On Equality Up-to Constraints over Finite Trees, Context Unification, and One-Step Rewriting

@inproceedings{Niehren1997OnEU,
  title={On Equality Up-to Constraints over Finite Trees, Context Unification, and One-Step Rewriting},
  author={Joachim Niehren and Manfred Pinkal and Peter Ruhrberg},
  booktitle={CADE},
  year={1997}
}
We introduce equality up-to constraints over finite trees and investigate their expressiveness. Equality up-to constraints subsume equality constraints, subtree constraints, and one-step rewriting constraints. We establish a close correspondence between equality up-to constraints over finite trees and context unification. Context unification subsumes string unification and is subsumed by linear second-order unification. We obtain the following three new results. The satisfiability problem of… 
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  • 2002
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References

SHOWING 1-10 OF 40 REFERENCES
Decidability of the Existential Theory of Infinite Terms with Subterm Relation
TLDR
It is proved that this problem of solving equations, inequalities, and atomic formulas built on the subterm relation in algebras of rational and infinite terms (trees) is decidable for any such algebra in a finite signature S with possible new free constants.
The First-Order Theory of One-Step Rewriting is Undecidable
TLDR
It is shown that there is no algorithm deciding the ∃*∀*-fragment of this first-order theory of one-step rewriting for an arbitrary rewrite system.
Linear Second-Order Unification
TLDR
This work presents a new class of second-order unification problems, which it is called linear, and describes a sound and complete procedure for this class of unification problems and proves termination for three different subcases of them.
Encompassment Properties and Automata with Constraints
We introduce a class of tree automata with constraints which gives an algebraic and algorithmic framework in order to extend the theorem of decidability of inductive reducibility. We use automata
Undecidability of the First Order Theory of One-Step Right Ground Rewriting
TLDR
The problem of decidability of the first order theory of one-step rewriting was stated in [CCD93] and in 1995 Ralf Treinen proved that the theory is undecidable.
A Unification Algorithm for Typed lambda-Calculus
  • G. Huet
  • Computer Science
    Theor. Comput. Sci.
  • 1975
A Complete Mechanization of Second-Order Type Theory
TLDR
A generalization of the resolution method for higher order logic is presented and it is established that the author's generalized resolution procedure is complete with respect to a natural notion of validity based on Henkin's general validity for type theory.
A Uniform Approach to Underspecification and Parallelism
We propose a unified framework in which to treat semantic underspecification and parallelism phenomena in discourse. The framework employs a constraint language that can express equality and subtree
Decidability of the purely existential fragment of the theory of term algebras
TLDR
It is established that the existential fragment of the theory of pure list structures in the language of NIL, CONS, CAR, CDR, =, ≤ (subexpression) is NP-complete.
Completion of Rewrite Systems with Membership Constraints
...
...