We establish syntactic cut-elimination for the one-variable fragment of the
modal mu-calculus. Our method is based on a recent cut-elimination technique by
Mints that makes use of Buchholz' Omega-rule.Comment: In Proceedings FICS 2012, arXiv:1202.317

Alwen Tiu

Brian Hill

Dale Miller

David Baelde

Dexter Kozen

Gaisi Takeuti

Gerhard Jäger

Grigori Mints

Kai Brünnler

Regimantas Pliuskevicius

Thomas Studer

Wilfried Buchholz

Zoltán Ésik

English

arXiv

Mints, Grigori

Studer, Thomas

Bern Open Repository and Information System (BORIS)

Submitted to:FICS 2012Cut-elimination for the mu-calculus with one variableGrigori MintsDept. of PhilosophyStanford UniversityUSAgmints@stanford.eduThomas StuderInst. of Computer Science and Appl. Math.University of BernSwitzerlandtstuder@iam.unibe.chWe establish syntactic cut-elimination for the one-variable fragment of the modal mu-calculus. Ourmethod is based on a recent cut-elimination technique by Mints that makes use of Buchholz’ Ω-rule.1 IntroductionThe propositional modal µ-calculus is a well-established modal fixed point logic that includes fixedpoints for arbitrary positive formulae. Thus it subsumes many temporal logics (with an always operator),epistemic logics (with a common knowledge operator), and program logics (with an iteration operator).Making use of the finite model property, Kozen [10] introduces a sound and complete infinitarysystem for the modal µ-calculus. In this system greatest fixed points are introduced by means of theω-rule that has a premise for each finite approximation of the greatest fixed point. Ja¨ger et al. [8] showby semantic methods that the cut rule is admissible in this kind of infinitary systems. So far, however,there is no syntactic cut-elimination procedure available for the modal µ-calculus. It is our aim in thispaper to present an effective cut-elimination method for the one-variable fragment of the µ-calculus.There are already a few results available on syntactic cut-elimination for modal fixed point logics.Most of them make use of deep inference where rules may not only be applied to outermost connectivesbut also deeply inside formulae. The first result of this kind has been obtained by Pliuskevicius [12] whopresents a syntactic cut-elimination procedure for linear time temporal logic. Bru¨nnler and Studer [2]employ nested sequents to develop a cut-elimination procedure for the logic of common knowledge. Hilland Poggiolesi [7] use a similar approach to establish effective cut-elimination for propositional dynamiclogic. A generalization of this method is studied in [3] where it is also shown that it cannot be extendedto fixed points that have a 2-operator in the scope of a µ-operator. Fixed points of this kind occur, forinstance, in CTL in the form of universal path quantifiers.Thus we need a more general approach to obtain syntactic cut-elimination for the modal µ-calculus.A standard proof-theoretic technique to deal with inductive definitions and fixed points is Buchholz’Ω-rule [4, 6]. Ja¨ger and Studer [9] present a formulation of the Ω-rule for non-iterated modal fixedpoint logic and they obtain cut-elimination for positive formulae of this logic. In order to overcome thisrestriction to positive formulae, Mints [11] introduces an Ω-rule that has a wider set of premises, whichenables him to obtain full cut-elimination for non-iterated modal fixed point logic.Mints’ cut-elimination algorithm makes use of, in addition to ideas from [5], a new tool presentedin [11]. It is based on the distinction, see [13], between implicit and explicit occurrences of formulae ina