A Constructor-Based Approach for Positive/Negative-Conditional Equational Specifications

@inproceedings{Wirth1992ACA,
  title={A Constructor-Based Approach for Positive/Negative-Conditional Equational Specifications},
  author={Claus-Peter Wirth and Bernhard Gramlich},
  booktitle={CTRS},
  year={1992}
}
We present a constructor-based approach for assigning appropriate semantics to algebraic specifications given by finite sets of positive/negative-conditional equations. Under the assumption of confluence of the reduction relation we define, the factor algebra of the ground term algebra modulo the congruence of this reduction relation is a minimal model which is (beyond that) the minimum of all models that do not identify more constructor ground terms than necessary. 
A Constructor-Based Approach to Positive/Negative-Conditional Equational Specifications
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This work presents a constructor-based approach for assigning appropriate semantics to algebraic specifications given by finite sets R of positive/negative-conditional equations with universally quantified first-order implications with a single equation in the succedent and a conjunction of positive and negative equations.
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Based on the presented macroADruleADconstruct, positive/negativeAD Conditional equational specifiADcations can be written on a higher level because a rewrite system translates the macroAD ruleADconstructsinto positive/ negativeADconditional equations.
On Notions of Inductive Validity for First-Oder Equational Clauses
We define and discuss various conceivable notions of inductive validity for first-order equational clauses. This is done within the framework of constructor-based positive/negative conditional
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This paper deene three types of semantics for such a hierarchical speciication: model-theoretic, operational, and rewrite-based semantics, and shows that all these semantics coincide, provided some restrictions are met.
Inductive theorem proving in theories specified by positive/negative-conditional equations
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The proposed inference system provides explicit induction hypotheses and can be instantiated with various wellfounded induction orderings and is well-suited for inductive theorem proving in the presence of partially defined functions.
Consistency of Recursive Definitions via Shallow Confluence of Non-Orthogonal Non-Terminating Conditional Term Rewriting Systems with any kind of Extra Variables
Recursive definitions can be adequately and conveniently modeled with left-linear conditional term rewriting systems, provided that non-termination, non-trivial critical pairs, and extra variables
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