Regular Properties and the Existence of Proof Systems AiML & - - PowerPoint PPT Presentation

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 by Modus Ponens from previously derived formulas. Not all proofs in a proof system have such a form: in natural deduction proofs can contain (discharged) assumptions. And resolution and Gentzen calculi are not even about formulas but about clauses and sequents. Given a logic, there often are (faithful) translations between the different proof 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 do not 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 logics establish 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 the frequency 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 interpolant depends only on the premiss or the conclusion: For all ϕ there are formulas ∃pϕ and ∀pϕ not containing p such that for all ψ not containing 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 uniform interpolation: ∃pϕ(p, ¯ q) =

• {ψ(¯

q) | ⊢ ϕ(p, ¯ q) → ψ(¯ q)} ∀pϕ(p, ¯ q) =

• {ψ(¯

q) | ⊢ ψ(¯ q) → ϕ(p, ¯ q)}.

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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 proof system 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 that class) for many such logics. The method also provide a uniform and modular way to prove uniform interpolation for classes of logics, including some logics for which this was unknown, such as KD. The class of proof systems is defined not in terms of concrete rules but in terms of the structural properties of rules. In this talk: classical modal logic with one modal operator. Language: ⊥, ∧, ∨, →,✷, 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 ✷Γ ≡df {✷ϕ | ϕ ∈ Γ} and ✷(Γ ⇒ ∆) ≡df (✷Γ ⇒ ✷∆) and (Γ ⇒ ∆) · (Π ⇒ Σ) ≡df (Γ, Π ⇒ ∆, Σ). Dfn A sequent calculus is a set of rules, where a rule R is an expression

• f the form

S1 . . . Sn S0 R (fr)

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

σS1 . . . σSn σS0 R

where σ is a substitution for the modal language. Dfn A nonaxiom rule (fr) is focussed if S0 contains a single nonboxed formula 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′ R(S)

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the proof system G3 Dfn A nonaxiom rule is focussed if the conclusion is a single, nonboxed formula and for every instance R and sequent S, R(S) is an instance of the 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 sequents such 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 a single, 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 · ✷S1 · ✷S0 R

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, ✷Γ ⇒ ✷ϕ, ∆ RK Γ, ϕ ⇒ Π, ✷Γ, ✷ϕ ⇒ ∆ RD

Example Rules that are not focussed (modal):

Γ, ψ → χ ⇒ ϕ → ψ Γ, χ ⇒ ∆ Γ, (ϕ → ψ) → χ ⇒ ∆ Γ, ✷ϕ ⇒ ϕ Π, ✷Γ ⇒ ✷ϕ, ∆ RGL

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results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of focussed and focussed modal rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. Proof All rules in the sequent calculus G3 are focussed. ⊣ Corollary The modal logics K (Ghilardi) and KD (Iemhoff) have uniform interpolation. A promised negative result: Corollary If a modal logic does not have uniform interpolation, then it does not have a terminating calculus that consists of focussed and focussed modal

• rules. 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 that class has uniform interpolation. Side benefit: Establishing uniform interpolation in a uniform, modular way, 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 Theorem A modal logic with a terminating calculus that consists of focussed and focussed modal rules has uniform interpolation. Proof idea Define for each rule R in the calculus and sequent S an expression ∀R

• pS. E.g. for focussed rules R:

R = (S1 . . . Sn/S0) ∀

R

pS0 ≡df ∀pS1 ∧ . . . ∧ ∀pSn

Inductively define ∀pS ≡df

{∀R pS | 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, if the premisses of a rule have a uniform interpolant, then so does the conclusion. Some details are omitted . . . ⊣ Uniform 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 and

• Hudelmaier. Not all rules of G4i are focussed.

Theorem (Iemhoff 2017) Any calculus that is an extension of G4i with focussed rules (in the sense of intermediate logic) has uniform interpolation. Proof Prove for the nonfocussed rules of G4i that if the premisses have a uniform interpolant, then so does the conclusion. Further proceed as in the classical case. ⊣ Corollary No intermediate logic except the 7 with uniform interpolation has such a calculus. Corollary When developing a calculus based on G4i for an intermediate logic without uniform interpolation, then some of the rules have to be not focussed.

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intuitionistic modal logic Work in progress. The logics are extensions of iK (only ✷, no diamond ✸). The sequent calculi are extensions of G4iK, which is G4i plus the rules

Γ ⇒ ϕ Π, ✷Γ ⇒ ✷ϕ, ∆ RK Γ ⇒ ϕ Π, ✷Γ, ψ ⇒ ∆ Π, ✷Γ, ✷ϕ → ψ ⇒ ∆ L✷

→

Lemma G4iK is terminating. Theorem Any logic with a calculus that is an extension of G4iK with focussed and focussed modal rules (in the sense of intuitionistic modal logic) 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 such that all rules that are not focussed (in the sense of intuitionistic modal logic) 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 uniform interpolation. Work in progress: Theorem (Tabatabai & Jalali 2018) Any logic with a terminating sequent calculus that extends the standard calculus 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 substructural logics. 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 focussed

and focussed modal rules.

• (intermediate & intuitionistic modal logics) a terminating calculus

that is an extension of G4iK by focussed and focussed modal rules.

• (substructural logics) a terminating calculus that is a

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 that the above also holds for the calculi extended by rules satisfying these properties, provided the whole calculus is terminating. (interpolant properties: variants of statements of the form “if the premisses 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 focussed

and focussed modal rules.

• (intermediate & intuitionistic modal logics) a terminating extension
• f 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 that the above also holds for the calculi extended by rules satisfying these properties, provided the whole calculus is terminating. (interpolant properties: variants of statements of the form “if the premisses have uniform interpolants, then so does the conclusion”) The above calculi are excluded for the many modal and intermediate and substructural logics without uniform interpolation.

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Finis

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