Издательство Springer, 1979, -251 pp.

Since the early part of this century, logicians have been interested in decision procedures for theories in the predicate calculus. By a theory, we mean the set of sentences true about a particular structure, or the set of sentences true about every structure in a particular class of structures. One of the first decision procedures discovered was Lowenheim's procedure for validity of sentences in the monadic predicate calculus. Another early example is Presburger's procedure for the first-order theory of the integers under addition.

Although some people felt that some theories might be undecidable, it wasn't until the thirties that it was possible to carefully state, let alone prove, such conjectures. It was first necessary to have formal notions of computation, as provided by the work of people such as Church, Kleene, Post and Turing. The first undecidabi1ity result was G?del's famous theorem that the first- order theory of the integers under the operations of addition and multiplication is not decidable. At the present time an enormous number of decidability and undecidabi1ity results are known; [ELTT65] is a good survey of these results.

For a long time, it was felt that one could say nothing stronger about a theory than that it was decidable. Although the concept of comparing the difficulty of undecidable (or nonrecursive) sets has existed for a long time, it was only recently that people have started investigating the relative decidability of recursive sets. It was in the sixties that the field of computational complexity theory began. The goal was to look at a recursive set and ask how much of a particular resource (usually time or storage) was needed to decide the membership problem for that set. It was necessary of course to have a precise model of computation to work with. Many different models have been defined today, but we shall use the Turing machine because of its elegance and simplicity; most of the following theorems also hold for most other existing models of computation. Time and space on a Turing machine will be defined in a straightforward way in Chapter one. The time or space used for a particular machine will be expressed as a function of the length of input: a machine operates within time (or space) f(n) if on all inputs of length n which are accepted, the time (or space) used is at most f(n) .

With this model, one can now look at previously existing algorithms (for instance decision procedures for logical theories) and ask how much time or space they use. One can use this model as a basis for comparing these algorithms, or for inventing new ones which are better in the very precise sense of using less time or space. One of the first of these upper bound results was Ferrante and Rackoff's theorem that the first order theory of the real numbers under addition can be decided by an algorithm operating in exponential space.

However, this model also gives us a precise way of saying that a particular set or theory requi res a certain amount of time or space. Here one is saying not only that a particular algorithm uses a lot of the resource, but that any algorithm for the membership problem must use a lot of the resource in question. These are called lower bound results. The techniques for showing that certain decidable sets require (for any algorithm) a lot of time or space to be decided, parallel earlier results showing that certain sets cannot be decided at all (by any algorithm). Whereas undecidabi1ity is usually proven by "arithmetization" of Turing machines, lower bounds are usually proven by efficient "arithmetization" of time or space bounded Turing machines, an idea having its beginnings in a paper by Stephen Cook. The first lower bound result for logical theories was Albert Meyer's theorem that the second order theory of the integers under successor requires an enormous (non-elementary recursive) amount of time and space. This was done in 1973.

As with decidability and undecidabi1ity results, there exist today many different upper and lower complexity bounds for many different types of sets, and in particular for many different logical theories. The purpose of this work is to present certain of the theorems and proofs about complexity bounds for logical theories of the first-order predicate calculus. Although the subjects chosen are those that the authors have personally worked on, the proof methods are typical of those commonly used in this area.

Precise definitions of the concepts discussed here are presented in Chapter one. Chapters two through five contain upper bound results, and Chapters six through nine contain lower bound results.

The authors would like to thank Albert Meyer for his help and encouragement, both during their studies and since. Most of these results would not have been obtained without his ideas and suggestions. The authors are also grateful to Teresa Miao for the typing of this manuscript. While at MIT, the authors were supported by NSF grant GJ-

34671. The first author would also like to thank the Tufts University Faculty Awards Committee. The second author thanks the Department of Computer Science of the University of Toronto, as well as the National Research Council of Canada.

Introduction and Background.

Ehrenfeucht Games and Decision Procedures.

Integer Addition - An Example of an Ehrenfeucht Game Decision Procedure.

Some Additional Upper Bounds.

Direct Products of Theories.

Lower Bound Preliminaries.

A Technique for Writing Short Formulas Defining Complicated Properties.

A Lower Bound on the Theories of Pairing Functions.

Some Additional Lower Bounds.

# Ferrante J., Rackoff C.W. The Computational Complexity of Logical Theories

## Комментарии

## Смотрите также

## Lamagna E.A. The Complexity of Monotone Functions

разное
Диссертация, Brown University, 1975, -110 pp.

An important open question in the field of computational complexity is the development of nontrivial lower bounds on the number of logical operations required to compute switching functions. Two important measures of functional complexity are combinational complexity, corres...

An important open question in the field of computational complexity is the development of nontrivial lower bounds on the number of logical operations required to compute switching functions. Two important measures of functional complexity are combinational complexity, corres...

15.12.2011 в 08:44
1.24 МБ
3 раза

## Papadimitriou C.M. Computational Complexity

разное
Издательство Addison-Wesley, 1994, -540 pp.

