Akira Maruoka

How to Produce BlockSum Instances with Various Levels of Difficulty

We propose a framework that yields instances of certain combinatorial puzzles. To explore such a frame- work, we focus on certain types of puzzles that ask an assignment of numbers to the cells of an n × n grid so that it satisfies certain constraints as well as the Latin square condition, that is, each row and column contains all of the numbers in {1,2 ,..., n}. Our algorithm based on the framework automatically yields puzzle instances whose diffi- culties to solve can be adjusted by means of puzzle inference rules built into the algorithm. Taking up BlockSum puzzle for example, we performed experiments to demonstrate that, as is expected, human solvers tend to solve puzzle instances correctly that are produced with easy inference rules, whereas they tend to fail to solve those produced with sophisticated rules.

Computational Complexity Based on Turing Machines

Among the problems that can be solved in principle by Turing machines, there exists a problem that requires one to run a modern supercomputer, say, for the life time of the earth. This fact naturally leads us to classify real-world problems into two types: tractable problems, which can be computed in a feasible amount of time, and intractable problems, which cannot be computed in a feasible amount of time. In order to classify problems we employ a Turing machine as a computational model, because it turns out that the time required to run a typical computer is somehow related to that required to run an equivalent Turing machine. First, we introduce a way of measuring the time required to compute problems. The class of tractable problems, denoted by P, consists of problems that can be computed in polynomial time on a Turing machine in terms of the measure. On the other hand, we often encounter in practice a problem that can be solved somehow by checking all the possible certificates for the problem instance. Finally, we introduce the other type of problems such that, once an appropriate certificate is given, checking the validity can be done in polynomial time. The class of this type of problems is denoted by NP, which includes P by definition.

Universality of Turing Machine and Its Limitations

The universal Turing machine is the one that, when given a description of Turing machine M and input w, can simulate the behavior of M with input w. In this chapter we construct the universal Turing machine. This means that the universal Turing machine can behave like any Turing machine with any input if the description of the Turing machine to be simulated is given together with its input. On the other hand, there exists a limit to the computational power of Turing machines. This limitation is shown by giving the halting problem, which any Turing machine can never solve. The halting problem asks whether, when a Turing machine M and an input w are given, M with input w will eventually halt or continue to run forever. As another example of a problem that any Turing machine cannot solve, we give the Post correspondence problem. The Post correspondence problem asks whether or not, when a collection of pairs of strings is given, there exists a sequence of the pairs (repetitions permitted) that has certain properties of a match.

Preliminaries to the Theory of Computation

In this chapter, we explain mathematical notions, terminologies, and certain methods used in convincing logical arguments that we shall have need of throughout the book.

Everything Begins with Computation

Computer science deals with the issue of what can and cannot be computed and, if possible, how it can be computed. We refer to what a computer does, whatever it is, as “computation.” What to compute is formalized as a problem, whereas how to compute it is formalized as a mechanical procedure or an algorithm. What is defined as a field within which an algorithm works is a computational model. Once a computational model is defined, a set of basic moves that are performed is fixed as one step. Under these settings, the theory of computation is intended to uncover the laws that govern computation, as physical sciences discover the laws that control physical phenomena.

Computational Complexity Based on Boolean Circuits

Computation performed by a Turing machine with time complexity t(n) can be thought of as being performed by a t(n)×t(n) table consisting of t(n)×t(n) cells. The ith row of the table transforms a configuration of the Turing machine at the ith step into the configuration at the next step. Each cell of the table can be regarded as a generalized gate that can be implemented by a certain number of Boolean gates under a suitable encoding. This circuit model which works as a counterpart to a Turing machine illustrates more directly how each configuration is transformed into the next configuration. By introducing this alternative circuit model, we can better understand the notion of nondeterminism discussed in Chap. 6 as well as the notion of NP-completeness which will be discussed in Chap. 10.

Context-Free Languages

Among what are called languages, there are artificial languages such as programming languages and natural languages such as English, German, French, and Japanese, etc. The languages that we study in this book belong to the former group, and in this chapter we study context-free languages in the former group. A regular language studied in the previous chapter is defined to be the language that a finite automaton accepts. In this chapter, we introduce a context-free grammar, and define a context-free language to be the one that a context-free grammar generates. The class of context-free languages includes the class of regular languages and, furthermore, contains nonregular languages like {0 n 1 n |n=0}. So the generating power of a context-free grammar exceeds the accepting power of a finite automaton. Corresponding to the pumping lemma for regular languages, we introduce a pumping lemma for context-free languages, which shows that a language does not belong to the class of context-free languages. Context-free grammar are also used to describe practical programming languages.

Monotone DNF Formula That Has a Minimal or Maximal Number of Satisfying Assignments

We consider the following extremal problem: Given three natural numbers n, mand l, what is the monotone DNF formula that has a minimal or maximal number of satisfying assignments over all monotone DNF formulas on nvariables with mterms each of length l? We first show that the solution to the minimization problem can be obtained by the Kruskal-Katona theorem developed in extremal set theory. We also give a simple procedure that outputs an optimal formula for the more general problem that allows the lengths of terms to be mixed. We then show that the solution to the maximization problem can be obtained using the result of Bollobas on the number of complete subgraphs when l= 2 and the pair (n,m) satisfies a certain condition. Moreover, we give the complete solution to the problem for the case l= 2 and m≤ n, which cannot be solved by direct application of Bollobass result. For example, when n= m, an optimal formula is represented by a graph consisting of

A maximum matching based heuristic algorithm for partial Latin square extension problem

lfloor{n/3}rfloor-1

Predicting like the best pruning of a decision tree based on the on-line DP (Algorithms and Theory of Computing)

copies of C 3 and one