The Relationship between Labelled Sequents and Relational Sequents
8.4. SEQUENT FOLDING
Proposition 8.18. Transitive folding x y1... yn z,Σ; Ax, Az,Γ⇒∆ x y1... yn z,Σ; Ax,Γ⇒∆ (L τ) x y1... yn z,Σ;Γ⇒∆, A x, Az x y1... yn z,Σ;Γ⇒∆, Az (R τ)
where n ≥0, is sound for relational sequents in Int∗/Geo.
Proof. Semantically from Corollary 8.12 on page 169. Remark 8.19. Note that the L τrule for n= 1 corresponds with how the accessibility relation is defined for Hintikka frames in completeness proofs of calculi for Int, e.g. [Fit69].
Remark 8.20. The proof of Proposition 8.17 can be directly translated to a relational calculus such as G3I∗ by using the LW and trans rules. With that, the proof of Propo- sition 8.18 can be directly translated by using the LW and L or R rules (the latter of which are admissible in G3I∗ by Lemma 5.100 on page 124).
Lemma 8.21 (Sequent Folding Correspondence). Let Γ0⇒∆0∈ SLS Then there exists Σ;Γ⇒∆ ∈ RLS such that Γ ⊂ Γ0
and∆ ⊂ ∆0, and if `Γ0⇒∆0, then `Σ;Γ⇒∆.
Proof. Γ0⇒∆0∈ RLS, trivially. Also by derivation, using the LW , L τ and R τ rules. Example 8.22. Γx,Γy,Γ0⇒∆0,∆x,∆y x y;Γx,Γy,Γ0⇒∆0,∆x,∆y (LW ) x y;Γx,Γ0⇒∆0,∆x,∆y (L τ) ∗ x y;Γx,Γ0⇒∆0,∆y (R τ) ∗
8.4.1. Grounded Sequents. Although the application of the transitive folding rules to a relational formula such as x y requires that slice y be a subset modulo permutation of slice x in the antecedent (or succedent), one can easily use weakening to reverse the subset relation between slices and apply the folding rules to the reverse relation y x. In other words, given any two labels x, y in a labelled sequent, either folding can be applied. One way of approaching the non-deterministic relationship between labelled sequents and relational sequents is to use a notion of a “normal form” of relational sequent, called grounded (relational) sequent, introduced below in Definition 8.23 on the next page.
172 8. THE RELATIONSHIP BETWEEN LABELLED SEQUENTS AND RELATIONAL SEQUENTS
Definition 8.23 (Grounded Sequent). Let Σ; Γ⇒∆ be a relational sequent. Σ; Γ⇒∆ is a grounded sequent iff all of the following hold:
(1) Σ,Γ⇒∆ is connected; (2) Γ ∪ ∆ , ∅; (3) x x < Σ; (4) if x y, y z ∈Σ+, then x z < Σ; (5) if x y ∈Σ+and Ax∈Γ, then Ay< Γ; (6) if x y ∈Σ+and Ay∈∆, then Ax< ∆; (7) ifΣ = x y,Σ0, then x y < Σ0; (8) ifΓ = Ax,Γ0, then Ax< Γ0; (9) if∆ = Ax,∆0, then Ax< ∆0.
We use the notation GRLS for the set of grounded sequents.
Remark 8.24. The definition of grounded sequent is a natural one. Conditions (3) through (9) require that there by no extraneous formulae in the sequent with respect to contraction, transitivity or folding. (This notion is applicable to logics that admit contrac- tion, reflexivity and transitivity. Substructural or subintuitionistic logics would require a different definition.)
Proposition 8.25. GRLS ⊂ RLSa.
Proof. By conditions 3 and 4 (where x= z) of Definition 8.23. Proposition 8.26. Let Σ; S ∈ GRLS. Then either lab(Γ,∆) ⊆ lab(Σ), or lab(Γ,∆) is a singleton andΣ = ∅.
Proof. Follows from condition 1 of Definition 8.23. Given a labelled sequent, we can obtain a grounded relational sequent:
Notation 8.27 (Sorted List). Let X be a finite poset. →−X denotes a sorted list of the elements in X. ←X−denotes the reversed list of→−X.
Remark 8.28. Any finite poset can be sorted by constructing a linear extension of it [Szp30]. One can also consider a poset as an acyclic digraph and use a topological sorting algorithm, e.g. [Kah62]. Note that cyclic digraphs may also be sorted, e.g. [Nuu94].
