Regular Properties and the Existence of Proof Systems

AiML & LATD

Bern, August 28, 2018

Rosalie Iemhoff

Utrecht University, the Netherlands

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The Existence of Proof Systems

AiML & LATD

Bern, August 28, 2018

Rosalie Iemhoff

Utrecht University, the Netherlands

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Regular Properties and the Existence of Proof Systems

AiML & LATD

Bern, August 28, 2018

Rosalie Iemhoff

Utrecht University, the Netherlands

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elementary questions

Proof systems are developed to . . .

◦ study properties of a logic: consistency, decidability, . . .

◦ model a form of reasoning: type theory, linear logic, . . .

◦ . . .

Does logic L has a useful proof system?

“useful” depends on the context: decidable, cut-free, normalizing, . . .

What is a proof system?

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Proof systems

A Hilbert system consists of

Axioms: ϕ1, . . . , ϕn

Rule Modus Ponens:

ϕ ϕ→ ψ

ψ

A proof is a sequence of formulas, which are either axioms or follow byModus Ponens from previously derived formulas.

Not all proofs in a proof system have such a form: in natural deductionproofs can contain (discharged) assumptions.

And resolution and Gentzen calculi are not even about formulas butabout clauses and sequents.

Given a logic, there often are (faithful) translations between the differentproof systems for the logic.

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existence of proof systems

Numerous positive results of the form:

This logic has such and such a proof system.

Few(er) negative results of the form:

This logic does not have such and such a proof system.

Examples of negative results:

◦ Based on the complexity of the logic.

◦ On specific proof systems.E.g. the work by Belardinelli & Jipsen & Ono, later extended by

Ciabattoni & Galatos & Terui, on the existence of cut-free sequent calculi.

E.g. the work by Negri on labelled sequent calculi.

...

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aim

To establish, for certain logics, that certain classes of proof systems donot exist.

In this talk:

◦ the logics are intermediate, modal, and intuitionistic modal logics;

◦ the proof systems are abstract versions of sequent calculi.

The method goes beyond these logics and proof systems.

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method

For a class of proof systems PS and a regular property RP of logicsestablish theorems of the form:

If a logic has a proof system in PS, then it has regular property RP.

Or, equivalently,

If a logic does not have RP, then it does not have a proof system in PS.

The strength of the method depends on the size of the class PS and thefrequency with which RP occurs among the considered logics.

In this talk:

◦ the logics are intermediate, modal, and intuitionistic modal logics;

◦ the proof systems are abstract versions of sequent calculi.

◦ the regular property is uniform interpolation.

Side benefit:Uniform interpolation in a uniform, modular way, and for new logics.

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uniform interpolation

Dfn A logic L has (Craig) interpolation if whenever ` ϕ→ ψ there is a χ

in the common language L(ϕ) ∩ L(ψ) such that ` ϕ→ χ and ` χ→ ψ.

A propositional (modal) logic has uniform interpolation if the interpolantdepends only on the premiss or the conclusion: For all ϕ there areformulas ∃pϕ and ∀pϕ not containing p such that for all ψ notcontaining p:

` ψ → ϕ iff ` ψ → ∀pϕ ` ϕ→ ψ iff ` ∃pϕ→ ψ.

∃pϕ is the right interpolant and ∀pϕ the left interpolant:

` ϕ→ ∃pϕ ` ∀pϕ→ ϕ.

Note A locally tabular logic that has interpolation, has uniforminterpolation:

∃pϕ(p, q̄) =∧{ψ(q̄) | ` ϕ(p, q̄)→ ψ(q̄)}

∀pϕ(p, q̄) =∨{ψ(q̄) | ` ψ(q̄)→ ϕ(p, q̄)}.

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uniform interpolation in modal and intermediate logics

Theorem (Pitts ’92)

IPC has uniform interpolation. (this was the inspiration for our approach)

Theorem (Shavrukov ’94)

GL has uniform interpolation.

Theorem (Ghilardi & Zawadowski ’95)

K has uniform interpolation. S4 does not.

Theorem (Bilkova ’06)

KT has uniform interpolation. K4 does not.

Theorem (Maxsimova ’77, Ghilardi & Zawadowski ’02)

There are exactly seven intermediate logics with (uniform) interpolation:

IPC, Sm, GSc, LC, KC, Bd2, CPC.

Van Gool & Metcalfe & Tsinakis 2017: general approach.

Pitts uses a terminating sequent calculus for IPC.(developed independently by Dyckhoff and Hudelmaier in ’92)

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aim

In the case of (intuitionistic) modal and intermediate logic, isolate a(large) class of proof systems and prove that any logic with a proofsystem in that class has uniform interpolation.

Since uniform interpolation is rare among modal and intermediate logics,this establishes the negative result (not having a proof system in thatclass) for many such logics.

