S. Mitra. Semantic unification for convergent systems. PhD thesis, University of Illinois at Urbana-Champaign, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is finitary i.e. there always exists a finite complete set of unifiers. Most decidability proofs are thus based on the fact that there exists a complete narrowing strategy whose search space is always finite. As concerns non finitary theories, a decidability result is established by S. Mitra in [23, 24] for constructor based rewrite systems, assuming that for every function symbol f there is at most one rewrite rule among the rules defining f , that does not have a data term as rhs. Moreover this rhs must contain only one function symbol and the subterm rooted by this function is flat in the ....

S. Mitra. Semantic Unification for Convergent System. Technical Report CS-R-941855, University of Illinois at Urbana-Champaign, 1994.

.... s # 1 , s m = s # m , L 1 , L n , #) Mutate: f(s 1 , s m ) t, L 1 , L n , #) # (s 1 = l 1 , s m = l m , r = t, #) where f(l 1 , l m ) r is a renamed rule in R. 6 The above rules are a subset of the transformation rules used in [DS87, MMR89, Mit94] for semantic unification. They yield a complete forwarddecomposition calculus for solving goals of the form s = N , where N is in ground normal form, provided that the underlying TRS is convergent and either variable preserving or left linear (see [DMS92] In the next section, we present a ....

S. Mitra. Semantic unification for convergent systems. PhD thesis, University of Illinois, 1994.

....theory is finitary, i.e. there always exists a finite complete set of unifiers. Most decidability proofs are thus based on the fact that there exists a complete narrowing strategy whose search space is always finite. As regards non finitary theories, a decidability result is established by Mitra [16, 21] for constructorbased rewrite systems, assuming that, for every function symbol f , there is at most one rewrite rule among the rules defining f that does not have a data term as the rhs. Moreover, this rhs must contain only one function symbol, and the subterm rooted by this function is flat in ....

Mitra, S. (1994). Semantic Unification for Convergent System. Technical Report CS-R-94-1855, University of Illinois at Urbana-Champaign.

....theory is finitary, i.e. there always exists a finite complete set of unifiers. Most decidability proofs are thus based on the fact that there exists a complete narrowing strategy whose search space is always finite. As regards non finitary theories, a decidability result is established by Mitra [16, 21] for constructorbased rewrite systems, assuming that, for every function symbol f , there is at most one rewrite rule among the rules defining f that does not have a data term as the rhs. Moreover, this rhs must contain only one function symbol, and the subterm rooted by this function is flat in ....

Mitra, S. (1994). Semantic Unification for Convergent System. Technical Report CS-R-94-1855, University of Illinois at Urbana-Champaign.

....: s m = s 0 m ; L 1 ; L n ; oe) Mutate: f(s 1 ; s m ) t; L 1 ; L n ; oe) s 1 = l 1 ; s m = l m ; r = t; oe) where f(l 1 ; l m ) r is a renamed rule in R. The above rules are a subset of the transformation rules which were used in [DS87, MMR89, Mit94] for semantic unification. They yield a complete forward decomposition calculus for solving goals of the form s = N , where N is in ground normal form, provided that the underlying TRS is convergent and either variable preserving or left linear [DMS92] In the next section, we will present a ....

S. Mitra. Semantic unification for convergent systems. PhD thesis, University of Illinois, 1994.

.... particular, restricted, confluent string rewriting systems [OND95] shallow theories [CHJ94] that is, particular trs s, in the rules of which variables may only occur at depth at most one. particular, restricted convergent (that is, terminating and confluent) trs s, as considered in [Hul80, KN87, Mit94] The reference which is most relevant and also closest related to our results, is [Mit94] There, decidability results for three different classes of convergent trs s are given: The E unification problem is decidable, if E is induced by 1. a convergent trs, in which the right hand side of each ....

....that is, particular trs s, in the rules of which variables may only occur at depth at most one. particular, restricted convergent (that is, terminating and confluent) trs s, as considered in [Hul80, KN87, Mit94] The reference which is most relevant and also closest related to our results, is [Mit94]. There, decidability results for three different classes of convergent trs s are given: The E unification problem is decidable, if E is induced by 1. a convergent trs, in which the right hand side of each rule is either a variable, a ground term, or a constructor only term. This is an ....

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S. Mitra. Semantic unification for convergent systems. Technical Report CS-R-94-1855, University of Illinois at Urbana--Champaign, 1994.

