Michael J. Maher. Complete axiomatizations of the algebras of the finite, rational, and infinite trees. Proc. 3rd IEEE LICS, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the di erent possible signatures of the language. This analysis captures the relationship between the satis ability of negative constraints and the shape of the interpretation structure. In the context of a CLP system, uninterpreted function symbols are typically manipulated as nite trees [19]. We develop a rst order theory which extends (general) ACI1 and corresponds to T( ACI1 i.e. the Herbrand Universe modulo the congruence relation imposed by ACI1 on the class of conjunctions of positive and negative constraints. This allows us to focus on the canonical domain of Herbrand ....

M. J. Maher. Complete Axiomatizations of the Algebras of Finite, Rational and In nite Trees. In Proc. 3 rd Symp. LICS (1988), 349-357.

....shown by noticing that in a type language with only one constant (i.e. the subtype relation is the same as equality. Thus we can simply turn any constraint t1 t2 into t1 = t2 . Since the first order theory of equality is decidable both for finite and regular trees (and infinite trees) [21], the theorem follows immediately. # 5 Decidability of FOT with Unary Symbols In this section, we show that if we restrict our type language to unary function symbols and constants, the first order theory is decidable. This result shows that the di#culty in the whole first order theory lies in ....

....There is also related work in term rewriting and constraint solving over trees in general [8, 10] The work in this paper is inspired by work in this area. Maher shows the first order theory of finite trees, infinite trees, and rational trees is decidable by giving a complete axiomatization [21]. Many researchers consider various order relations among trees, similar to the subtype orders. Venkataraman study the first order theory of subterm ordering over finite trees. The existential fragment is shown to be NP complete and to be undecidable [43] Muller et al. study the first order ....

M.J. Maher. Complete axiomatizations of the algebras of the finite, rational and infinite trees. In Proceedings of the Third IEEE Symposium on Logic in Computer Science, pages 348--357, Edinburgh, UK, 1988. Computer Society Press.

....is understood that OE and may have free variables. Given a constraint system, a constraint OE is called satisfiable if OE 6j= Delta (i.e. there is at least one model of Delta in which OE is satisfiable) For examples we will use the finite tree constraint system H (often called Herbrand) [16, 19] underlying conventional logic programming. The signature of H consists of infinitely many function symbols for every arity, and the theory of H (known as Clark s Equality Theory) is given by the schemes: f(y) x =y =g(y) f 6= g) x =f( Delta Delta Delta x Delta Delta Delta) ....

Michael J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite trees. In Proceedings of the 3rd Annual Symposium on Logic in Computer Science, pages 348--457, Edinburgh, Scotland, July 1988.

.... FT is complete, all three models are elementarily equivalent (i.e. satisfy exactly the same first order formulae) While the feature graph model captures intuitions common in linguistically motivated investigations, the feature tree model provides the connection to the tree constraint systems [9, 10, 16, 17] employed in logic programming. Our proof of FT s completeness will exhibit a simplification algorithm that computes for every feature description an equivalent solved form from which the solutions of the description can be read of easily. For a closed feature description the solved form is ....

....of the proof of Theorem 8.3 in [20] one can show that FT 0 is undecidable. 4 Outline of the Completeness Proof The completeness of FT will be shown by exhibiting a simplification algorithm for FT. The following lemma gives the overall structure of the algorithm, which is the same as in Maher s [17] completeness proof for the theory of constructor trees. Lemma 4.1 Suppose there exists a set of so called prime formulae such that: 1. every sort constraint Ax, every feature constraint xfy, and every equation x = y such that x 6= y is a prime formula 2. is a prime formula, and there is no ....

M. J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite trees. In Proceedings of the 3rd Annual Symposium on Logic in Computer Science, pages 348--457, Edinburgh, Scotland, July 1988.

.... used to show decidability and classification of boolean algebras [40, 44] Presburger arithmetic [36] decidability of products [30, 13] 28, Chapter 12] and algebraically closed fields [43] Directly relevant to our work are quantifierelimination techniques for term algebras [28, Chapter 23] [27, 41]. Several extensions of term algebras have been shown decidable using quantifier elimination. 9] gives a terminating term rewriting system for quantifier elimination in term algebras with membership constraints, 38] gives quantifier elimination for term algebras with queues, 6] presents ....

M. J. Maher. Complete axiomatizations of the algebras of the finite, rational, and infinite trees. Proc. 3rd IEEE LICS, 1988.

....step in solving the problem described above is to build an appropriate algebraic framework. Such a framework is provided by universal algebra in the case of first order trees. Formally, these are the elements of the free algebra over a given signature of function symbols (finite or infinite, cf. [Mah88]) This framework yields immediately a good notion of automata. In fact, as Courcelle has shown in [Cou89, Cou92] universal algebra provides a framework for a rich variety of trees. Clearly, it is that work that inspired our notion of the algebra underlying feature trees. We introduce this ....

Michael J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite trees. In LICS, pages 348--457, July 1988.

....of set inclusion (t 1 =t 2 t 1 t 2 t 2 t 1 ) Actually, even the firstorder theories of equality constraints over trees and of equality constraints over non empty sets of trees coincide. This follows from the complete axiomatization of the first order theory of equality constraints over trees [18, 19, 12] since its axioms also hold over non empty sets of trees (but don t over possibly empty sets) There exists a natural interpretation of INES constraint over tree like structures that we call tree prefixes. In a different context [6] tree prefixes are called Bohm trees (without binders) Tree ....