derivation with cut. If an occurrence of a formula is traceable to the endsequent of the derivation, thenit is called explicit. If it is traceable to a cut-formula, then it is an implicit occurrence.Implicit and explicit occurrences of greatest fixed points are treated differently in the translationof the induction rule to the infinitary system. An instance of the induction rule that derives a sequent2 Cut-elimination for the mu-calculus with one variableνX .A,B goes to an instance of the ω-rule if νX .A is explicit. Otherwise, if νX .A is traceable to a cut-formula, the induction rule is translated to an instance of the Ω-rule that is preserved until the last stageof cut-elimination. At that stage, called collapsing, the Ω-rule is eliminated completely.In the present paper we show that this method can be extended to a µ-calculus with iterated fixedpoints. Hence we obtain complete syntactic cut-elimination for the one-variable fragment of the modalµ-calculus. Our infinitary system is completely cut-free in the sense that there are not only no cut rulesin the system but also no embedded cuts. Thus our cut-free system enjoys the subformula property. Thisis in contrast to the recent cut-elimination results by Baelde [1] and by Tiu and Momigliano [14] for thefinitary systems µMALL and Linc−, respectively, where the ν-introduction rule and the co-induction rulecontain embedded cuts, which results in the loss of the subformula property.2 Syntax and semanticsWe first introduce the language L . We start with a countable set PROP of atomic propositions pi andtheir negations pi. We use P to denote an arbitrary element of PROP. Moreover, we will use a specialvariable X .Definition 1. Operator forms A,B, . . . are given by the following grammar:A :== pi | pi | X | A∧A | A∨A | 2A | 3A | µX .A | νX .A.Formulae F are defined by:F :== pi | pi | F ∧F | F ∨F | 2F | 3F | µX .A | νX .A.The fixed point operators µ and ν bind the variable X and, therefore, we will talk of free and boundoccurrences of X . Hence a formula is an operator form without free occurrences of X .The negation of an operator form is inductively defined as follows.1. ¬pi := pi and ¬pi := pi2. ¬X := X3. ¬(A∧B) := ¬A∨¬B and ¬(A∨B) := ¬A∧¬B4. ¬2A :=3¬A and ¬3A :=2¬A5. ¬µX .A := νX .¬A and ¬νX .A := µX .¬ANote that negation is well-defined: the negation of an X-positive operator form is again X-positivesince we have ¬X := X . Thus, for example,¬µX .2(pi∧X) := νX .¬2(pi∧X) := νX .3¬(pi∧X) := νX .3(¬pi∨¬X) := νX .3(pi∨X).For an arbitrary but fixed atomic proposition pi we set > := pi∨ pi. If A is an operator form, then wewrite A(B) for the result of simultaneously substituting B for every free occurrence of X in A. We willalso use finite iterations of operator forms, given as followsA0(B) := B and Ak+1(B) := A(Ak(B)).G. Mints & T. Studer 3Γ,P,¬P Γ,µX .A,¬µX .AΓ,A,BΓ,A∨B (∨)Γ,A Γ,BΓ,A∧B (∧)Γ,A3Γ,2A,Σ(2)Γ,A(µX .A)Γ,µX .A(clo)¬A(B),B¬µX .A,B (ind)Γ,A Γ,¬AΓ(cut)Figure 1: System M3 System MSystem M derives sequents, that are finite sets of formulae. We denote sequents by Γ,Σ and use thefollowing notation: if Γ := {A1, . . . ,An}, then 3Γ := {3A1, . . . ,3An}, System M consists of the axiomsand rules given in Figure 1.4 System MωSystem Mω is an infinitary cut-free system for the modal µ-calculus with one variable. It consists of theaxioms and rules given in Figure 2.