....of shallow theories [CHJ94] A shallow theory is induced by a TRS that is R restricted in the following way: in the right hand sides of the rules of R, variables may occur only at depth at most 1. 2. Particular, restricted convergent (that is, terminating and confluent) TRSs, as considered in [Hul80, KN87, Mit94]. 3. Particular, restricted, confluent, constructor based, and linear TRSs [LR96] 4. Finite, length reducing, and confluent Thue systems [NO90] Let us now elaborate on these results and show that the induced classes of equational theories are incomparable with the class of TRSs for which we ....

....there are shallow theories that are not in E mon Hom . On the other hand, monadic tree homomorphisms allow rules of the form s # t, such that s and t have common nonshallow variables. 4.1. 2 Results by Mitra The reference that is most relevant and also most closely related to our results is [Mit94]. There, decidability results for three di#erent classes of convergent TRSs are given: the E unification problem is decidable if E is induced by: 1. A convergent TRS, in which the right hand side of each rule is either a variable, a ground term, or a constructor only term. This is an extension of ....

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S. Mitra. Semantic unification for convergent systems. Technical Report CS-R-94-1855, University of Illinois at UrbanaChampaign, 1994.

....[Nie96] which are an extension of shallow theories [CHJ94] Shallow theories are equational theories induced by particular trs s, in the rules of which variables may only occur at depth at most one. 2. particular, restricted convergent (that is, terminating and confluent) trs s, as considered in [Hul80, KN87, Mit94] 3. finite, length reducing, and confluent Thue systems [NO90] 4. particular, restricted, confluent, constructor based, and linear trs s [LR96] Let us now elaborate on these results and show that the induced classes of equational theories are incomparable with the class of trs s for which we ....

....there are shallow theories which are not in E mon Hom . On the other hand, monadic tree homomorphisms allow rules of the form s t, such that s and t have common non shallow variables. 12 4. 2 Results by Mitras The reference which is most relevant and also closest related to our results, is [Mit94]. There, decidability results for three different classes of convergent trs s are given: The E unification problem is decidable, if E is induced by (A) a convergent trs, in which the right hand side of each rule is either a variable, a ground term, or a constructor only term. This is an ....

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S. Mitra. Semantic unification for convergent systems. Technical Report CSR -94-1855, University of Illinois at Urbana--Champaign, 1994.

.... rule in R This forward decomposition calculus is a complete procedure for solving goals s = N , where N is in ground normal form, and the underlying TRS is canonical and variable preserving or left linear [DMS92] The above rules are a subset of the transformation rules which were used in [DS87, MMR89, Mit90, Mit94] for semantic unification. However, this approach does not explicitly exploit the fact that the right hand side of an initial goal has to be in ground normal form. In the next section we will present a calculus which takes more advantage of this since it starts with the result on the right hand ....

S. Mitra. Semantic Unification for Convergent Systems. PhD thesis, University of Illinois, 1994.

....is finitary i.e. there always exists a finite complete set of unifiers. Most decidability proofs are thus based on the fact that there exists a complete narrowing strategy whose search space is always finite. As concerns non finitary theories, a decidability result is established by S. Mitra in [23, 24] for constructor based rewrite systems, assuming that for every function symbol f there is at most one rewrite rule among the rules defining f , that does not have a data term as rhs. Moreover this rhs must contain only one E Unification by Means of Tree Tuple Synchronized Grammars 5 function ....

S. Mitra. Semantic Unification for Convergent System. Technical Report CS-R-94-1855, University of Illinois at Urbana-Champaign, 1994.

.... 1980] ffl Every non ground right side is a constructor term [Dershowitz et al. 1992] ffl Every non ground right side is a proper subterm of its left side [Narendran, Pfenning and Statman, 1997] ffl Every non ground right side is either a constructor term or a proper subterm of its left side [Mitra, 1994]. ffl Every right side is composed of constructors and proper subterms of its left side [Mitra, 1994] ffl All variables are shallow on the left side [Christian, 1992] ffl The system is linear and every variable that appears on both sides is shallow on both sides (convergence is unnecessary) ....

.... right side is a proper subterm of its left side [Narendran, Pfenning and Statman, 1997] ffl Every non ground right side is either a constructor term or a proper subterm of its left side [Mitra, 1994] ffl Every right side is composed of constructors and proper subterms of its left side [Mitra, 1994]. ffl All variables are shallow on the left side [Christian, 1992] ffl The system is linear and every variable that appears on both sides is shallow on both sides (convergence is unnecessary) Nieuwenhuis, 1998] ffl The system is linear and the right side of every f rule is either a ....

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Mitra, S.: Semantic unification for convergent systems. Ph.D. thesis, Dept. of Computer Science, University of Illinois, Urbana, IL, Tech. Rep. UIUCDCS-R-94-1855 (1994).

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S. Mitra. Semantic unification for convergent systems. PhD thesis, University of Illinois at Urbana-Champaign, 1994.

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