Jancars formal system for deciding bisimulation of first-order grammars and its non-soundness

(these intermediate objects T , LAB A will ease the definition of ACT below). We define a first-order grammar G = (N , A, R) by: N := {A, A ′ , A ′′ , B, B ′ , B ′′ , C, D, E, L 1 } 1 mailing adress:LaBRI and UFR Math-info, Université Bordeaux1 351 Cours de la libération -33405-Talence Cedex. email:ges@labri.u-bordeaux.fr; fax: 05-40-00-66-69; URL:http://dept-info.labri.u-bordeaux.fr/∼ges/ and the set of rules R consists of the following:

A ′ (v)

x -→ A ′′ (v) (7)

A ′′ (v)

x -→ D(v) (9)

Let us name rule r i (for 1 ≤ i ≤ 14), the rule appearing in order i in the above list. We define a map LAB T : R → T by: LAB T (r i ) is the terminal letter used by the given rule r i . Subsequently we define ACT(r i ) := LAB A (LAB T (r i )). Namely, ACT maps all the rules r 1 , . . . , r 12 onto a, r 13 on b and r 14 on ℓ 1 .

We consider the formal systems J (T 0 , T ′ 0 , S 0 , B) defined in page 22 of [Jan10], which are intended to be sound and complete for the bisimulation-problem for non-deterministic firstorder grammars. Let us denote by T the set of all terms over the ranked alphabet N ∪ {L i | i ∈ N} ∪ {⊥} (here the symbols L i have arity 0).

The notion of finite prefix of a D-strategy is mentionned p. 23, line 11. We assume it has the following meaning

for some n ∈ N and some D-strategy S ′ w.r.t. (T, T ′ ).

In order to make clear that the above notion is effective, we consider the following notion of D-q-strategy (Defender’s quasi-strategy).

Note that a D-strategy is a D-q-strategy where, condition DQ4 is replaced by: DQ'4: ∀α ∈ S, NEXT((T,

A winning D-strategy, is a D-q-strategy where condition DQ4 is replaced by: DQ"4: ∀α ∈ S, NEXT((T, T ′ ), α) ∈∼ 1 and the set

Lemma 1. Every finite prefix of a strategy is a D-q-strategy.

Proof: Let S ′ be a D-strategy w.r.t. (T, T ′ ) and

for some n ∈ N, S ′ D-strategy w.r.t. (T, T ′ ). DQ1: Since S ′ is non-empty and prefix-closed (ε, ε) ∈ S ′ , hence (ε, ε) ∈ S ′ ∩ S(R × R) ≤n . DQ2: S ′ and (R × R) ≤n are both prefix-closed, hence their intersection is also prefix-closed. DQ3: S ′ ⊆ PLAYS(T, T ′ ) and S ⊆ S ′ , hence S ⊆ PLAYS(T, T ′ ) DQ4: ∀α ∈ S, NEXT((T, T ′ ), α) / ∈∼ 1 or [NEXT((T, T ′ ), α) ∈∼ 1 and the set

If |α| < n, the above property holds in S. If |α| = n, the property α\S = {(ε, ε)} holds. In all cases DQ4 is fulfilled. ✷ Definition 3. We define the extension ordering over P((R × R) * ) as follows: for every

which is maximal in S 1 for the prefix ordering and such that , β α.

Lemma 2. Let T, T ′ ∈ T. The extension ordering over the set of all D-q-strategies w.r.t. (T, T ′ ), is inductive.

Proof: We recall that a partial order ≤ over a set E is inductive iff, every totally ordered subset of E has some upper-bound. One can check that, if P is a set of D-q-strategies w.r.t. (T, T ′ ), which is totally ordered by ⊑, then the set S := s∈P s is still a D-q-strategy and fulfills:

Hence the extension ordering over the set of D-q-strategies w.r.t.

Proof: Direct implication: Let S ′ be a D-strategy w.r.t. (T, T ′ ) and

for some n ∈ N and some S ′ which is a D-strategy w.r.t. (T, T ′ ).

Suppose that S fulfills conditions (1)(2). By Lemma 2, Zorn’s lemma applies on the set of Dq-strategies w.r.t. (T, T ′ ): there exists a maximal D-q-strategy S ′ (for the extension ordering) such that S ⊑ S ′ . Since S ′ is maximal, if α ∈ S ′ and α\S = {(ε, ε)}, NEXT((T, T ′ ), α) / ∈∼ 1 . Thus, instead of the weak property DQ4, S ′ fulfills the property:

Hence S ′ is a strategy w.r.t. (T, T ′ ).

Let us prove the reverse inclusion. Let α ∈ S ′ ∩ (R × R) ≤n . Let β be the longuest word in PREF(α) ∩ S.

If β = α, then α ∈ S, as required.

Otherwise α ∈ S ′ -S. By condition E2 of definition 3, there exists some β ∈ S, which is maximal in S for the prefix ordering and such that

Maximality of β implies, by condition (2) of the lemma, that,

Since β ≺ α we are sure that |β| < n so that

This last statement contradicts the fact that β\S ′ is a D-strategy, w.r.t NEXT((T, T ′ ).β) which is non-reduced to {(ε, ε)} (since it posesses β -1 α).

We can conclude that α ∈ S. Finally: This follows immediately from the characterisation given by Lemma 3.

For every T 0 , T ′ 0 ∈ T, S 0 finite prefix of strategy w.r.t (T 0 , T 0 ) and finite B ⊆ T × T, is defined a formal system J (T 0 , T ′ 0 , S 0 , B) The set of judgments of all the systems are the same. But the axiom and one rule (namely R7), is depending on the parameters (T 0 , T ′ 0 , S 0 , B).

A judgment has one of the three forms: FORM 1:

where m ∈ N, and T, T ′ ∈ T are regular terms and S is a finite prefix of a strategy. w.r.t.

(T, T ′ ) (D-strategies are defined p.20, lines 27-30; finite prefixes are mentionned, though in a fuzzy way. at p. 23, line 11; we shall apply here Definition 1).

FORM 2:

) where m ∈ N, (T, T ′ , S), (T 1 , T ′ 1 , S 1 ) fulfilling the a