Book Reviews A Proof Theory for General Unification . By Wayne Snyder . Birkhauser, 1991 . vi+17 5 pages . ISBN 0-8176-3593-9 . $28 .00 . Snyder's book A Proof Theory for General Unification focuses on two important topics o f equational proof theory, unification in arbitrary equational theories and higher-order unification . Unification in specific theories is not part of the monograph . The emphasis is on the descriptio n and the formal justification of computing complete sets of unifiers by means of transformatio n rules, a technique originating from Herbrand and revived much later by Montanari, Colmerauer , Kirchner and others . The book is well written, rather pleasant to read, and surely helpful fo r those who want to know more on this subject . Although it is self contained, a good backgroun d in the area of logic programming and/or automated deduction is recommended . Applicatio n areas are mentioned in the introduction . An application to logic programming or automate d theorem proving in the form of a system description would have been helpful for the averag e reader. Chapter 2 presents the problems and gives ideas on the techniques used . Chapter 3 gives the necessary algebraic and logical background . References to recent surveys, by Dershowit z and Jouannaud, or by Klop, would have been helpful to the reader willing to know more in th e area of equational logic . Chapter 4 introduces the problem of E-unification, complete sets o f E-unifiers . It then describes narrowing and proves its completeness when E is given in the for m of a convergent set of rewrite rules . This subsection is strongly inspired by Hullot 's work, datin g from 1980 . Chapter 5 addresses the problem of enumerating a complete set of E-unifiers for a n arbitrary E. To this end, a set of transformation rules is given which is an improvement ove r the British Museum method by enumerating a small subset of the set of all proofs . Although this method is still highly inefficient, its completeness proof is highly non-trivial and conducte d precisely. Chapter 6 introduces a second, improved set of transformation rules, and proves agai n its completeness by using ground completion techniques . Chapter 7 gives a treatment of higher order unification in terms of transformation rules . Again, two sets of transformations are given . The first, for computing higher-order unifiers, may yield a search tree infinitely branching a t some nodes . The second avoids this difficulty by adopting the notion of solved form introduced by Huet under the name of "pre-unifiers" . Chapter 8 concludes the monograph and reviews briefly recent results in the area . Jean-Pierre Jouannau d Universite de ParisâSu d Orsay, France
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Book Review: A Proof Theory for General Unification. By Wayne Snyder. (Birkhauser, 1991. vi+175 pages. ISBN 0-8176-3593-9. $28.00)
Reviewer-Jouannaud, Jean-Pierre
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, Volume 25 (2)
Association for Computing Machinery – Jun 1, 1994
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