Claudio Moraga: A Passion for Multi-Valued Logic and Soft Computing by Rudolf Seising & Héctor Allende-Cid

Author:Rudolf Seising & Héctor Allende-Cid
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(W2)For each i-interval I, we have , where . Further more, for any .

(W3)In every i-interval I, all transitions reading from register i read data value . That is, for all , if then .

(W4)If I and J are intervals with the same data value then I and J do neither overlap nor meet.

Proposition 12.3.1

For every DGRA R and every data word w it holds that if and only if there exists a valid witness structure W for R and w.

Proof

(if) From it is straightforward to construct a sequence : The and can just be taken from and the register assignments can be defined as follows: , for every register number i. For every and register number i, is just for the unique i-interval I from with .

It remains to show that is actually an accepting run of R on w. To verify this, we check that conditions (i)–(iii) from Definition 12.2.1 are satisfied. Condition (i) holds because is a pre-run. Condition (ii) holds by construction of . We show that for each conditions (iiia)–(iiie) hold (with and as i and S, respectively):Condition (iiia) holds since is a pre-run.

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