to characterize size dependent effects near the threshold. Finally, we comment on the relationship between thresholds and computational complexity.