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Nt1310 Unit 8 Final Report

Decent Essays

Given a set of equation axioms and a set of related reduction orderings, the standard completion procedure KB [5] orients the equations into rewrite rules and tries generating complete TRS equivalent to the input equation axioms. The appropriate given reduction orderings lead the procedure to success while the others to diverge or fail which makes it hard to test all candidate orderings in sequence or physically paralleled environment. The multi-context completion procedure called MKB [6] takes a finite set of reduction orderings O = {≻1, . . . , ≻n} and a set of equation axioms E0 as input. A set of convergent rewrite rules Rc is the expected output. We denote n parallel processes as {P1 , . . . , Pn } to cover the reduction orderings set …show more content…

Notice that the symmetric cases are implicitly GC: N∪{⟨s:t,∅,∅,∅⟩}⊢N …show more content…

Definition 2.2: Let n = ⟨s : t,B1,B2,E⟩ be a node and i ∈ I be an index. The E-projection Epro[n, i] of n onto i is a set of equations defined by: {{s ↔ t}, if i ∈ E, Epro[n, i] = ∅, otherwise. Definition 2.3: The R-projection Rpro[n,i] of n onto i is a set of rules defined by:  {s→t}, ifi∈B1, Rpro[n,i]= {t→s}, if i∈B2,  ∅, otherwise. Definition 2.4: The extended version for a set N of nodes is defined by: ∪∪ Epro[N, i] = Epro[n, i], Rpro[N, i] = Rpro[n, i] n∈N n∈N The state sequence of MKB is change as N0 ⊢ N1 ⊢ ··· ⊢ Nc, where the initial N0 is N0 ={⟨s:t,∅,∅,I⟩|s↔t∈E0}. At the beginning, we have (Epro[N0, i]; Rpro[N0, i]) = (E0; ∅) for all i ∈ I. With the completion procedure going on, if we get a state where Epro[Nc,i] is empty and all critical pairs of Rpro[Nc, i] have been created, means the process Pi succeeds, and the Rpro[Nc,i] would be a complete TRS covering the input E0

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