EQUIVALENCE OF LABELLED SEQUENTS

labelled multisets:

π· ∅=_{de f} ∅

π·(Ax,Γ)=de f (π·Ax),(π·Γ)

and to labelled sequents:

π·(Γ⇒∆)=de f (π·Γ)⇒(π·∆)

LetRx1,...,xnbe ann-ary relational formula. We define the application of a permutation

πto ann-ary relational formula asπ·Rx1,...,xn=de f R(π·x1),...,(π·xn). Clearly,π·(xy)=de f (π·x)(π·y).

This is extended naturally to multisets of relational formulae.

Terminology5.110. When discussing permutations of labels, explicit reference to the

setT of labels will be omitted when it is obvious from the context.

Notation5.111. Permutations will be represented by π(with a possible subscript or

prime mark), so that the notation for the application of a permutation, e.g. π·Γ, will not be confused with the notation forninstances of schematic variable, e.g.n·Γ.

Proposition5.112. π·(Γ∪Γ0)=(π·Γ)∪(π·Γ0).

Proof. By induction on the size of Γ0. The base case is trivial. For the induction step, we assume that π·(Γ∪Γ0)=(π·Γ)∪(π·Γ0) holds for smaller multisets. Using Defini- tion 5.109, π·Γ∪π·(Γ0,Ax)=π·Γ∪(π·Γ0,π·Ax) =π·Γ∪(π·Γ0∪π·Ax) =(π·Γ∪π·Γ0)∪π·Ax =π·(Γ∪Γ0),π·Ax =π·((Γ∪Γ0)∪ {Ax}) =π·(Γ∪(Γ0∪ {Ax}))=π·(Γ∪(Γ0,Ax))

132 5. LABELLED SEQUENT CALCULI

Corollary5.113. π·(StS0)=(π·S)t(π·S0).

Proof. From Definition 5.10 on page 106. Definition5.114 (Composition of Permutations). The compositionof two permuta-

tionsπandπ0is denoted byπ◦π0. We define the application of a composed permutation to a labelled formula as

(π◦π0)·Ax=de f π·(π0·Ax)

This definition is extended naturally to labelled multisets and labelled sequents, as well as to relational formulae and multisets of relational formulae.

P_roposition5.115. The set of permutations on a given set of of labels Y forms a group under composition—that is, composition is closed and associative, each permutation π has an inverseπ−1, and that the composition of a permutation with its inverse is equivalent to the identity permutationπid.

Proof. See [Pit03].

Remark5.116. In [Pit03], it is shown that permutations can be represented by lists of

pairs of names, e.g. (x y), such that

(x y)·z=de f                      y ifz= x x ifz=y z otherwise

and π(x y)·z=de f π·((x y)·z). Composition then corresponds with appending lists of pairs.

Definition 5.117 (π-permutable). Let π be a permutation, and x,y be labels. Then

x→_πyiffπ·x=y.

This definition is extended to labelled multisets and labelled sequents in the obvious way.

Proposition5.118 (Permutation Substitution). Letπbe a permutation such that

5.5. EQUIVALENCE OF LABELLED SEQUENTS 133

Proof. Follows from Definition 5.117 on the preceding page. Proposition5.119. IfΓ→π∆, then∆→_π−1 Γ.

Proof. Follows from Proposition 5.115 on the facing page. Definition 5.120 (Equivalence modulo permutation). Two multisets of labelled for-

mula, Γ,∆areequivalent modulo permutation, written asΓ≈∆, iffthere existsπsuch thatΓ→_π∆.

Two sequentsΓ₁⇒∆₁ andΓ₂⇒∆₂ areequivalent modulo permutation, written as (Γ₁⇒∆₁)≈(Γ₂⇒∆₂), iffthere existsπsuch thatΓ₁→_πΓ₂and∆₁→_π∆₂.

This notation is extended naturally to relational and prefix sequents.

