Motivating examples Confluence (abstract rewriting)

the usual rules of elementary arithmetic form abstract rewriting system. example, expression (11 + 9) × (2 + 4) can evaluated starting either @ left or @ right parentheses; however, in both cases same result obtained eventually. suggests arithmetic rewriting system confluent one.

a second, more abstract example obtained following proof of each group element equalling inverse of inverse:

this proof starts given group axioms a1-a3, , establishes 5 propositions r4, r6, r10, r11, , r12, each of them using earlier ones, , r12 being main theorem. of proofs require non-obvious, if not creative, steps, applying axiom a2 in reverse, thereby rewriting 1 ⋅ in first step of r6 s proof. 1 of historical motivations develop theory of term rewriting avoid need such steps, difficult find unexperienced human, let alone computer program.

if term rewriting system confluent , terminating, straightforward method exists prove equality between 2 expressions (a.k.a. terms) s , t: starting s, apply equalities left right long possible, obtaining term s’. obtain t term t’ in similar way. if both terms s’ , t’ literally agree, s , t (not surprisingly) proven equal. more important, if disagree, s , t cannot equal. is, 2 terms s , t can proven equal @ all, can method.

the success of method doesn t depend on sophisticated order in apply rewrite rules, confluence ensures sequence of rule applications lead same result (while termination property ensures sequence reach end @ all). therefore, if confluent , terminating term rewriting system can provided equational theory,

not tinge of creativity required perform proofs of term equality; task hence becomes amenable computer programs. modern approaches handle more general abstract rewriting systems rather term rewriting systems; latter special case of former.

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