PreprintIn this paper, we present a propositional sequent calculus containing disjoint copies of classical and intuitionistic logics. We prove a cut-elimination theorem and we establish a relation between this system and linear logic

Laurent, Olivier

Nour, Karim

Hal - Université Grenoble Alpes

Parametric mixed sequent calculus

HAL Descartes

HAL Université de Savoie

Parametric mixed sequent calculusKarim Nour, Olivier LaurentTo cite this version:Karim Nour, Olivier Laurent. Parametric mixed sequent calculus. Preprint. 2005. <hal-00382689>HAL Id: hal-00382689https://hal.archives-ouvertes.fr/hal-00382689Submitted on 11 May 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.PARAMETRIC MIXED SEQUENT CALCULUSOlivier LAURENT Karim NOURUniversite´ Denis Diderot Universite´ de SavoiePPS - Case 7014 LAMA - Equipe de logique2, place Jussieu Campus scientifique75251 Paris Cedex 05 - France 73376 Le Bourget du Lac - FranceAbstractIn this paper, we present a propositional sequent calculus containing dis-joint copies of classical and intuitionistic logics. We prove a cut-eliminationtheorem and we establish a relation between this system and linear logic.1 IntroductionThe systems which represent classical and intuitionistic logics have often been stud-ied separately. For each of these logics we establish its proper properties. Althoughthe systems which represent intuitionistic logic are sub-systems of those which rep-resent classical logic, we usually do not find in the literature studies of the propertiesof these systems at the same time. We mainly find many external translations fromclassical to intuitionistic logic. The main novelty of the system which we look foris that intuitionistic and classical logics appear as fragments. For instance a proofof an intuitionistic formula may use classical lemmas without any restriction. Thisapproach is radically different from the one that consists in changing the rule of thegame when we want to change the logic. We want only one logic which, dependingon its use, may appear classical or intuitionistic.J.-Y. Girard presented in [2] a single sequent calculus (denoted LU) common toclassical, intuitionistic and linear logics. Each formula is given with a polarity: pos-itive, neutral and negative. For each connective the rules depend on the polarityof the formulas. On the other hand the system LU has a cut-elimination theoremand then the sub-formula property. Although the system LU is an answer to ourquestion, we seek a simpler and more intuitive system only for classical and intu-itionistic logics.The usefulness of finding a system which mixes classical and intuitionistic logicsis more and more recognized. For example, J.-L. Krivine and the second authorintroduced a second order mixed logic in order to type storage and control opera-tors in λC-calculus (see [3]). Indeed, they needed intuitionistic logic to characterizethe operational behavior of the storage operators and classical logic for the controloperators. The theoretical properties of this system are not difficult to prove be-cause the only connectives are → and ∀. Recently, C. Raffalli introduced in [5] asecond order mixed logic which is slight extension of that of [3] in order to extracta program for some classical proofs. He applied his method to extract a programfrom a classical proof of Dickson’s lemma.In the paper [4], the second author and A. Nour presented a propositional logic withall connectives (denoted PML) containing three kinds of variables: minimal, intu-itionistic and classical. The absurdity rules are restricted to the formulas containingthe corresponding variables. They introduced for the system PML a Kripke seman-tics and they showed a completeness theorem. They deduced from this theorem avery significant result which is the following: “for a formula to be derivable in a logic,1it is necessary that the formula contains at least a variable which corresponds to thislogic”. They also presented a sequent calculus version of this system. The systemspresented in [4] are not satisfactory because they do not have cut-elimination results.We propose in this paper another approach to solve this problem. We fix a setof formulas P which represents intuitively the set of formulas on which we can doclassical reasoning. A sequent is a pair of multisets of formulas, denoted Γ ⊢ ∆;Π,where ∆ ⊆ P and Π contains at most one formula. We introduce a list of classicalrules on the multiset ∆ and intuitionistic rules on Π. Certain rules will requireconditions on the membership of some formulas to the set P . We prove a cut-elimination theorem and thus deduce the sub-formula property. We show how tocode systems LK and LJ in our system which is coded in system LL.The paper is organized as follows. In section 2 we present the rules of our system.We prove in section 3 the cut elimination properties of the system. The codings ofsystems LK and LJ in our system are given in section 4. We present in section 5 acoding of the system in system LL.2 The MLP sequent calculusDefinition 2.11. The set of formulas is defined by the following grammar:F ::= 0 | ⊥ | X | F ∧ F | F ∨ F | F → Fwhere X ranges over a set of propositional variables V.2. Let P be a subset of formulas. A P-sequent is a pair of multisets of formulas,denoted Γ ⊢ ∆;Π, where ∆ ⊆ P and Π contains at most one formula. Theset ∆ is called the body and Π (the space after “;”) is called the stoup. AP-derivation may be constructed according to one of the rules below.AXIOM/CUTSA ⊢;AaxΓ ⊢ ∆;A Γ′, A ⊢ ∆′; ΠΓ,Γ′ ⊢ ∆,∆′; Πcut1Γ ⊢ ∆, A; Π Γ′, A ⊢ ∆′;Γ,Γ′ ⊢ ∆,∆′; Πcut2STRUCTUREΓ ⊢ ∆;A A∈PΓ ⊢ ∆, A;derΓ, A,A ⊢ ∆;ΠΓ, A ⊢ ∆;ΠclΓ ⊢ ∆, A,A; ΠΓ ⊢ ∆, A; ΠcrΓ ⊢ ∆;ΠΓ, A ⊢ ∆;ΠwlΓ ⊢ ∆;Π A∈PΓ ⊢ ∆, A; ΠwrLOGIC2∆⊆PΓ, 0 ⊢ ∆;Π0⊥⊢;⊥Γ, A,B ⊢ ∆;C A 6∈P and B 6∈PΓ, A ∧B ⊢ ∆;C∧1lΓ ⊢ ∆;A Γ′ ⊢ ∆′;BΓ,Γ′ ⊢ ∆,∆′;A ∧B∧1rΓ, A,B ⊢ ∆;Γ, A ∧B ⊢ ∆;∧2lΓ ⊢ ∆, A; Γ′ ⊢ ∆′, B;Γ,Γ′ ⊢ ∆,∆′;A ∧B∧2rΓ ⊢ ∆;A Γ′ ⊢ ∆′, B;Γ,Γ′ ⊢ ∆,∆′;A ∧B∧3rΓ ⊢ ∆, A; Γ′ ⊢ ∆′;BΓ,Γ′ ⊢ ∆,∆′;A ∧B∧4rΓ, A ⊢ ∆;C Γ, B ⊢ ∆;C A 6∈P and B 6∈PΓ, A ∨B ⊢ ∆;C∨1lΓ ⊢ ∆;AΓ ⊢ ∆;A ∨B∨1rΓ ⊢ ∆;BΓ ⊢ ∆;A ∨B∨2rΓ, A ⊢ ∆; Γ, B ⊢ ∆;Γ, A ∨B ⊢ ∆;∨2lΓ ⊢ ∆, A;Γ ⊢ ∆;A ∨B∨3rΓ ⊢ ∆, B;Γ ⊢ ∆;A ∨B∨4rΓ, B ⊢ ∆;C Γ′ ⊢ ∆′;A B 6∈PΓ,Γ′, A→ B ⊢ ∆,∆′;C→1lΓ, A ⊢ ∆;BΓ ⊢ ∆;A→ B→1rΓ, B ⊢ ∆; Γ′ ⊢ ∆′;AΓ,Γ′, A→ B ⊢ ∆,∆′;→2lΓ, A ⊢ ∆, B;Γ ⊢ ∆;A→ B→2rΓ, B ⊢ ∆; Γ′ ⊢ ∆′, A; ΠΓ,Γ′, A→ B ⊢ ∆,∆′; Π→3lWe write Γ ⊢P ∆;Π if the P-sequent Γ ⊢ ∆;Π is derivable in system MLP .Remark 2.1 1. The conditions which we add on the set P in some left logicalrules are necessary to obtain a cut-elimination theorem. Indeed without theseconditions the cuts on the principal formulas of the rules (∧ir 2 ≤ i ≤ 4 and∧1l) or (∨ir 3 ≤ i ≤ 4 and ∨1l) or (→2r and →1l) cannot be eliminate.2. We can remove the rules ∧3r and ∧4r and replace the rule ∧1lby the followingrule:Γ, A,B ⊢ ∆;C A 6∈P or B 6∈PΓ, A ∧B ⊢ ∆;C∧1lIn this new system, the results of sections 3 and 4 remain true but not thoseof section 5.3. We chose an additive “or” to facilitate the embedding of our system in LL.The results of sections 3 and 4 