Γ,P,¬PΓ,A,BΓ,A∨B (∨)Γ,A Γ,BΓ,A∧B (∧)Γ,A3Γ,2A,Σ(2)Γ,A(µX .A)Γ,µX .A(clo)Γ,Ai(>) for all natural numbers iΓ,νX .A(ω)Figure 2: System Mω5 System Mω,ΩkIn order to embed M into Mω , we need a family of intermediate systems Mω,Ωk that include additionalrules to derive greatest fixed points that later will be cut away.The languageLΩ extendsL by a new connective ν ′ to denote those greatest fixed points. Formally,LΩ is given as follows. Operator forms ofLΩ are defined like operator forms ofL with the additionalcase1. If A is an operator form, then ν ′X .A is also an operator form.A formula of LΩ is an LΩ operator form without free occurrence of X . A formula is a greatest fixedpoint if it has the form νX .A or ν ′X .A.4 Cut-elimination for the mu-calculus with one variableDefinition 2. The level lev(A) of an operator form A is the maximal nesting of fixed point operators inA. Formally we set:1. lev(P) := lev(X) := 0 for all P in PROP2. lev(A∧B) := lev(A∨B) := max(lev(A), lev(B))3. lev(2A) := lev(3A) := lev(A)4. lev(µX .A) := lev(νX .A) := lev(ν ′X .A) := lev(A)+1The level of a sequent is the maximum of the levels of its formulae. We say a formula (sequent) isk-positive if for all ν ′X .A occurring in it we have lev(ν ′X .A)< k.When working in Mω,Ωk , we will use the following notation: the formula A′ is obtained from A byreplacing all occurrences of νX in A with ν ′X .Let k ≥ 0. System Mω,Ωk consists of the axioms and rules of Mω (formulated in LΩ) and the addi-tional rules: cut, Ωh, and Ω˜h. The cut rule is given as followsΓ,A′ Γ,(¬A)′Γ(cut),where A is a formula with lev(A)≤ k. The rules Ωh and Ω˜h , where 1≤ h≤ k, are informally describedas follows:· · ·Mω,Ωk−1 0 ∆, (µX .A)′∆, Γ · · ·ΩhΓ, (¬µX .A)′andΓ, (µX .A)′ · · ·Mω,Ωk−1 0 ∆, (µX .A)′∆, Γ · · ·Ω˜hΓwhere lev((¬µX .A)′) = h and ∆ ranges over h-positive sequents such that there is a cut-free proof of thesequent ∆, (µX .A)′ in Mω,Ωk−1 .Definition 3. We use Mω,Ωk 0 Γ to express that there is a cut-free derivation of Γ in Mω,Ωk .In a more formal notation we can state the Ωh-rule as follows. If for every h-positive sequent ∆Mω,Ωk−1 0 ∆, (µX .A)′ =⇒ Mω,Ωk ∆, Γ,thenMω,Ωk Γ, (¬µX .A)′,and similarly for Ω˜h.Note that System Mω,Ω0 does not include Ωh- or Ω˜h-rules. Hence we immediately get the followinglemma.Lemma 4. Let Γ be anL sequent. We haveMω,Ω0 0 Γ =⇒ Mω Γ.G. Mints & T. Studer 56 EmbeddingIn this section we present a translation from M-proofs into Mω,Ωk -proofs. First we establish an auxiliarylemma.Lemma 5. For all natural numbers h≤ k we have the following.1. If lev(µX .A) = h, then Mω,Ωk 0 µX .A, ¬µX .A.2. If lev(A) = h, then Mω,Ωk 0 Γ, A′ =⇒ Mω,Ωk 0 Γ, A.3. If lev(µX .A) = h, then Mω,Ωk 0 µX .A, (¬µX .A)′.4. If lev(A) = h, then Mω,Ωk 0 B,C =⇒ Mω,Ωk 0 (¬A)(B), A(C).5. If lev(A) = h, then Mω,Ωk 0 B,C′ =⇒ Mω,Ωk 0 (¬A)(B), A′(C′).Proof. The five statements are shown simultaneously by induction on h. For space considerations weshow only one particular case of the second statement, which is shown by induction on the derivation ofΓ, A′ and a case distinction on the last rule. Assume the last rule is an instance of Ωh with main formulaA′. We have A′ = (νX .A0)′ with lev(A0) < h. By the premise of the Ωh-rule we have for all h-positivesequents ∆Mω,Ωk−1 0 ∆,(µX .¬A0)′ =⇒ Mω,Ωk 0 ∆,Γ. (1)Trivially we haveMω,Ωk 0 >,Γ. (2)We also haveMω,Ωk−1 0 >,(µX .¬A0)′from which we get by the induction hypothesis for the fifth claim of this lemmaMω,Ωk−1 0 A0(>),(¬A0)′((µX .¬A0)′).An application of clo yieldsMω,Ωk−1 0 A0(>),(µX .