Regular Properties and the Existence of Proof Systems AiML & LATD Bern, August 28, 2018 Rosalie Iemhoff Utrecht University, the Netherlands 1 / 23
The Existence of Proof Systems AiML & LATD Bern, August 28, 2018 Rosalie Iemhoff Utrecht University, the Netherlands 2 / 23
Regular Properties and the Existence of Proof Systems AiML & LATD Bern, August 28, 2018 Rosalie Iemhoff Utrecht University, the Netherlands 3 / 23
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? 4 / 23
Proof systems A Hilbert system consists of Axioms: ϕ 1 , . . . , ϕ n ϕ ϕ → ψ Rule Modus Ponens: 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. 5 / 23
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. . . . 6 / 23
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. 7 / 23
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. 8 / 23
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 ∃ p ϕ and ∀ p ϕ not containing p such that for all ψ not containing p: ⊢ ψ → ϕ iff ⊢ ψ → ∀ p ϕ ⊢ ϕ → ψ iff ⊢ ∃ p ϕ → ψ. ∃ p ϕ is the right interpolant and ∀ p ϕ the left interpolant: ⊢ ϕ → ∃ p ϕ ⊢ ∀ p ϕ → ϕ. Note A locally tabular logic that has interpolation, has uniform interpolation: � ∃ p ϕ ( p , ¯ q ) = { ψ (¯ q ) | ⊢ ϕ ( p , ¯ q ) → ψ (¯ q ) } � ∀ p ϕ ( p , ¯ q ) = { ψ (¯ q ) | ⊢ ψ (¯ q ) → ϕ ( p , ¯ q ) } . 9 / 23
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. S 4 does not. Theorem (Bilkova ’06) KT has uniform interpolation. K 4 does not. Theorem (Maxsimova ’77, Ghilardi & Zawadowski ’02) There are exactly seven intermediate logics with (uniform) interpolation: IPC , Sm, GSc, LC, KC, Bd 2 , CPC . Van Gool & Metcalfe & Tsinakis 2017: general approach. Pitts uses a terminating sequent calculus for IPC . (developed independently by Dyckhoff and Hudelmaier in ’92) 10 / 23
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: ⊥ , ∧ , ∨ , → , ✷ , p 1 , p 2 , . . . . 11 / 23
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 of the form S 1 . . . S n R S 0 ( fr) for certain sequents S 0 , . . . , S n (that may be empty). An instance R of a rule is of the form σ S 1 . . . σ S n R σ S 0 where σ is a substitution for the modal language. Dfn A nonaxiom rule (fr) is focussed if S 0 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 ′ . . . S · S ′ 1 n R ( S ) S · S ′ 0 12 / 23
the proof system G 3 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 G 3 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 ⇒ ∆ 13 / 23
the proof systems for modal logic Dfn A nonaxiom rule R = ( S 1 . . . S n / S 0 ) is focussed if S 0 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 ◦ S 1 · S 0 S 2 · ✷ S 1 · ✷ S 0 R where S 0 contains a single formula, that is boxed, S 2 is of the form (Π ⇒ ∆) , S 1 contains only multisets, and ◦ S 1 denotes S 1 or � S 1 . Example Focussed (modal) rules: Γ ⇒ ϕ, ψ, ∆ Γ ⇒ ϕ Γ , ϕ ⇒ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R K Π , ✷ Γ , ✷ ϕ ⇒ ∆ R D Γ ⇒ ϕ ∨ ψ, ∆ Example Rules that are not focussed (modal): Γ , ψ → χ ⇒ ϕ → ψ Γ , χ ⇒ ∆ � Γ , ✷ ϕ ⇒ ϕ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R GL Γ , ( ϕ → ψ ) → χ ⇒ ∆ 14 / 23
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 G 3 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 K 4 and S 4. Interplay: Semantics (algebraic logic) and proof theory. 15 / 23
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. 16 / 23
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 R = ( S 1 . . . S n / S 0 ) ∀ pS 0 ≡ df ∀ pS 1 ∧ . . . ∧ ∀ pS n � {∀ R Inductively define ∀ pS ≡ df 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. 17 / 23
Recommend
More recommend