The method also provide a uniform and modular way to prove uniforminterpolation for classes of logics, including some logics for which this wasunknown, such as KD.

The class of proof systems is defined not in terms of concrete rules but interms of the structural properties of rules.

In this talk: classical modal logic with one modal operator. Language:⊥,∧,∨,→,2, p1, p2, . . . .

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the proof systems

The proof systems are sequent calculi, where a sequent is an expression(Γ⇒ ∆), where Γ and ∆ are multisets, interpreted as (

∧Γ→

∨∆).

Dfn 2Γ ≡df {2ϕ | ϕ ∈ Γ} and 2(Γ⇒ ∆) ≡df (2Γ⇒ 2∆) and

(Γ⇒ ∆) · (Π⇒ Σ) ≡df (Γ,Π⇒ ∆,Σ).

Dfn A sequent calculus is a set of rules, where a rule R is an expressionof the form

S1 . . . Sn

S0R

(fr)

for certain sequents S0, . . . ,Sn (that may be empty). An instance R of arule is of the form

σS1 . . . σSn

σS0R

where σ is a substitution for the modal language.

Dfn A nonaxiom rule (fr) is focussed if S0 contains a single nonboxedformula and for every instance R = (S ′

1 . . . S′n/S

′0) and sequent S the

following is an instance of R:

S · S ′1 . . . S · S ′

n

S · S ′0

R(S)

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the proof system G3

Dfn A nonaxiom rule is focussed if the conclusion is a single, nonboxedformula and for every instance R and sequent S , R(S) is an instance ofthe rule.

Dfn All the rules in G3 that are not axioms are focussed:

Γ, p ⇒ p,∆ Γ,⊥ ⇒ ∆

Γ⇒ ϕ,∆

Γ,¬ϕ⇒ ∆

Γ, ϕ⇒ ∆

Γ⇒ ¬ϕ,∆

Γ, ϕ⇒ ∆ Γ, ψ ⇒ ∆

Γ, ϕ ∨ ψ ⇒ ∆

Γ⇒ ϕ,ψ,∆

Γ⇒ ϕ ∨ ψ,∆...

Dfn A calculus is terminating if there is a well-founded order on sequentssuch that in every rule the premisses come before the conclusion, and . . .

In general, the cut rule does not belong to a terminating calculus:

Γ⇒ ϕ,∆ Γ, ϕ⇒ ∆

Γ⇒ ∆

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the proof systems for modal logic

Dfn A nonaxiom rule R = (S1 . . . Sn/S0) is focussed if S0 contains asingle, nonboxed formula and for every instance R and sequent S , R(S)is an instance of the rule.

Axioms (Γ, p ⇒ p,∆), (Γ,⊥ ⇒ ∆) and (Γ⇒ >,∆) are focussed.

A focussed modal rule is of the form

◦S1 · S0

S2 ·2S1 ·2S0R

where S0 contains a single formula, that is boxed, S2 is of the form(Π⇒ ∆), S1 contains only multisets, and ◦S1 denotes S1 or �S1.

Example Focussed (modal) rules:

Γ⇒ ϕ,ψ,∆

Γ⇒ ϕ ∨ ψ,∆Γ⇒ ϕ

Π,2Γ⇒ 2ϕ,∆RK

Γ, ϕ⇒Π,2Γ,2ϕ⇒ ∆

RD

Example Rules that are not focussed (modal):

Γ, ψ → χ⇒ ϕ→ ψ Γ, χ⇒ ∆

Γ, (ϕ→ ψ)→ χ⇒ ∆

�Γ,2ϕ⇒ ϕ

Π,2Γ⇒ 2ϕ,∆RGL

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results for modal logic

Theorem (Iemhoff 2016)

A logic with a terminating calculus that consists of focussed and focussedmodal rules has uniform interpolation.

Corollary (well-known)

Classical propositional logic CPC has uniform interpolation.Proof All rules in the sequent calculus G3 are focussed. aCorollaryThe modal logics K (Ghilardi) and KD (Iemhoff) have uniforminterpolation.

A promised negative result:

CorollaryIf a modal logic does not have uniform interpolation, then it does nothave a terminating calculus that consists of focussed and focussed modalrules. Examples are K4 and S4.

Interplay: Semantics (algebraic logic) and proof theory.

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so far

Aim: Isolate a (large) class of proof systems and prove that any(intuitionistic) modal and intermediate logic with a proof system in thatclass has uniform interpolation.

Side benefit: Establishing uniform interpolation in a uniform, modularway, and for new logics.

So far: a uniform way to prove uniform interpolation for modal logics,where the proof systems consist of focussed and focussed modal rules.

