# Computability, Complexity and Languages: Fundamentals of Theoretical Computer Science (Computer Science and Applied Mathematics)

## Martin Davis

Language: English

Pages: 425

ISBN: 0122063805

Format: PDF / Kindle (mobi) / ePub

This introductory text covers the key areas of computer science, including recursive function theory, formal languages, and automata. It assumes a minimal background in formal mathematics. The book is divided into five parts: Computability, Grammars and Automata, Logic, Complexity, and Unsolvability.

* Computability theory is introduced in a manner that makes maximum use of previous programming experience, including a "universal" program that takes up less than a page.

* The number of exercises included has more than tripled.

* Automata theory, computational logic, and complexity theory are presented in a flexible manner, and can be covered in a variety of different arrangements.

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occurrences of w as a part of u [e.g., #{bab,ababab) = 2]. Also, let # ( 0 , L 0 = 0. Prove that #(u,v) is primitive recursive. 7. Show that UPCHANGE„ , and DOWNCHANGE„ , are primitive recursive. 8. For n > 2, show that when u is calculated with respect to base n notation, u < Llog/? u\ + 1 for all u e N. 2. A Programming Language for String Computations From the point of view of string computations, the language 6? seems quite artificial. For example, the instruction V <- V + 1 which is so

that are computable in S^n for every n. The general results in the next section will 124 Chapter 5 Calculations on Strings Y " J END % 1 ' 1X 0 ends sf X TEST X BEGIN Carry propayates L i *'' /? X Y - xiY ends sn ' X - X" Y * ST Y * TEST X x - 0 * END I x ends s, t i X - X" Y - SY / Figure 2.2. Flow chart for computing x + 1 in ^ . make it clear that these two examples are the only bit of programming in <9"n that we shall need to carry out explicitly. We want to

programming system. [See Exer cise 5.4 in Chapter 4 for the definition of acceptable programming systems.] 4.* Give an upper bound on the length of the shortest S?x program which computes the function 4>V(JC) defined in Chapter 4. [See Exercise 3.6 in Chapter 4.] 129 4. Post-Turing Programs 4. Post-Turing Programs In this section, we will study yet another programming language for string manipulation, the Post-Turing language ^ Unlike ò^n , the language F has no variables. All of the

simulated easily by quintuples. But a quadruple requiring a "print" necessitates using a quintuple which causes a motion after the "print" has taken place. The final list of quintuples undoes the effect of this unwanted motion. The extra states qK+ x,..., q2K serve to "remember" that we have gone a square too far to the right. ■ Finally, we will complete another circle by proving Theorem 1.4. Any partial function that can be computed by a quintuple Turing machine can be computed by a Post-Turing