On Hypersequents and Labelled Sequentslengrand/Events/Dyckhoff/Slides/Rothenberg.pdfOn Hypersequents and Labelled Sequents Translating Labelled Sequent Proofs to Hypersequent Proofs

On Hypersequents and Labelled SequentsTranslating Labelled Sequent Proofs to Hypersequent Proofs

Robert Rothenberg 1 2

1School of Computer ScienceUniversity of St Andrews

2Interactive Information, LtdEdinburgh

Workshop in Honour of Roy DyckhoffSt Andrews, 18-19 November 2011

Extensions of Gentzen-style Sequent Calculi

Extensions to Gentzen-style sequent calculi obtained by changingto specific syntactic features [Paoli] in order to control proof searchfor non-classical logics, such as:

I Labelled Systems

I Multiple Sequents (e.g. higher-order sequents, hypersequents)

I Multi-sided Sequents

I Multi-arrow Sequents (e.g. sequents of relations)

I Multi-comma Systems (e.g. Display Logics)

I Deep Inference Systems (e.g. Calculus of Structures)

Many systems are hybrids of these, such as nested sequents orrelational hypersequents.

Why Compare Formalisms?

I Interface vs implementation (automated proof assistants)

I Translating proofs of meta properties.

I Novel and interesting rules obtained from other formalisms.

I Formal criteria for comparing formalisms.

I Illuminate the meaning of particular syntactic features.

I Use abstraction to conceive of new extensions? (akin tojuggling notation...)

I Develop a hierarchy of the strength of proof systems.

Why Compare Labelled Sequents and Hypersequents?

I Folklore about relationship, but no published formalcomparison beyond specific calculi (mainly for S5).

I There are labelled and hypersequent calculi for overlappingsets of logics. (Here we look at some Int∗ logics.)

I A comparison of the rules for some logics suggests arelationship. . .

Labelled Systems

I First labelled systems apparently introduced by [Kanger, 1957]for S5 and [Maslov, 1967] for Int.

I The language of formulae is extended with a language ofannotations to control inference, e.g.

Γ⇒∆, Ay

Γ⇒∆,�Ax R�

where y is fresh for the conclusion.

I Additional kinds of formulae based on labels may be used forcontrolling inference, e.g. Rxy.

I Easily obtained using the relational semantics of a logic.

Syntax of Labelled Sequents

I Formulae in a sequent are annotated with labels, e.g. Ax.

Γx11 , . . . ,Γxn

n ⇒∆x11 , . . . ,∆xn

n

I Sequents may also contain relational formulae whichindicate a relationship between labels , e.g. Rxy.

Rxi1xj1 , . . . ,Rxikxjk ,Γx11 , . . . ,Γxn

n ⇒∆x11 , . . . ,∆xn

n

I In some calculi, labels may be complex expressions, or maycontain variables. . .

I . . . relational formulae may be n-ary, occur on either side, oreven be “first class” and combined with formulae, e.g.Rxy ∧ (A ∨B)x.

The Simple Relational Calculus G3I

I A labelled calculus with atomic labels and binary relations.I A fragment of the calculus G3I from [Negri, 2005]:

Rxy,Σ;P x,Γ⇒∆, P y

Rxy,Σ; (A⊃B)x,Γ⇒∆, Ay Rxy,Σ; (A⊃B)x, By,Γ⇒∆

Rxy,Σ; (A⊃B)x,Γ⇒∆L⊃

Rxy,Σ;Ay,Γ⇒∆, By

Σ; Γ⇒∆, (A⊃B)xR⊃

The rules for ∧, ∨ and ⊥ are standard.I The pure relational rules (or “ordering rules”):

Rxx,Σ; Γ⇒∆

Σ; Γ⇒∆refl

Rxz,Rxy,Ryz,Σ; Γ⇒∆

Rxy,Ryz,Σ; Γ⇒∆trans

A Similar Calculus for BiInt

[Pinto & Uustalu, 2009] give a similar calculus for BiInt, with(aside from the dual of⊃) contraction as a primitive rule andreplacing the axiom with

