Translating Labelled Sequent Proofs to Hypersequent Proofs

On Hypersequents and Labelled Sequents Translating Labelled Sequent Proofs to Hypersequent Proofs

Robert Rothenberg 1 2

1School of Computer Science University of St Andrews

2Interactive Information, Ltd Edinburgh

Workshop in Honour of Roy Dyckhoﬀ St Andrews, 18-19 November 2011 Extensions of Gentzen-style Sequent Calculi

Extensions to Gentzen-style sequent calculi obtained by changing to speciﬁc syntactic features [Paoli] in order to control proof search for 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 or relational 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 to juggling notation...)

I Develop a hierarchy of the strength of proof systems. Why Compare Labelled Sequents and Hypersequents?

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

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

I A comparison of the rules for some logics suggests a relationship. . . 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 of annotations to control inference, e.g. Γ⇒∆,Ay x R Γ⇒∆, A where y is fresh for the conclusion.

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

I Easily obtained using the relational semantics of a logic. Syntax of Labelled Sequents

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

x1 xn x1 xn Γ1 ,..., Γn ⇒∆1 ,..., ∆n

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

x1 xn x1 xn Rxi1 xj1 ,..., Rxik xjk , Γ1 ,..., Γn ⇒∆1 ,..., ∆n

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

I . . . relational formulae may be n-ary, occur on either side, or even 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⇒∆ L⊃ Rxy, Σ; (A⊃B)x, Γ⇒∆

Rxy,ˆ Σ; Ay, Γ⇒∆,By R⊃ Σ; Γ⇒∆, (A⊃B)x The rules for ∧, ∨ and ⊥ are standard. I The pure relational rules (or “ordering rules”):

Rxx, Σ; Γ⇒∆ Rxz, Rxy, Ryz, Σ; Γ⇒∆ reﬂ trans Σ; Γ⇒∆ Rxy, Ryz, Σ; Γ⇒∆ 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 and replacing the axiom with

Σ; Ax, Γ⇒∆,Ax

Rxy, Σ; Ax,Ay, Γ⇒∆ Rxy, Σ; Γ⇒∆,Ax,Ay L R Rxy, Σ; Ax, Γ⇒∆ mono Rxy, Σ; Γ⇒∆,Ay mono

The mono rules are derivable in G3I using cut, e.g.: . . Rxy, Σ; Ax, Γ⇒∆,Ay Rxy, Σ; Ax,Ay, Γ⇒∆ cut Rxy, Σ; Ax, Γ⇒∆ Geometric Rules

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

[z/¯ˆ y¯]Σ1, Σ0, Γ⇒∆ ... [z/¯ˆ y¯]Σn, Σ0, Γ⇒∆ Σ0, Γ⇒∆

where the variables z¯ˆ do not occur free in the conclusion, and each Σi is a multiset of atoms. I Geometric rules can be added to G3-style calculi without aﬀecting admissibility of cut, weakening or contraction. [Negri 2005] [Simpson 1994]. I A geometric implication [Palmgren 2002?] is a formula of the 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 geometric implications. 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,ˆ Ryz,ˆ Rwx, Rwy, Σ; Γ⇒∆ dir Rwx, Rwy, Σ; Γ⇒∆

I Adding the “linearity rule” yields a calculus for GD: Rxy, Σ; Γ⇒∆ Ryx, Σ; Γ⇒∆ lin Σ; Γ⇒∆

I Adding the “symmetry” rule yields a calculus for Cl: Rxy, Ryx, Σ; Γ⇒∆ sym Rxy, Σ; Γ⇒∆

I Weakening, contraction and cut admissibility is preserved. Hypersequents

I Attributed to [Avron] although similar calculi occur in earlier work 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 its components, 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 active component in each premiss, and one principal component in the conclusion. External rules are (structural) rules which are not internal rules. I The standard external rules are 0 0 0 H H|Γ⇒∆|Γ⇒∆ H|Γ ⇒∆ |Γ⇒∆|H EW EC EP H|Γ⇒∆ H|Γ⇒∆ H|Γ⇒∆|Γ0⇒∆0|H0

where H, H0 denote the side components.

I The hyperextention of a sequent calculus is its extension as a hypersequent calculus by adding hypercontexts to rules and the standard external rules. A Hyperextention of a Calculus for Int

H|Γ⇒∆, ⊥ Ax L⊥ R⊥ Γ,P⇒P, ∆ Γ, ⊥⇒∆ H|Γ⇒∆

H|Γ,A⇒∆ H|Γ,B⇒∆ H|Γ⇒A, ∆ H|Γ⇒B, ∆ L∨ R∨1 R∨2 H|Γ,A ∨ B⇒∆ H|Γ⇒A ∨ B, ∆ H|Γ⇒A ∨ B, ∆

H|Γ⇒∆,A H|Γ,B⇒∆ H|Γ,A⇒B L⊃ R⊃ H|Γ,A⊃B⇒∆ H|Γ⇒A⊃B, ∆

H|Γ⇒∆ H|Γ, Γ0, Γ0⇒∆, ∆0, ∆0 W C H|Γ, Γ0⇒∆, ∆0 H|Γ, Γ0⇒∆, ∆0

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⇒ LQ H|Γ1⇒ |Γ2⇒

I Adding the communication rule yields a calculus for GD:

H|Γ1, Γ2⇒∆1 H|Γ1, Γ2⇒∆2 Com H|Γ1⇒∆1|Γ2⇒∆2

I Adding the split rule yields a calculus for Cl:

H|Γ1, Γ2⇒∆1, ∆2 S H|Γ1⇒∆1|Γ2⇒∆2 The Labelled and Hypersequent Rules Look Similar

Hypersequent Rule Relational Rule

H|Γ1, Γ2⇒ Rxz,ˆ Ryz,ˆ Rwx, Rwy,Σ; Γ⇒∆

H|Γ1⇒|Γ2⇒ Rwx, Rwy,Σ; Γ⇒∆

H|Γ1, Γ2⇒∆1 H|Γ1, Γ2⇒∆2 Rxy, Σ; Γ⇒∆ Ryx, Σ; Γ⇒∆

H|Γ1⇒∆1|Γ2⇒∆2 Σ; Γ⇒∆

H|Γ1, Γ2⇒∆1, ∆2 Rxy, Ryx, Σ; Γ⇒∆

H|Γ1⇒∆1|Γ2⇒∆2 Rxy, Σ; Γ⇒∆ Components roughly correspond to labels, and relational formula roughly 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 component for x (r. y) as a subset of the antecedent (r. succedent) of the 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 labels into 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. The following 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 multiple components in accordance with monotonicity, just as labelled rules do. I But the following rule is not valid for Int:

H|A, Γ⇒∆, ∆0|A, Γ, Γ0⇒∆0 L ⊆ H|A, Γ⇒∆, ∆0|Γ, Γ0⇒∆0

I A simple counterexample is

A⇒A ∧ B|A, B⇒A ∧ B L ⊆ A⇒A ∧ B|B⇒A ∧ B

which is valid for GD = Int + (A⊃B) ∨ (B ⊃A). The Translation Requires Communication

Theorem Labelled proofs in G3I∗ (that do not contain ordering rules with principal relational formulae) can be translated into hypersequent proofs in a corresponding calculus augmented with the Com rule,

H|Γ⇒∆, ∆0|Γ, Γ0⇒∆0 H|Γ, Γ0⇒∆|Γ0⇒∆, ∆0 Com H|Γ⇒∆|Γ0⇒∆0

I Labelled rules and proofs for some logics Int∗ can be translated into hypersequent proofs for GD∗. I The restriction on ordering rules has to do with the admissibility of cut. A rule such as Ryx, Rxy, Ryz;Γ⇒∆ Rzy, Rxy, Ryz;Γ⇒∆ bd Rxy, Ryz;Γ⇒∆ 2

translates to hypersequent rules with duplicated metavariables in the conclusion, and that may aﬀect cut admissibility.( ?) Translation of Proofs

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

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

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

I The reﬂ, trans and mono rules are ignored as they are implicit in the translation. (?) Monotonicity Rules

Lemma The rules H|A, Γ⇒∆, ∆0|A, Γ, Γ0⇒∆0 H|Γ⇒∆, ∆0,A|Γ, Γ0⇒∆0,A L⊂ R⊂ H|A, Γ⇒∆, ∆0|Γ, Γ0⇒∆0 ∼ H|Γ⇒∆, ∆0|Γ, Γ0⇒∆0,A ∼

are derivable using Com. Proof. H|A, Γ⇒∆, ∆0|A, Γ, Γ0⇒∆0 W H|A, Γ, Γ0⇒∆, ∆0|A, Γ, Γ0⇒∆0 (RS) H|A, Γ, Γ0⇒∆, ∆02 C H|A, Γ⇒∆, ∆0|A, Γ, Γ0⇒∆0 H|A, Γ, Γ0⇒∆, ∆0 W EW H|A, Γ⇒∆, ∆02|A, Γ, Γ0⇒∆0 H|A, Γ, Γ0|Γ0⇒∆2, ∆0 Com H|A, Γ⇒∆, ∆0|Γ, Γ0⇒∆0

⊂ The proof of R ∼ is similar. Parallel Hypersequent Rules

Lemma The rule

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

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

The dual rule R ∧ ? is similarly derivable.

AL⊃? rule is also derivable, using the derived monotonicity rules. An Example Translation

Ryy, Rxy; Bx,By , (B ⊃ C)y⇒By Ryy; Cy , (B ⊃ C)y⇒Cy Ryy, Rxy; Bx,By , (B ⊃ C)y⇒Cy Rxx, Rxy; Ax⇒Ax,Cy Rxy; Bx,By , (B ⊃ C)y⇒Cy Rxy; Ax⇒Ax,Cy Rxy; Bx, (B ⊃ C)y⇒Cy Rxy;( A ∨ B)x, (B ⊃ C)y⇒Ax,Cy

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 S5 and N3, [P. Girard, 2005].

I Relationship between labelled calculi and nested sequents for modal logics [Fitting, 2010]. Related Work (2)

I Obtaining labelled calculi from non-labelled (e.g. Hilbert and sequent) 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-classical logics [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 in the presence of Com?

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

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