....sets of trees, over tree prefixes, and over trees coincide (i.e. of the structures P (Tree) Prefix and Tree) Independently, A. Colmerauer observed this for P (Tree) and Tree (pers. comm. 10 Proof. This follows from the fact that all axioms of the complete axiomatization of trees [18, 19, 12] are valid for non empty sets of trees. This holds for the axioms of the form 8y9 x(x 1 =f 1 (x y) xn=fn (x y) Validity of the other axioms is immediate since they are already contained in A with inclusion replaced for equality. 2 In contrast, first order formulae over inclusion ....

M. J. Maher. Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees. In Proc. 3 LICS, pp. 348--457. IEEE, 1988.

....be recursively enumerable. 2 A related problem is the satisfiability problem for the feature graph algebra F : Is it decidable whether a constraint has a solution in F We conjecture that the satisfiability 33 problem for F is decidable. Evidence for this conjecture comes from recent results [7, 29, 30] showing that related problems for the ground term algebra and the algebra of rational trees are decidable. Next we show that coherence of feature terms with respect to models of recursive sort equations is undecidable. Lemma 8.4 Let B be a sort, b and c be two distinct atoms, and f 1 , f 2 and ....

M. J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite trees. In Proceedings of the 3rd Annual Symposium on Logic in Computer Science, pages 348--457, Edinburgh, Scotland, July 1988.

....be recursively enumerable. 2 A related problem is the satisfiability problem for the feature graph algebra F : Is it decidable whether a constraint has a solution in F We conjecture that the satisfiability 33 problem for F is decidable. Evidence for this conjecture comes from recent results [7, 29, 30] showing that related problems for the ground term algebra and the algebra of rational trees are decidable. Next we show that coherence of feature terms with respect to models of recursive sort equations is undecidable. Lemma 8.4 Let B be a sort, b and c be two distinct atoms, and f 1 , f 2 and ....

M. J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite trees. Research report, IBM, Thomas J. Watson Research Center, P.O. Box 704, Yorktown Heights, NY 10598, U.S.A., 1988.

....system, and again apply the memorization technique. Entailment tests for feature constraints, which refine equational constraints for infinite trees, have been treated in [ST92a, AKPS92] In most of those contexts rational and infinite trees can not be distinguished by means of logical formulae [BS92, Mah88]. Membership constraints over sets of finite trees have been considered in [CD91, Uri92] The case of finite feature trees is discussed in [NP93] In these works (generalized) tree automata or regular equation systems with least fixpoint solutions are used. The proposed simplification algorithms ....

....again for the entailment check. Proposition 4. 2 If OE is in normal form, then every assignment of the non constrained variables of OE can be extended to a solution of OE in M: 89C(OE)OE : The proof reduces immediately to the case of equational constraints only, which has been solved in [Mah88]. Normal forms of equational constraints can be obtained by the well known unification rules in Figure 2. We obtain normal forms of arbitrary constraints in four steps. First we calculate a normal form j of the equational part. Second we apply to the membership part and call the result . Third ....

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Michael J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite trees. In Proceedings of the Third Annual Symposium on Logic in Computer Science, pages 348--357. IEEE Computer Society, 1988.

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Michael J. Maher. Complete axiomatizations of the algebras of the finite, rational, and infinite trees. Proc. 3rd IEEE LICS, 1988.

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M. J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite tree. In Proceedings of the 3rd IEEE Symposium on Logic in Computer Science, pages 348--357. IEEE Computer Society Press, 1988.

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M. J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite tree. In Proceedings of the Third Annual Symposium on Logic in Computer Science, pages 348--357. IEEE Computer Society Press, 1988.

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M. J. Maher. Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees. In Proc. Third IEEE Symp. on Logic In Computer Science, pages 348--357. Computer Science Press, New York, 1988.

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M. J. Maher. Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees. In Proc. Third IEEE Symp. on Logic In Computer Science, pages 348--357. Computer Science Press, New York, 1988.

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M. J. Maher. Complete axiomatizations of the algebras of finite, rational and infinite trees. In Proceedings of the 3rd Annual Symposium on Logic in Computer Science, pages 348--359. Computer Society Press, 1988.

No context found.

M. J. Maher. Complete axiomatizations of the algebras of the finite, rational, and infinite trees. Proc. 3rd IEEE LICS, 1988.

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M. J. Maher, Complete axiomatizations of the algebras of finite, rational and infinite trees, in: Proceedings, Third Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, Edinburgh, Scotland, 1988, pp. 348-- 357.

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M. J. Maher, Complete axiomatizations of the algebras of finite, rational and infinite trees, in: Proceedings, Third Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, Edinburgh, Scotland, 1988, pp. 348-- 357.

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M. J. Maher, Complete axiomatizations of the algebras of finite, rational and infinite trees, in: Proceedings, Third Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, Edinburgh, Scotland, 1988, pp. 348-- 357.

No context found.

M. J. Maher, Complete axiomatizations of the algebras of finite, rational and infinite trees, in: Proceedings, Third Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, Edinburgh, Scotland, 1988, pp. 348-- 357.

No context found.

M. J. Maher. Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees. In Proceedings of 3rd Symposium Logic in Computer Science (1988), 349--357.

No context found.

M. J. Maher. Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees. In Proc. 3 rd Symp. LICS (1988), 349--357.

No context found.

M.J. Maher. Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees. In Proc. of Third IEEE Symp. on Logic In Computer Science, pages 348--357. Computer Science Press, New York, 1988.

No context found.

M.J. Maher. Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees. In Proc. of Third IEEE Syrup. on Logic In Computer Science, pages 348-357. Computer Science Press, New York, 1988.

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