Σ;Ax,Γ⇒∆, Ax

Rxy,Σ;Ax, Ay,Γ⇒∆

Rxy,Σ;Ax,Γ⇒∆Lmono

Rxy,Σ; Γ⇒∆, Ax, Ay

Rxy,Σ; Γ⇒∆, Ay Rmono

The mono rules are derivable in G3I using cut, e.g.:

....Rxy,Σ;Ax,Γ⇒∆, Ay Rxy,Σ;Ax, Ay,Γ⇒∆

Rxy,Σ;Ax,Γ⇒∆cut

Geometric Rules

I A geometric rule is a G3-style rule of the form

[ˆz/y]Σ1,Σ0,Γ⇒∆ . . . [ˆz/y]Σn,Σ0,Γ⇒∆

Σ0,Γ⇒∆

where the variables ˆz do not occur free in the conclusion, andeach Σi is a multiset of atoms.

I Geometric rules can be added to G3-style calculi withoutaffecting admissibility of cut, weakening or contraction.[Negri 2005] [Simpson 1994].

I A geometric implication [Palmgren 2002?] is a formula ofthe form ∀x.(A⊃B), without⊃, ∀ in subformulae of A,B.They are constructively equivalent to:

∀x.((P10 ∧ . . .∧Pk0)⊃∃y.((P11 ∧ . . .∧Pk1

)∨ . . .∨(P1n ∧ . . .∧Pkn)))

I Frame conditions of many logics in Int∗ are geometricimplications.

Extending G3I for Geometric Intermediate Logics

I Adding rules that correspond to frame conditions of logics. . .

I Adding the “directedness” rule yields a calculus for Jan:

Rxz,Ryz,Rwx,Rwy,Σ; Γ⇒∆

Rwx,Rwy,Σ; Γ⇒∆dir

I Adding the “linearity rule” yields a calculus for GD:

Rxy,Σ; Γ⇒∆ Ryx,Σ; Γ⇒∆

Σ; Γ⇒∆lin

I Adding the “symmetry” rule yields a calculus for Cl:

Rxy,Ryx,Σ; Γ⇒∆

Rxy,Σ; Γ⇒∆sym

I Weakening, contraction and cut admissibility is preserved.

Hypersequents

I Attributed to [Avron] although similar calculi occur in earlierwork by [Beth], [Sambin & Valentini], [Pottinger].

I A hypersequent is a non-empty list/multiset of sequents

Γ1⇒∆1 | . . . | Γn⇒∆n

called its components.

I A hypersequent H is true in an interpretation I iff one of itscomponents, Γi⇒∆i ∈ H is true in that interpretation, i.e.

( ∧∧Γ1⊃ ∨∨∆1) ∨ . . . ∨ ( ∧∧Γn⊃ ∨∨∆n)

Syntax of Hypersequents

I Internal rules are (structural) rules which have one activecomponent in each premiss, and one principal component inthe conclusion. External rules are (structural) rules which arenot internal rules.

I The standard external rules are

HH|Γ⇒∆

EWH|Γ⇒∆|Γ⇒∆

H|Γ⇒∆EC

H|Γ′⇒∆′|Γ⇒∆|H′

H|Γ⇒∆|Γ′⇒∆′|H′ EP

where H,H′ denote the side components.

I The hyperextention of a sequent calculus is its extension asa hypersequent calculus by adding hypercontexts to rules andthe standard external rules.

A Hyperextention of a Calculus for Int

Γ, P⇒P,∆Ax

Γ,⊥⇒∆L⊥

H|Γ⇒∆,⊥H|Γ⇒∆

R⊥

H|Γ, A⇒∆ H|Γ, B⇒∆

H|Γ, A ∨B⇒∆L∨

H|Γ⇒A,∆

H|Γ⇒A ∨B,∆R∨1

H|Γ⇒B,∆

H|Γ⇒A ∨B,∆R∨2

H|Γ⇒∆, A H|Γ, B⇒∆

H|Γ, A⊃B⇒∆L⊃

H|Γ, A⇒B

H|Γ⇒A⊃B,∆R⊃

H|Γ⇒∆

H|Γ,Γ′⇒∆,∆′ WH|Γ,Γ′,Γ′⇒∆,∆′,∆′

H|Γ,Γ′⇒∆,∆′ C

plus the dual ∧ rules and standard external rules and(hyperextended) cut.

