Varieties of Formal Languages
Published on 1. October 2011
Book
Paperback/Softback
148 pages
978-1-4612-9300-2 (ISBN)
Description
The theory of finite automata and of rational languages could be likened to the ground floor of a huge building under construction which is theoretical computer science. The metaphor would indicate first that it can be entered on the ground level and secondly that it is more convenient to go through it in order to reach the higher levels. It is also the first purely mathematical theory to emerge from the needs and intuitions of computer science in the wider sense. In fact, at the end of the 1950s Kleene, who was intrigued by electronic models of the nervous system which were then very fashionable, proposed characterizing feasible calculations by means of a system making use of a single bounded memory. This led him to discover what are now called rational languages which are the subject of the present book by J. E. Pin. Subsequent work has revealed that this class is a particularly fundamental mathematical entity in the study of finite systems, for they appear quite naturally starting from considerations as diverse as those of restricted logical systems or the standard rational functions of analysis. From the start, one of the principal problems was found to be a problem of classification, or rather of hierarchization. J. Rhodes showed that the compo sition of automata preserved the associated groups and McNaughton discovered that the existence of non-trivial groups of this kind was intimately related to the presence of loops within the system of calculation. The development by S.
More details
Series
Edition
Softcover reprint of the original 1st ed. 1986
Language
English
Place of publication
NY
United States
Target group
Professional and scholarly
Research
Illustrations
148 p.
Dimensions
Height: 23.5 cm
Width: 15.5 cm
ISBN-13
978-1-4612-9300-2 (9781461293002)
DOI
10.1007/978-1-4613-2215-3
Schweitzer Classification
Other editions
Additional editions
Book
05/1986
Kluwer Academic / Plenum Publishers
€109.28
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Content
Relations.- 1. Semigroups, languages and automata.- 1. Semigroups..- 1.1. Semigroups, monoids, morphisms.- 1.2. Idempotents, zero, ideal.- 1.3. Congruences.- 1.4. Semigroups of transformations.- 1.5. Free semigroups.- 2. Languages.- 2.1. Words.- 2.2. Automata.- 2.3. Rational and recognizable languages.- 2.4. Syntactic monoids.- 2.5. Codes.- 2.6. The case of free semigroups.- 3. Explicit calculations.- 3.1. Syntactic semigroup of L = A*abaA* over the alphabet A = {a,b}.- 3.2. The syntactic monoid of L = {a2,aba,ba}* over the alphabet A = {a,b}.- Problems.- 2. Varieties.- 1. Varieties of semigroups and monoids.- 1.1. Definitions and examples.- 1.2. Equations of a variety.- 2. The variety theorem.- 3. Examples of varieties.- Problems.- Chapters 3. Structure of finite semigroups.- 1. Green's relations.- 2. Practical calculation.- 3. The Rees semigroup and the structure of regular D-classes.- 4. Varieties defined by Green's relations.- 5. Relational morphisms and V-morphisms.- Problems.- 4. Piecewise-testable languages and star-free Languages.- 1. Piecewise-testable languages; Simon's theorem.- 2. Star-free languages; Schützenberger's theorem.- 3. ?-trivial and ?-trivial languages.- Problems.- 5. Complementary results.- 1. Operations.- 1.1. Operations on languages.- 1.2. Operations on monoids.- 1.3. Operations on varieties.- 2. Concatenation hierarchies.- 2.1. Locally testable languages.- 2.2. General results on concatenation hierarchies.- 2.3. Straubing's hierarchy.- 2.4. Brzozowski's hierarchy.- 2.5. Connection between the hierarchies of Straubing and Brzozowski.- 2.6. The group-languages hierarchy.- 2.7. Hierarchies and symbolic logic.- 3. Relations with the theory of codes.- 3.1. Restriction of the operations star and plus.- 3.2. Varieties described by codes.- 3.3. Return to the operation V*W.- 4. Other results and problems.- 4.1. Congruences.- 4.2. The lattice of varieties.- Bibliographic notes.