remain true if we add the two following leftrules:Γ, A ⊢ ∆;C Γ, B ⊢ ∆; A 6∈PΓ, A ∨B ⊢ ∆;C∨3lΓ, A ⊢ ∆; Γ, B ⊢ ∆;C B 6∈PΓ, A ∨B ⊢ ∆;C∨4l34. If we consider the connector ¬ as primitive, we can add the following rules:Γ ⊢ ∆;AΓ,¬A ⊢ ∆;¬1lΓ, A ⊢ ∆;Γ ⊢ ∆;¬A¬rΓ ⊢ ∆, A; ΠΓ,¬A ⊢ ∆;Π¬2lIf ⊥ ∈ P, we derive ¬A ⊢P ;A→ ⊥ and A→ ⊥ ⊢P ;¬A.3 Properties of system MLPTheorem 3.1 The Hauptsatz holds for MLP .Proof The degree of a cut-rule in a P-derivation is the pair of integers (l, k)where l is the length of the cut-formula and k is defined by:• k = 3, if it is the rule cut2.• k = 2, if it is the rule cut1 and the cut-formula of the stoup of the left premiseis the principal formula of a logical rule.• k = 1, if it is the rule cut1 and the cut-formula of the stoup of the left premiseis the principal formula of a logical rule but not the cut-formula of the rightpremise.• k = 0, if it is the rule cut1 and the cut-formulas are the principal formulas oflogical rules.The order we consider on degrees is the lexicographic order.The degree of a P-derivation is the finite list of increasing degrees of its cuts. Weconsider also the lexicographic order on these degrees.Let D be a P-derivation. We will explain how to reduce a cut in D to obtain aP-derivation of smaller degree.We consider a cut of degree (l, k) where its premises are derivable without thecut-rules.• If k = 0, we replace this cut by other cuts of degrees (l′, k′) where l′ < l.• If k = 1, we move up the left premise in the P-derivation of the right premiseat the places where the cut-formula was the principal formula of a logical rule.We thus replace this cut by other cuts of degrees (l, 0).• If k = 2, we move up the right premise in the P-derivation of the left premiseat the places where the cut-formula was the principal formula of a logical rule.We thus replace this cut by other cuts of degrees (l, 1) or (l, 0).• If k = 3, we move up the right premise in the P-derivation of the left premiseat the places where the cut-formula was introduced using the rules der, wr or0. We thus replace this rule cut2 by other rules cut1 of degrees (l, k′) where0 ≤ k′ ≤ 2.We notice that in each case the degree of the obtained P-derivation decreasesstrictly.Corollary 3.1 The MLP has the sub-formula property.Proof By theorem 3.1. 4Corollary 3.2 1. If ⊢P ;A ∨B, then ⊢P ;A or ⊢P A; or ⊢P ;B or ⊢P B;.2. If A,B 6∈ P and ⊢P ;A ∨B, then ⊢P ;A or ⊢P ;B.Proof We consider a normal derivation of ⊢P ;A∨B and we look at the last usedrule. 4 Codings of LK and LJ in MLPWe consider systems LK and LJ constructed respectively over the sets of variablesV ∪ {⊥} and V ∪ {0}. We suppose that ⊥ (resp. 0) is the symbol for the absurdityof LK (resp. LJ). We will give some conditions to code separately systems LK andLJ in MLP .Definition 4.1 A set of formulas S is said to be stable ifffor every c ∈ {∧,∨,→}, if AcB ∈ S, then A,B ∈ STheorem 4.1 Let K be a stable set such that K ⊆ P and 0 6∈ K.If Γ,∆ ⊆ K, then Γ ⊢P ∆; iff Γ ⊢LK ∆.Proof ⇒ : If we replace “;” by “,”, the rules of MLP are rules of LK.⇐ : We consider a derivation D of Γ ⊢LK ∆. We check that we can move up therules used in D without putting formulas in the stoups. The rule cut and the leftrules of LK correspond to the rules cut2, cl, wl, ⊥, ∧2l,∨2land→3l(without stoups).The axiom and the right rules of LK correspond to ax, cr, wr, ∧2r, ∨3r , ∨4r and →2rusing the rule der. Remark 4.1 1. The set K = P of all formulas on V ∪{⊥} satisfies the hypoth-esis of theorem 4.1.2. For every formula A on V ∪ {⊥}, let KA be the finite set of the sub-formulasof A. If KA ⊆ P, then ⊢P A; iff ⊢LK A.Theorem 4.2 Let I be a stable set such that I ∩ P = ∅ and ⊥6∈ I.If Γ, A ⊆ I, then Γ ⊢P ;A iff Γ ⊢LJ A.Proof ⇒ : We consider a normal P-derivation D of Γ ⊢P ;A. The fact thatΓ, A ⊆ I allows to move up the rules using in D without putting formulas in thebodies. Then the only rules used in D are intuitionistic rules.