Extensions for Some Intermediate Logics

I Adding the LQ rule yields a calculus for Jan:

H|Γ1,Γ2⇒H|Γ1⇒ |Γ2⇒

LQ

I Adding the communication rule yields a calculus for GD:

H|Γ1,Γ2⇒∆1 H|Γ1,Γ2⇒∆2

H|Γ1⇒∆1|Γ2⇒∆2Com

I Adding the split rule yields a calculus for Cl:

H|Γ1,Γ2⇒∆1,∆2

H|Γ1⇒∆1|Γ2⇒∆2S

The Labelled and Hypersequent Rules Look Similar

Hypersequent Rule Relational Rule

H|Γ1,Γ2⇒H|Γ1⇒|Γ2⇒

Rxz,Ryz,Rwx,Rwy,Σ; Γ⇒∆

Rwx,Rwy,Σ; Γ⇒∆

H|Γ1,Γ2⇒∆1 H|Γ1,Γ2⇒∆2

H|Γ1⇒∆1|Γ2⇒∆2

Rxy,Σ; Γ⇒∆ Ryx,Σ; Γ⇒∆

Σ; Γ⇒∆

H|Γ1,Γ2⇒∆1,∆2

H|Γ1⇒∆1|Γ2⇒∆2

Rxy,Ryx,Σ; Γ⇒∆

Rxy,Σ; Γ⇒∆

Components roughly correspond to labels, and relational formularoughly correspond to subset relations.

Translation of Labelled Sequents to Hypersequents

I We want a translation of proofs in labelled systems like G3I∗to (familiar) hypersequent systems.

I Each label corresponds to a component.I Relations are translated using monotonicity: Rxy is translated

by including the antecedent (r. succedent) of the componentfor x (r. y) as a subset of the antecedent (r. succedent) ofthe component for y (r. x). e.g.,

Rxy,Ax, By⇒Cx, Dy 7→ A⇒C,D | A,B⇒D

The process is called transitive unfolding.

I The translation makes an explicit relationship between labelsinto an implicit relationship between components.

Labelled Calculi are More Expressive than Hypersequents

I The two labelled sequents,

Rxy,Rxz; Γx⇒ Rxy,Ryz; Γx⇒

both translate to the same hypersequent,

Γ⇒ | Γ⇒ | Γ⇒

I What do relations mean w.r.t. hypersequents? e.g. Thefollowing holds for Int models:

Rxy; (A ∨B)x, (B⊃C)y⇒Ax, Cy

but the corresponding hypersequent is not derivable for Int:

A ∨B⇒A,C | A ∨B,B⊃C⇒C

Hypersequents and Monotonicity

I Ideally, we’d like hypersequent rules to act on multiplecomponents in accordance with monotonicity, just as labelledrules do.

I But the following rule is not valid for Int:

H|A,Γ⇒∆,∆′|A,Γ,Γ′⇒∆′

H|A,Γ⇒∆,∆′|Γ,Γ′⇒∆′ L ⊆

I A simple counterexample is

A⇒A ∧B|A,B⇒A ∧B

A⇒A ∧B|B⇒A ∧BL ⊆

which is valid for GD = Int + (A⊃B) ∨ (B⊃A).

The Translation Requires Communication

TheoremLabelled proofs in G3I∗ (that do not contain ordering rules withprincipal relational formulae) can be translated into hypersequentproofs in a corresponding calculus augmented with the Com rule,

H|Γ⇒∆,∆′|Γ,Γ′⇒∆′ H|Γ,Γ′⇒∆|Γ′⇒∆,∆′

H|Γ⇒∆|Γ′⇒∆′ Com

I Labelled rules and proofs for some logics Int∗ can betranslated into hypersequent proofs for GD∗.

I The restriction on ordering rules has to do with theadmissibility of cut. A rule such as

Ryx,Rxy,Ryz; Γ⇒∆ Rzy,Rxy,Ryz; Γ⇒∆

Rxy,Ryz; Γ⇒∆bd2

translates to hypersequent rules with duplicated metavariablesin the conclusion, and that may affect cut admissibility. (?)

Translation of Proofs

I Note that this work is about translating proofs of arbitrarylabelled sequents (with relations) into hypersequents.

I The communication rule allows us to derive hypersequentvariants of the labelled rules.