To come:

◦ extend the method to intermediate and intuitionistic modal logics,

◦ explain the proof method, in particular its modularity.

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proof method

TheoremA modal logic with a terminating calculus that consists of focussed andfocussed modal rules has uniform interpolation.

Proof idea Define for each rule R in the calculus and sequent S an

expression ∀RpS . E.g. for focussed rules R:

R = (S1 . . .Sn/S0) ∀RpS0 ≡df ∀pS1 ∧ . . . ∧ ∀pSn

Inductively define ∀pS ≡df

∨{∀RpS | R a rule instance with conclusion S}.

For free sequents S , ∀pS is defined separately.

Prove with induction along the order that for any rule in the calculus, ifthe premisses of a rule have a uniform interpolant, then so does theconclusion.

Some details are omitted . . . aUniform and modular proof.

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intermediate logic

Similar to the classical case, but far more complicated: ∃p and ∀p .

One needs a terminating calculus for IPC. Use G4i by Dyckhoff andHudelmaier. Not all rules of G4i are focussed.

Theorem (Iemhoff 2017) Any calculus that is an extension of G4i withfocussed rules (in the sense of intermediate logic) has uniforminterpolation.

Proof Prove for the nonfocussed rules of G4i that if the premisses have auniform interpolant, then so does the conclusion. Further proceed as inthe classical case. a

Corollary No intermediate logic except the 7 with uniform interpolationhas such a calculus.

Corollary When developing a calculus based on G4i for an intermediatelogic without uniform interpolation, then some of the rules have to benot focussed.

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intuitionistic modal logic

Work in progress.

The logics are extensions of iK (only 2, no diamond 3).

The sequent calculi are extensions of G4iK, which is G4i plus the rules

Γ⇒ ϕ

Π,2Γ⇒ 2ϕ,∆RK

Γ⇒ ϕ Π,2Γ, ψ ⇒ ∆

Π,2Γ,2ϕ→ ψ ⇒ ∆L2→

Lemma G4iK is terminating.

Theorem Any logic with a calculus that is an extension of G4iK withfocussed and focussed modal rules (in the sense of intuitionistic modallogic) has uniform interpolation. This holds in particular for iK and iKD.

Modularity of the proof: Six properties of rules are isolated such that:

Theorem Any logic with a calculus that is an extension of G4iK suchthat all rules that are not focussed (in the sense of intuitionistic modallogic) satisfy the six properties, has uniform interpolation.

Question: Which intuitionistic modal logics have uniform interpolation?

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substructural logics

Theorem (Alizadeh & Derakhshan & Ono 2014)

FLe and FLew and various predicate substructural logics have uniforminterpolation.

Work in progress:

Theorem (Tabatabai & Jalali 2018)

Any logic with a terminating sequent calculus that extends the standardcalculus for FLe and consists of focussed axioms and semi-analytic rules(in the sense of substructural logics) has uniform interpolation.

For the negative results, use:

Theorem (Marchioni & Metcalfe 2012)

Craig interpolation fails for certain classes of semilinear substructurallogics.

Theorem (Urquhart 1993)

Failure of Craig interpolation in relevant logics.

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positive summary

A logic has uniform interpolation if it has

◦ (classical modal logic) a terminating calculus consisting of focussedand focussed modal rules.

◦ (intermediate & intuitionistic modal logics) a terminating calculusthat is an extension of G4iK by focussed and focussed modal rules.

◦ (substructural logics) a terminating calculus that is asingle-conclusion extension of (the standard calculus for) FLe bysemi-analytic rules.

The notion of focussed (modal) rule in the first two cases is not the same.

In all cases there are a finite number of interpolant properties such thatthe above also holds for the calculi extended by rules satisfying theseproperties, provided the whole calculus is terminating.

(interpolant properties: variants of statements of the form “if thepremisses have uniform interpolants, then so does the conclusion”)

Uniform interpolation can be shown for:

K, KD, IPC, iK, iKD, FLe, FLew, . . .

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negative summary

A logic without uniform interpolation cannot have as proof system

◦ (classical modal logic) a terminating calculus consisting of focussedand focussed modal rules.

◦ (intermediate & intuitionistic modal logics) a terminating extensionof G4iK by focussed and focussed modal rules.

◦ (substructural logics) a terminating single-conclusion extension of(the standard calculus for) FLe by semi-analytic rules.

The notion of focussed (modal) rule in the first two cases is not the same.

In all cases there are a finite number of interpolant properties such thatthe above also holds for the calculi extended by rules satisfying theseproperties, provided the whole calculus is terminating.

(interpolant properties: variants of statements of the form “if thepremisses have uniform interpolants, then so does the conclusion”)

The above calculi are excluded for the many modal and intermediate andsubstructural logics without uniform interpolation.

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Finis

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