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Abstract: Publisher Summary This chapter discusses the concepts needed for defining the complexity classes. A complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form—the set of problems that can be solved by an abstract machine M using O(f(n)) of resource R , where n is the size of the input. The simpler complexity classes are defined by various factors. The type of computational problem in which the most commonly used problems are decision problems. However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise problems, etc. The most common model of computation is the deterministic Turing machine, but many complexity classes are based on nondeterministic Turing machines, etc.

Abstract: We look at several problems from areas such as network flows, game theory, artificial intelligence, graph theory, integer programming and nonlinear programming and show that they are related in that any one of these problems is solvable in polynomial time if all the others are, too. At present, no polynomial time algorithm for these problems is known. These problems extend the equivalence class of problems known as P-Complete. The problem of deciding whether the class of languages accepted by polynomial time nondeterministic Turing machines is the same as that accepted by polynomial time deterministic Turing machines is related to P-Complete problems in that these two classes of languages are the same if each P-Complete problem has a polynomial deterministic solution. In view of this, it appears very likely that this equivalence class defines a class of problems that cannot be solved in deterministic polynomial time.

Abstract: Abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations. When the application domain is described by a logical theory, we speak about logic- based abduction. Candidates for abductive explanations are usually subjected to minimality criteria such as subset-minimality, minimal cardinality, minimal weight, or minimality under prioritization of individual hypotheses. This paper presents a comprehensive complexity analysis of relevant problems related to abduction on propositional theories. They show that the different variations of abduction provide a rich collection of natural problems populating all major complexity classes between P and Σ 3 P , Π 3 P in the refined polynomial hierarchy. More precisely, besides polynomial, NP-complete and co-NP-complete abduction problems, abduction tasks that are complete for the classes Δ i P , Δ i P [O(logn), Σ i P , and Π i P , for i=2,3, are identified.

Abstract: This paper considers a generalization, called the Shannon switching game on vertices, of a familiar board game called Hex. It is shown that determining who wins such a game if each player plays perfectly is very hard; in fact, if this game problem is solvable in polynomial time, then any problem solvable in polynomial space is solvable in polynomial time. This result suggests that the theory of combinational games is difficult.