I We proceed by transitive unfolding the premisses of eachlabelled inference and then applying the hypersequent variantof the inference rule, to obtain a conclusion that is thetransitive unfolding of the conclusion of the labelled inference.

I The refl, trans and mono rules are ignored as they are implicitin the translation. (?)

Monotonicity Rules

LemmaThe rules

H|A,Γ⇒∆,∆′|A,Γ,Γ′⇒∆′

H|A,Γ⇒∆,∆′|Γ,Γ′⇒∆′ L⊂∼H|Γ⇒∆,∆′, A|Γ,Γ′⇒∆′, A

H|Γ⇒∆,∆′|Γ,Γ′⇒∆′, AR⊂∼

are derivable using Com.

Proof.

H|A,Γ⇒∆,∆′|A,Γ,Γ′⇒∆′

H|A,Γ⇒∆,∆′2|A,Γ,Γ′⇒∆′ W

H|A,Γ⇒∆,∆′|A,Γ,Γ′⇒∆′

H|A,Γ,Γ′⇒∆,∆′|A,Γ,Γ′⇒∆′ W

H|A,Γ,Γ′⇒∆,∆′2 (RS)

H|A,Γ,Γ′⇒∆,∆′ C

H|A,Γ,Γ′|Γ′⇒∆2,∆′ EW

H|A,Γ⇒∆,∆′|Γ,Γ′⇒∆′ Com

The proof of R⊂∼ is similar.

Parallel Hypersequent Rules

LemmaThe rule

H|A,Γ1⇒∆1| . . . |A,Γk⇒∆k H|B,Γ1⇒∆1| . . . |B,Γk⇒∆k

H|A ∨B,Γ1⇒∆1| . . . |A ∨B,Γk⇒∆kL ∨ ?

where Γi ⊆ Γi+1 and ∆i+1 ⊆ ∆i, is derivable using Com.

The dual rule R ∧ ? is similarly derivable.

A L⊃? rule is also derivable, using the derived monotonicity rules.

An Example Translation

Rxx,Rxy;Ax⇒Ax, Cy

Rxy;Ax⇒Ax, Cy

Ryy,Rxy;Bx, By, (B⊃C)y⇒By Ryy;Cy, (B⊃C)y⇒Cy

Ryy,Rxy;Bx, By, (B⊃C)y⇒Cy

Rxy;Bx, By, (B⊃C)y⇒Cy

Rxy;Bx, (B⊃C)y⇒Cy

Rxy; (A ∨ B)x, (B⊃C)y⇒Ax, Cy

A⇒A,C|A,B⊃C⇒C

A⇒A,C|A,B⊃C⇒C

B⇒A,C|B,B⊃C⇒B,C B⇒A,C|C,B⊃C⇒C

B⇒A,C|B,B⊃C⇒C

B⇒A,C|B,B⊃C⇒C

B⇒A,C|B,B⊃C⇒C

A ∨ B⇒A,C|A ∨ B,B⊃C⇒C

Related Work (1)

I Hypersequents and labelled calculi for S5, [Avron, 1996], etc.

I Hypersequents and Display Logics for specific logics,[Wansing, 1998], and labelled calculi for S5, [Restall, 2006].

I Hypersequents and labelled calculi for A and L,[Metcalfe et al, 2002].

I Starred sequents, hypersequents and indexed sequents for S5and N3, [P. Girard, 2005].

I Relationship between labelled calculi and nested sequents formodal logics [Fitting, 2010].

Related Work (2)

I Obtaining labelled calculi from non-labelled (e.g. Hilbert andsequent) calculi, [Gabbay, 1996].

I Obtaining (hyper)sequent rules from Hilbert-style axioms[Ciabattoni et al, 2008].

I Syntactic conditions for cut admissibility [Ciabattoni et al,2009].

I Labelled sequent calculi with geometric rules, for non-classicallogics [Negri, 2005], spec. for intermediate logics [Dyckhoff &Negri, 2010 (MS)].

Open Questions and Future Work

I Do rules with non-linear conclusions (e.g. bd2) admit cut inthe presence of Com?

I Can hypersequent proofs of single components be transformedso that they do not have Com, for logics weaker than GD?

I Can transformation of labelled proofs into hypersequentproofs give a technique for parallelising programs?