8.4. SEQUENT FOLDING 173
Notation 8.29. Let→−X = x1,..., xn be a sorted list. xi5 xj∈
− →
X denotes that xi, xj∈ X
and i ≤ j. Informally, xioccurs before xjin
− →
X or is equal to xj.
Definition 8.30 (Sorted List of Relations). Let Σ be an acyclic multiset of relational formulae, and let→−V be the sorted list of labels inΣ, ordered by the relational formulae in Σ.
Let→−Σ be the lexicographically sorted list of relational formulae in Σ, ordered by the labels in→−V. That is, (a c)5 (b d) ∈→−Σ iff a 5 b ∈→−V, and (a c)5 (a d) ∈→−Σ iff c5 d ∈→−V. ←Σ denotes the reverse of− →−Σ.
Lemma 8.31. Let S0∈ SLS. If ` S0, then then there existsΣ;S ∈ GRLS such that from ` S0, `Σ;S .
Proof. By derivation, using the following procedure:
(1) Apply the rules LC and RC until conditions 8 and 9 above are met. (2) Apply the EC rule to remove equivalent slices (and non-determinism).
(3) Let the resulting sequent be S00. For all labels x1,..., xn∈ lab(S00), use LW to
add the context formula w x1,...,w xn, where w#S00. This adds a common
root label, and meets condition 1.
(4) Let S00= Γ⇒∆. For all pairs of labels x,y ∈ lab(S00), whenever bothΓ x⊂ ∼ Γ y and∆ y⊂
∼ ∆ x, use LW to add formulae x y to the relational context. (5) Apply the transnrule, where applicable, to meet condition 4.
(6) For each relation x y in the list←Σ, if whenever Γ− x⊂
∼ Γ y, apply the Lτrule until condition 5 is met. (Note that folding is applied in the order that relational formulae occur in←Σ.)−
(7) For each relation x y in the list→−Σ, if whenever ∆ y⊂
∼ ∆ x, apply the Rτrule until condition 6 is met. (Note that folding is applied in the order that relational formulae occur in→−Σ.)
Proposition 8.32. The procedure given in Lemma 8.31 results in a grounded sequent.
174 8. THE RELATIONSHIP BETWEEN LABELLED SEQUENTS AND RELATIONAL SEQUENTS
Proposition 8.33. The procedure given in Lemma 8.31 on the previous page termi- nates.
Proof. By induction on the size of the sequent.
Remark 8.34. Note that the procedure given in Lemma 8.31 on the preceding page assumes that the labelled sequent and its resulting grounded sequent are derivable. So the notion them having the same model or countermodel is irrelevant.
Example 8.35. From the labelled sequent (A∨ B)x,(A∨ B)y⇒ Ax, Bx, By, we can derive a grounded sequent:
(A ∨ B)x,(A ∨ B)y⇒ Ax, Bx, By
w x,w y, x y;(A ∨ B)x,(A ∨ B)y⇒ Ax, Bx, By (LW )+
w x,w y, x y;(A ∨ B)x⇒ Ax, Bx, By (L τ)
w x,w y, x y;(A ∨ B)x⇒ Ax, By (R τ)
Corollary 8.36 (Unique Grounding). Let S0∈ SLS. If ` S0, then then there exists a unique equivalence class of sequents modulo permutation of labelsΣ;S ∈ GRLS such that `Σ;S .
Proof. From Lemma 8.31 on the previous page, there exists a sequentΣ;S ∈ GRLS such that `Σ;S . Suppose that there are two such sequents, Σ1; S1andΣ2; S2.
Let S001 ∈ SLS and S002 ∈ SLS be the resulting sequents after applying contractions (steps 1 and 2). The application of the LC and RC rules yield the same result, whereas the application of EC allows for a choice of principal labels. However, EC can only be applied to labels x and y iff S0 x≈ S0 y, so by induction on the number of instances of EC, S001 ≈ S002.
The choice of a root label is not fixed, but by virtue of the sequents being equivalent modulo permutation of labels, the relational contexts resulting from adding root labels in step 3 will be equivalent modulo permutation.
The set of relational formulae added in step 4 will also be equivalent modulo permu- tation of labels, by virtue of both sequents having the an equivalent modulo permutation of labels set of slices.