### Turing Machines for Dummies why representations do matter

"... Abstract. Various methods exists in the literature for denoting the configuration of a Turing Machine. A key difference is whether the head position is indicated by some integer (mathematical representation) or is specified by writing the machine state next to the scanned tape symbol (intrinsic repr ..."

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Abstract. Various methods exists in the literature for denoting the configuration of a Turing Machine. A key difference is whether the head position is indicated by some integer (mathematical representation) or is specified by writing the machine state next to the scanned tape symbol (intrinsic representation). From a mathematical perspective this will make no difference. However, since Turing Machines are primarily used for proving undecidability and/or hardness results these representations do matter. Based on a number of applications we show that the intrinsic representation should be preferred 1. 1 The Turing Machine model Given the nature of the meeting I expect that the dummies mentioned in my title will not be present in the audience. Still I believe that it is useful to start with a description of the Turing Machine model as we are supposed to know it. The simplest version of the Turing machine is defined in mathematical terms

### Alternation Alternation

"... ABSTRACT. Alternation is a generalization of nondeterminism in which existential and universal quantitiers can alternate during the course of a computation, whereas in a nondeterministic computation there are only existential quantifiers. Alternating Turing machines are defined and shown to accept p ..."

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ABSTRACT. Alternation is a generalization of nondeterminism in which existential and universal quantitiers can alternate during the course of a computation, whereas in a nondeterministic computation there are only existential quantifiers. Alternating Turing machines are defined and shown to accept precisely the recursively enumerable sets. Complexity classes of languages accepted by time- (space-) bounded alternating Turing machines are characterized in terms of complexity classes of languages accepted by space- (time-) bounded deterministic Turing machines. In particular, alternating polynomial time is equivalent to deterministic polynomial space and alternating polynomial space is equivalent to deterministic 'exponential time. Subrecursive quantifier hierarchies are defined in terms of time- or space-bounded alternating Tufing machines by bounding the number of alternations allowed during computations. Alternating finite-state automata are defined and shown to accept only regular languages, although, in general, 2 2 states are necessary and sufficient to simulate a k-state alternating finite automaton deterministically. Finally, it is shown that alternating pushdown automata are strictly more powerful than nondeterministic pushdown automata.

### Evaluation of Different Solutions to the Left-Right-Neighbour and Priority Queue Problems (Extended Abstract)

, 1998

"... Abstract In this thesis we study two problems defined on elements drawn from a bounded universe of size M. The problems are: the problem of maintaining a dynamic set of elements, in order to answer queries about the left and right neighbour of an arbitrary element from the universe and the related P ..."

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Abstract In this thesis we study two problems defined on elements drawn from a bounded universe of size M. The problems are: the problem of maintaining a dynamic set of elements, in order to answer queries about the left and right neighbour of an arbitrary element from the universe and the related Priority Queue problem. Several different solutions, such as Split-Tagged-Tree, vEB Stratified Tree and heap, are compared, implemented and tested. We also introduce a simplification of the Split-Tagged-Tree and use it to solve the Priority Queue problem. In some of our solutions we use a model of computation called RAMBO which extends the RAM model with a special memory block. Under this model the simplified Split-Tagged-Tree uses 3M + O(lg M) bits and permits constant time operations on the Priority Queue. In comparison, under the same model the general Split-Tagged-Tree uses 5M + O(lg M) bits and permits queries of the Left-Right-Neighbour problem to be answered in constant time. The test results show that we can solve the Priority Queue problem efficiently using the simplified Split-Tagged-Tree and a special hardware implementing the RAMBO.

### Parallel Computation: Models and Complexity Issues

, 1996

"... This chapter is an introduction to the area of parallel computation written in accordance with the guidelines for the CRC HANDBOOK ON ALGORITHMS AND THEORY OF COMPUTATION where it will appear. References to chapters refer to other chapters in this book. This research partially supported by Nationa ..."

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This chapter is an introduction to the area of parallel computation written in accordance with the guidelines for the CRC HANDBOOK ON ALGORITHMS AND THEORY OF COMPUTATION where it will appear. References to chapters refer to other chapters in this book. This research partially supported by National Science Foundation grant CCR-9209184; a Fulbright Scholarship, Senior Research Award; and a Spanish Fellowship for Scientific and Technical Investigations. Address during 1995--96: Departament Llenguatges i Sistemes Inform`atics, Universitat Polit`ecnica de Catalunya, Pau Gargallo 5, 08028 Barcelona, Spain. y This research partially supported by the Natural Sciences and Engineering Research Council of Canada grant OGP 38937. 1 Introduction Parallel computation is the branch of computational complexity theory concerned with the development and analysis of parallel computing models and the techniques for solving and classifying problems on such models. Despite technology that continuous...

### Analysis Of Pram Instruction Sets From A Log Cost Perspective

"... The log cost measure has been viewed as a more reasonable method of measuring the time complexity of an algorithm than the unit cost measure. The more widely used unit cost measure becomes unrealistic if the algorithm handles extremely large integers. Parallel machines have not been examined under t ..."

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The log cost measure has been viewed as a more reasonable method of measuring the time complexity of an algorithm than the unit cost measure. The more widely used unit cost measure becomes unrealistic if the algorithm handles extremely large integers. Parallel machines have not been examined under the log cost measure. In this paper, we investigate the Parallel Random Access Machine under the log cost measure. Let the instruction set of a basic PRAM include addition, subtraction, and Boolean operations. We relate resource-bounded complexity classes of log cost PRAMs to complexity classes of Turing machines and circuits. We also relate log cost PRAMs with different instruction sets by simulations that are much more efficient than possible in the unit cost case. Let LCRCW k (CRCW k ) denote the class of languages accepted by a log cost (unit cost) basic CRCW PRAM in O(log k n) time with polynomial in n number of processors. We position the log cost PRAM in the hierarchy of paralle...

### unknown title

"... A model of computation based on random ac-cess machines operating in parallel and sharing a common memory is presented. The computational power of this model is related to that of tradi-tional models. In particular, deterministic par-allel RAM's can accept in polynomial time exactly the sets ac ..."

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A model of computation based on random ac-cess machines operating in parallel and sharing a common memory is presented. The computational power of this model is related to that of tradi-tional models. In particular, deterministic par-allel RAM's can accept in polynomial time exactly the sets accepted by polynomial tape bounded Turing machines; nondeterministic RAM's can ac-cept in polynomial time exactly the sets accepted by nondeterministic exponential time bounded Turing machines. Similar results hold for other classes. The effect of limiting the size of the common memory is also considered. The speed of serial computers has increased