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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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