⇐ : The rules of LJ correspond to the rules ax, cut1, cl, wl, 0, ∧1l , ∧1r, ∨1l, ∨1r, ∨2r ,→1land →1r (without bodies). Remark 4.2 1. Let P = {⊥}. The set I of all formulas on V ∪ {0} satisfiesthe hypothesis of theorem 4.2.2. Let A be a formula on V ∪ {0}. We have ⊢{⊥};A iff ⊢LJ A.To code, at the same time, the two systems, it is necessary to realize all the condi-tions of theorems 4.1 and 4.2. We give an example of such a system.Definition 4.2 1. We suppose that we have two disjoint sets of propositionalvariables: Vi = {Xi, Yi, Zi, ...} the set of intuitionistic variables and Vc ={Xc, Yc, Zc, ...} the set of classical variables. Let V = Vi ∪ Vc.2. If A is a formula, we denote by var(A) the set of variables and constants ofA.53. Let F˜ be the set of all formulas,K˜ = {F ∈ F˜ / var(F ) ⊆ Vc ∪ {⊥}} the set of classical formulas,I˜ = {F ∈ F˜ / var(F ) ⊆ Vi ∪ {0}} the set of intuitionistic formulas andP˜ = F˜ − I˜.Corollary 4.1 1. If Γ,∆ ⊆ K˜, then Γ ⊢P˜ ∆; iff Γ ⊢LK ∆.2. If Γ, A ⊆ I˜, then Γ ⊢P˜ ;A iff Γ ⊢LJ A.Proof We use theorems 4.1 and 4.2. 5 Coding of MLP in LLDefinition 5.1 1. We define the following two translations b and t from MLPto LL:• if A 6∈ P, then b(A) = t(A)• if A ∈ P, then b(A) =?t(A)and• t(0) = t(⊥) = 0• t(X) =!X, for every X ∈ V• t(A ∧B) =!b(A)⊗!b(B)• t(A ∨B) =!b(A)⊕!b(B)• t(A→ B) =!(t(A)−◦b(B))2. If Γ = A1, ..., An, then t(Γ) = t(A1), ..., t(An).Lemma 5.1 1. If Γ, t(A), t(A) ⊢LL ∆, then Γ, t(A) ⊢LL ∆.2. If Γ ⊢LL ∆, then Γ, t(A) ⊢LL ∆.3. If t(Γ), A ⊢LL?t(∆), then t(Γ), ?A ⊢LL?t(∆).4. If t(Γ) ⊢LL?t(∆), A, then t(Γ) ⊢LL?t(∆), !A.Proof See [1]. Theorem 5.1 If Γ ⊢P ∆;Π, then t(Γ) ⊢LL?t(∆), t(Π)Proof By induction on a P-derivation of Γ ⊢P ∆;Π. We look at the last ruleused.• For the rules ax, cut1, der, cr, wr, 0, ⊥, ∧1l, ∨1land →1l, the proof is easy.• For the rules cl and wl, we use 1. and 2. of lemma 5.1.• For the rules cut2, ∧2l , ∨2land →2l, we use 3. of lemma 5.1.• For the rules ∧ir (1 ≤ i ≤ 4), ∨il(1 ≤ i ≤ 4) and →il(1 ≤ i ≤ 2), we use 4. oflemma 5.1.6• For the rule →3l, we have, by induction hypothesis, t(Γ), b(B) ⊢LL?t(∆) andt(Γ′) ⊢LL p?t(∆′), b(A), t(Π). We deduce thatt(Γ), b(B) ⊢LL?t(∆) t(A) ⊢ t(A)t(Γ), t(A)−◦b(B), t(A) ⊢?t(∆)t(Γ), !(t(A)−◦b(B)), t(A) ⊢?t(∆)t(Γ), !(t(A)−◦b(B)), b(A) ⊢?t(∆) t(Γ′) ⊢LL?t(∆′), b(A), t(Π)t(Γ), t(Γ′), t(A→ B) ⊢?t(∆), ?t(∆′), t(Π)References[1] V. Danos, J.-B. Joinet and H. Schellinx, A new deconstructive logic: LinearLogic, Journal of Symbolic Logic, vol. 62, num. 3, pp. 755-807, 1997.[2] J.-Y. Girard, On the unity of logic, Annals of pure and applied logic, vol. 59,pp. 201–217, 1993.[3] K. Nour, Mixed logic and storage operators, Archive for Mathematical Logic,vol. 39, pp. 261–280, 2000.[4] K. Nour and A. Nour; Propositional mixed logic: its syntax and semantics,Journal of Applied Non-Classical Logics, vol. 13, pp. 377-390, 2003.[5] C. Raffali, Getting results from programs extracted from classical proofs, Theo-retical Computer Science, vol. 323, pp. 49-70, 2004.7

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Parametric mixed sequent calculus

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