Theory of Finite Trees Revisited : Application of Model - Theoretic Algebra ?
نویسنده
چکیده
The theory of nite trees in nite signature is axiomatized by the simple set of axioms E : 1. 8x x 6 = t(x) for every non-variable term t(x) containing x, 2. 8x 8y (f(x) = f(y) , x = y) for every f 2 , 3. 8x 8y (f(x) 6 = g(y)) for diierent f, g 2 , plus the usual equality axioms, plus the following Domain Closure Axiom: 8x _ f2 9z (x = f(z)) (DCA) postulating that every element of a model is in the range of some (perhaps 0-ary) function, i.e., there are no isolated elements. The theory E (DCA) has numerous applications in Automated deduction, Constraint solving, Uniication theory, Logic programming, Database theory. It was proved complete by Maher Mah88] using the straightforward quantiier elimination, and also by Lescanne and Comon CL89] by a direct transformational method. Earlier Kunen Kun87] proved that E (without (DCA)) is complete in the case of innnite signatures with constants (this case is much more simple), again by elimination of quantiiers, and also sketched the proof by using saturated models CK73]. Here we give a new indirect completeness proof of the theory E (DCA) for any nite signature using the elegant A. Robinson's model completeness test CK73, Kei77, Mac77]. The notion of model completeness is one of the central in the classical Model theory, and has numerous applications. We give applications of other model-theoretic methods, as categoricity in power and nonstandard models for completeness proofs. The aim of the paper consists also in promoting a nice logic into Computer Science applications.
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