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Complexity and decidability. (Complexité et décidabilité.) (French) Zbl 0787.03030
Mathématiques & Applications (Berlin). 12. Paris: Springer-Verlag. 200 p. (1993).
This book is the supporting text of the course taught by the author at the University of Caen. It includes main results in complexity theory concerning first-order logic. The author emphasizes on the main negative results concerning the complexity of Boolean calculus, first-order logic and of arithmetic. The theory is presented in the standard model of Turing machines. The model is largely studied in the first six chapters of the book. Thus, the reader may not be familiar with complexity and even logic theories. We note the elegant manner the author succeeds in giving a rigorous and at the same time easy to understand presentation of some difficult results.
Here are the chapters of the book: 1. Algorithmes et machines de Turing, 2. Simulation d’algorithmes, 3. Changement de réprésentation, 4. Fonctions récursives, 5. Machines universelles, 6. Machines non déterministes, 7. Complexité du calcul booléen, 8. Complexité des logiques du premier ordre, 9. Complexité de l’arithmétique, 10. Complexité de l’addition des entiers.
Reviewer: D.Lucanu (Iaşi)

03D15 Complexity of computation (including implicit computational complexity)
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science
03D10 Turing machines and related notions
03B25 Decidability of theories and sets of sentences
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
03D20 Recursive functions and relations, subrecursive hierarchies
03F30 First-order arithmetic and fragments
68Q10 Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.)