It is well known that for many NP-complete problems, such as K-Sat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P = NP). This paper shows that NP-complete problems can be summarized by at least one "order parameter", and that the hard problems occur at a critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for K-colorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability of a solution changes abruptly from near 0 to near 1. It is the high density of wellseparated almost solutions (local minima) at this boundary that cause search algorithms to "thrash". This boundary is a type of phase transition and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transition or it occurs for bounded N (and so bounds the cost). These results suggest a way of deciding if a problem is in P or NP and why they are different. 1

: We describe a detailed experimental investigation of the phase transition for several different classes of randomly generated satisfiability problems. We observe a remarkable consistency of features in the phase transition despite the presence in some of the problem classes of clauses of mixed lengths. For instance, each of the problem classes considered has a sharp transition from satisfiable to unsatisfiable problems at a critical value. In addition, there is a common easyhard -easy pattern in the difficulty of the problems, with the hardest problems being associated with the phase transition. However, the difficulty of problems of mixed clause lengths is much more variable than that of fixed clause length. Indeed, whilst the median difficulty of random problems of mixed clause lengths can be orders of magnitude easier than that of equivalently sized problems of fixed clause length, the hardest problems of mixed clause lengths can be orders of magnitude harder than the hardest equi...

Recent work by Birnbaum & Lozinskii [1999] demonstrated that a clever yet simple extension of the well-known DavisPutnam procedure for solving instances of propositional satisfiability yields an efficient scheme for counting the number of satisfying assignments (models). We present a new extension, based on recursively identifying connected constraint-graph components, that substantially improves counting performance on random 3-SAT instances as well as benchmark instances from the SATLIB and Beijing suites. In addition, from a structure-based perspective of worst-case complexity, while polynomial time satisfiability checking is known to require only a backtrack search algorithm enhanced with nogood learning, we show that polynomial time counting using backtrack search requires an additional enhancement: good learning. Introduction Many practical problems from a variety of domains, most notably planning [Kautz & Selman 1996], have been efficiently solved by formulating ...

Very recently many researchers, with backgrounds in parallel computing, started to develop hybrids of traditional genetic algorithms. The main departure from standard genetic algorithms is that these new methods incorporate specific heuristics for the problem at hand (drawing on a tradition which has roots outside the genetic framework) and which we apply within a stochastic game that exerts a selective pressure. The heuristics are used for periods of individual optimization, that is when agents do not interact. New computational results for the Traveling Salesman Problem will be presented in this paper. The approach is prepared to include Tabu Search techniques, introducing a new crossover operator (which is called Random Respectful Corner Recombination) and a special pair of a topology and set of rules for the interaction between agents. The approach has a natural parallelism and a feature called superlinear speed-up will also be discussed.

The process of understanding a source code in a high-level programming language is a complex cognitive task. The provision of helpful decision aid subsystems would be of great benefit to software maintainers. Given a library of program plan templates, generating a partial understanding of a piece of software source code can be shown to correspond to the construction of mappings between segments of the source code and particular program plans represented in a library of domain source programs (plans). These mappings can be used as part of the larger task of reverse engineering source code, to facilitate many software engineering tasks such as software reuse, and for program maintenance. We present a novel model of program understanding using constraint satisfaction. The model composes a partial global picture of source program code by transforming knowledge about the problem domain and the program structure into constraints. These constraints facilitate the efficient construction of ma...

When dealing with autonomous robots, the definition of the term "autonomy" is important. Autonomy implies internally an organisation of self-contained and self-referential processes. In this paper, we are dealing with the methodological impact of considering what the robot can actually know. In this context, the idea of combining reactive behaviours with goaloriented mechanisms and of using representations to do the latter remains valid but has to be reformulated. First, we distinguish the internal sensory-motor processes from the behaviours produced when the formers are interacting with an environment. Second, a representation cannot be a representation of an external world, inaccessible to the robot anyway, but only of the interaction history as perceived by the robot itself. In this contribution, we propose such a representation as a graph encoding of the interaction history and show its usages for interpreting, structuring and planning without having to refer to any external enviro...

It is well known that for many NP-complete problems, such as K-Sat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P 6= NP ). This paper shows that NP-complete problems can be summarized by at least one "order parameter ", and that the hard problems occur at a critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for K-colorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability of a solution changes abruptly from near 0 to near 1. It is the high density of wellseparated almost solutions (local minima) at this boundary that cause search algorithms to "thrash". This boundary is a type of phase transition and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transition or it occurs for bounded N (and so bound...

It is well known that satisfiability of random sets of propositional clauses undergoes phase transition while the clause-to-variable ratio of the sets increases. We introduce another parameter of sets of clauses, impurity, and show that the satisfiability undergoes a phase transition as a function of impurity. This phenomenon supports a conjecture that various properties (such as random graph connectivity, perfect integer partition) exhibit phase transition under control of several different syntactic parameters. Keywords: SAT, phase transition, impurity

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