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Decision Procedure for the Word Problem

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Decision Procedure for the Word Problem

PPP

PPPPPPPPPPPPPPP

PPP

PPPPPPPPPPPPPPP

.

?

? 6

-

yes

yes Success

Orient equations in E

Does R terminate?

Is R confluent?

no

no

Give up Completion

Tasks

1. How can one prove termination of TRSs?

2. How can one prove confluence of TRSs?

3. How can one perform completion of TRSs?

(2)

Structural Induction:

ϕ(0) ∧ (∀y ∈ IN. ϕ(y) ⇒ ϕ(y + 1)) ⇒ ∀x ∈ IN. ϕ(x) (∀x ∈ V. ϕ(x)) ∧

(∀ti ∈ T (Σ,V). ∀f ∈ Σ. ϕ(t1)∧..∧ϕ(tn) ⇒ ϕ(f(t1, .., tn))) ⇒ ∀t ∈ T (Σ,V). ϕ(t)

ϕ(ǫ) ∧ (∀π′ ∈ IN∗. ∀i ∈ IN. ϕ(π′) ⇒ ϕ(iπ′)) ⇒ ∀π ∈ IN∗. ϕ(π) ϕ(ǫ) ∧ (∀π′ ∈ IN∗. ∀i ∈ IN. ϕ(π′) ⇒ ϕ(π′i)) ⇒ ∀π ∈ IN∗. ϕ(π)

Noetherian Induction:

( ∀m ∈ M. ( ∀k ∈ M. m ≻ k ⇒ ϕ(k) ) ⇒ ϕ(m) ) ⇒ ∀n ∈ M. ϕ(n) if ≻ is well founded

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