This book is an introduction to the theory of computational complexity at a level appropriate for a beginning graduate or advanced undergraduate course. Computational complexity is the area of computer science that contemplates the reasons why some problems are so hard to solve by ...

This book is an introduction to the theory of computational complexity at a level appropriate for a beginning graduate or advanced undergraduate course. Computational complexity is the area of computer science that contemplates the reasons why some problems are so hard to solve by ...

28.10.2011 в 01:39
4.5 МБ
10 раз

## Savage J.E. Models of Computation. Exploring the Power of Computing

разное
Издательство Addison-Wesley, 1998, -699 pp.

Theoretical computer science treats any computational subject for which a good model can be

created. Research on formal models of computation was initiated in the 1930s and 1940s by

Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages,...

Theoretical computer science treats any computational subject for which a good model can be

created. Research on formal models of computation was initiated in the 1930s and 1940s by

Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages,...

28.10.2011 в 09:46
4.2 МБ
24 раза

## Jukna S. Boolean Function Complexity. Advances and Frontiers

разное
Издательство Springer, 2011, -633 pp.

Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterlo...

Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterlo...

24.10.2011 в 23:18
2.98 МБ
10 раз

## Paterson M.S. (ed.) Boolean Function Complexity

разное
Издательство Cambridge University Press, 1992, -211 pp.

Complexity theory attempts to understand and measure the intrinsic difficulty of computational tasks. The study of Boolean Function Complexity reaches for the combinatorial origins of these difficulties. The field was pioneered in the 1950's by Shannon, Lupanov and...

Complexity theory attempts to understand and measure the intrinsic difficulty of computational tasks. The study of Boolean Function Complexity reaches for the combinatorial origins of these difficulties. The field was pioneered in the 1950's by Shannon, Lupanov and...

28.10.2011 в 01:39
1.16 МБ
3 раза

## Creignou N., Kolaitis P.G., Vollmer H. (eds.) Complexity of Constraints. An Overview of Current Research Themes

разное
Издательство Springer, 2008, -326 pp.

In October 2006, the editors of this volume organized a Dagstuhl Seminar on Complexity of Constraints at the Schloss Dagstuhl Leibniz Center for Informatics inWadern, Germany. This event consisted of both invited and contributed talks by some of the approximately 40 participants, as...

In October 2006, the editors of this volume organized a Dagstuhl Seminar on Complexity of Constraints at the Schloss Dagstuhl Leibniz Center for Informatics inWadern, Germany. This event consisted of both invited and contributed talks by some of the approximately 40 participants, as...

17.12.2011 в 10:43
2.41 МБ
5 раз

## Buss S. Circuit Complexity and Computational Complexity

разное
University of California, 1992, -154 pp.

These lecture notes were written for a topics course in the Mathematics Department at the University of California, San Diego during the winter and spring quarters of 1992.

Introduction to circuit complexity.

Theorems of Shannon and Lupanov giving upper and lower bounds of cir...

These lecture notes were written for a topics course in the Mathematics Department at the University of California, San Diego during the winter and spring quarters of 1992.

Introduction to circuit complexity.

Theorems of Shannon and Lupanov giving upper and lower bounds of cir...

24.10.2011 в 23:03
573.37 КБ
6 раз

## Edelsbrunner H. Algorithms in Combinatorial Geometry

разное
Издательство Springer, 1987, -439 pp.

Computational geometry as an area of research in its own right emerged in the early seventies of this century. Right from the beginning, it was obvious that strong connections of various kinds exist to questions studied in the considerably older field of combinatorial geometry. For ...

Computational geometry as an area of research in its own right emerged in the early seventies of this century. Right from the beginning, it was obvious that strong connections of various kinds exist to questions studied in the considerably older field of combinatorial geometry. For ...

15.10.2011 в 20:13
4.52 МБ
11 раз

## Vollmer H. Introduction to Circuit Complexity. A Uniform Approach

разное
Издательство Springer, 1995, -287 pp.

This introductory textbook presents an algorithmic and computability based approach to circuit complexity. Intertwined with the consideration of practical examples and the design of efficient circuits for these, a lot of care is spent on the formal development of the computation mod...

This introductory textbook presents an algorithmic and computability based approach to circuit complexity. Intertwined with the consideration of practical examples and the design of efficient circuits for these, a lot of care is spent on the formal development of the computation mod...

31.01.2012 в 00:59
7.78 МБ
22 раза

## Singh A. Elements of Computation Theory

разное
Издательство Springer, 2009, -428 pp.

The foundation of computer science is built upon the following questions:

What is an algorithm?

What can be computed and what cannot be computed?

What does it mean for a function to be computable?

How does computational power depend upon programming constructs?

The foundation of computer science is built upon the following questions:

What is an algorithm?

What can be computed and what cannot be computed?

What does it mean for a function to be computable?

How does computational power depend upon programming constructs?

08.12.2011 в 20:58
4.5 МБ
15 раз