Proposition5.121. Let S₁≈S₂, where S₁,S₂#S₃. Then S₁tS₃≈S₂tS₃.

Proof. ∃π.π·S₁=S₂ π·S₃=S₃ π·(S₁tS₃)=(π·S1)t(π·S₃) =S₂tS₃ Lemma 5.122 (Equivalence Relation). ≈ is an equivalence relation on labelled se-

quents. That is,≈is (a) reflexive, (b) symmetric and (c) transitive.

Proof. (a) Using the identity permutation. (b) Using the inverse permutation (Proposi- tion 5.119). (c) By composition of permutations (Proposition 5.115 on the facing page). Proposition5.123. IfΓ≈∆, then|Γ|=|∆|.

Proof. From Definition 5.120.

Definition5.124 (Subset Modulo Permutation). Γ⊂_∼∆iffthere existsΓ0such that

Γ0_≈Γ

andΓ0⊆∆.

134 5. LABELLED SEQUENT CALCULI

Proof. Straightforward.

Lemma5.126. IfΓ⊂_∼∆and∆⊂_∼Γ, thenΓ≈∆.

Proof. SupposeΓ⊂

∼∆and ∆⊂∼Γ. By Definition 5.124 on the previous page, there exists

Γ0

and∆0such thatΓ0≈Γ,Γ0⊆∆,∆0≈∆and∆0⊆Γ. So there existsΓ00and∆00such that Γ0_∪Γ00_{= ∆}

and∆0∪∆00= Γ. ThenΓ0≈∆0∪∆00 and∆0≈Γ0∪Γ00. From Proposition 5.123 on the preceding page,Γ00= ∆00=∅. SoΓ0≈∆0, which meansΓ≈∆. Remark5.127. M. J. Gabbay has suggested [Gab10] that there may be a connection

between the formal properties of labels, particularly with respect to equivalence mod- ulo permutation, and the fresh name quantifier N from [GP01] or the∇ quantifier from [MT02]. This is an area for future investigation.

5.6. Conclusion

In this chapter we have introduced the notation and terminology of simply labelled and relational calculi, along with example calculi for logics inInt∗/Geo. We have also super- ficially examined a variant of relational calculi from [Vig00] that restricts relation rules to separate branches of the proof, as well as prefix calculi, which absorb relations into the labels themselves. This is only a superficial survey of the kinds of labelled calculi dis- cussed in this thesis. We do not have a theory to describe the relative strength of various kinds of labelled sequent calculi, which we consider a topic for a separate thesis in itself. We also introduced the notation of a relational logic by defining a logic in terms of relational sequents that are derived by a particular calculus, e.g. Int as the set of all relational sequents derivable byG3I. This notion is “good enough” to discuss whether a relational sequent calculus for a particular logic such asIntiscompletefor the relational logic—that is, can it derive sequents such as x y; (A∨B)x⇒Ay,Bx. An alternative method of defining a logic such asIntby extending the language ofIntto have labelled formulae, and to incorporate relational formulae as atomic formulae. A corresponding Hilbert system forIntwould be defined by using labelled forms of the axioms forIntand adding corresponding axioms for the persistence property (i.e.,xy∧Ax⊃Ay), reflexivity and transitivity. It is not clear that such an axiomatisation would be advantageous over the simpler method of definingIntwith respect to a calculus that is known to be sound and complete.

5.6. CONCLUSION 135

We have also introduced a notion of equivalence modulo permutation of labels, which is akin to α-equivalence for bound variables. While this is an unsurprising result, we believe it to be novel, and to suggest a more general notion of equivalence of variables that includes unbound variables. (As will be shown in Chapters 6 and 8, there is a cor- respondence between labels and unbound first-order variables when translating sequents into first-order formulae corresponding to the truth conditions on intermediate Kripke frames.) Equivalence modulo permutation will be useful for showing the correspondence between hypersequents and simply labelled sequents in the next chapter.

CHAPTER 6

The Relationship between Hypersequents and Labelled Sequents