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Conditional rewrite rules: Confluence and termination. (English) Zbl 0658.68031
The authors consider conditional rewrite rules: $\bigwedge^{n}_{i=0}(t_ i\@s_ i)\quad \Rightarrow \quad (t\to s)\quad,\quad where\quad \@\in (=,|,\to \to \}\quad,$ and four corresponding term rewriting systems (TRS);
0 for $$n=0,$$
I for $$n=0$$ and $$\@$$ is $$=,$$
II for $$n=0$$ and $$\@$$ is $$|$$, and
III for $$n=0$$ and $$\@$$ is $$\to \to;$$
also the particular case $$III_ n$$ when s are normal closed forms (they contain neither variables nor rules). $$\@$$ operations define a reduction relation which could be circular; the problem is solved by a fixed point construction; this approach enables the authors to show that the whole type 0 syntactic theory is suitable also to I and $$III_ n$$ types without using a hierarchy of TRSs. The authors consider the confluence and termination problem for some known strategies: full substitution, leftmost reduction, parallel outermost reduction. They analyse the decidability of normal forms and give some criteria.
Reviewer: V.Calmatuianu

##### MSC:
 68Q65 Abstract data types; algebraic specification 68T99 Artificial intelligence
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##### References:
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