¬A0)′.By (1) we getMω,Ωk 0 A0(>),Γ. (3)Note that (2) and (3) are the first two premises of an instance of ω . By further iterating this we obtainfor all iMω,Ωk 0 Ai0(>),Γ.Hence an application of ω yieldsMω,Ωk 0 νX .A0,Γ.We will need a certain form of the induction rule in Mω,Ωk , which we are going to derive next. Wewrite Σ[(µX .A)′ := B] for the result of simultaneously replacing in every formula in Σ every occurrenceof (µX .A)′ with B.6 Cut-elimination for the mu-calculus with one variableLemma 6. Let A be an operator form with lev(νX .A)≤ k. Let ∆,Σ1,Σ2 be h-positive sequents and let Bbe a formula with lev(B)≤ k. Assume thatMω,Ωk (¬A(B))′, B and Mω,Ωk (¬A(B))′, B′.Then we have, ifMω,Ωk−1 0 ∆, Σ1, Σ2thenMω,Ωk ∆, Σ1[(µX .A)′ := B], Σ2[(µX .A)′ := B′].Lemma 7. Let A be an operator form with lev(νX .A) ≤ k. Further let B be an arbitrary formula withlev(B)≤ k. Assume thatMω,Ωk (¬A(B))′, B and Mω,Ωk (¬A(B))′, B′.Then we haveMω,Ωk (¬µX .A)′, B and Mω,Ωk (¬µX .A)′, B′.Proof. Let h = lev(νX .A). In view of our assumptions and the previous lemma we know that for allh-positive sequents ∆Mω,Ωk−1 0 ∆, (µX .A)′ =⇒ Mω,Ωk ∆, B.Hence by an application of the Ωh-rule we conclude Mω,Ωk (¬µX .A)′, B. Similarly, we can deriveMω,Ωk (¬µX .A)′, B′.Theorem 8. Let Γ be a sequent of L . Assume M Γ and assume further for any sequent ∆ occurringin that proof we have lev(∆)≤ k. Then we have Mω,Ωk Γ.Proof. An operation σ on sequents is called ’-operation if σ(Γ,A1, . . . ,An) = Γ,A′1, . . . ,A′n. The result ofapplying σ to a sequent Γ is denoted Γσ .To establish the theorem, we show by induction on the depth of the M-proof that for all ’-operationsσ , we have Mω,Ωk Γσ . We distinguish the following cases for the last rule.1. Γ is an axiom different from Γ0,µX .A,¬µX .A. Then Γσ is an axiom of Mω,Ωk , too.2. Γ is Γ0,µX .A,¬µX .A. Then Γσ follows either by the first or the third claim of Lemma 5 dependingon whether ¬µX .A is replaced by σ or not.3. The last rule is an instance of ∧, ∨, 2 or clo. We can apply the same rule in Mω,Ωk .4. The last rule is a cutΓ,A Γ,¬AΓ.We extend the current ’-operation σ to a ’-operation τ such that (Γ,A)τ = Γσ ,A′ and (Γ,¬A)τ =Γσ ,(¬A)′ By the induction hypothesis for the ’-operation τ we obtain Mω,Ωk Γσ ,A′ as well asMω,Ωk Γσ ,(¬A)′. With an instance of cut we get Mω,Ωk Γσ .5. The last rule is an instance of the induction rule. Then the endsequent has the form ¬µX .A, Bwhich is νX .¬A, B. There are two possible cases.(a) The principal occurrence of νX .¬A is not changed by σ . By the induction hypothesis we canderive (¬A(B))′, Bσ and (¬A(B))′, B′. We obtain our claim by the following proof.G. Mints & T. Studer 7· · ·I.H.(¬A(B))′, BσI.H.(¬A(B))′, B′>,B′L. 5(¬A)(>),(A(B))′cut(¬A)(>),B′...(¬A)i(>),B′L. 5(¬A)i+1(>),(A(B))′cut(¬A)i+1(>),Bσ · · ·ωνX .¬A, Bσ(b) The principal occurrence of νX .¬A is changed by σ . Let τ1,τ2 be ’-operations such that(¬A(B),B)τ1 = (¬A(B))′,Band(¬A(B),B)τ2 = (¬A(B))′,B′.By the induction hypothesis for τ1 and τ2 we obtainMω,Ωk (¬A(B))′, B and Mω,Ωk (¬A(B))′, B′.We apply Lemma 7 and conclude Mω,Ωk (¬µX .A)′, Bσ .7 Cut eliminationWe eliminate instances of cut in the standard way, see for instance [5, 11], by pushing them up thederivation. When an instance of cut with cut formulae (µX .A)′ and (¬µX .A)′ meets the instance of Ωhthat introduces (¬µX .A)′, this pair of inferences is replaced by Ω˜h.Lemma 9 (Cut-elimination). If Mω,Ωk Γ, then Mω,Ωk 0 Γ.The cut-elimination process terminates in a formally cut-free derivation that may contain instancesof Ω˜h-rules. Now we show that these instances of Ω˜h also can be eliminated.Lemma 10 (Collapsing). Let Γ be an (h+1)-positive sequent. If Mω,Ωk 0 Γ, then Mω,Ωh 0 Γ.Proof. By transfinite induction on the derivation in Mω,Ωk . The only interesting case is when the last ruleis an instance of Ω˜l for h < l ≤ k as followsΓ, (µX .A)′ · · ·Mω,Ωl−1 0 ∆, (µX .A)′∆, Γ · · ·Ω˜lΓNote that Γ,(µX .A)′ is l-positive. Thus by the induction hypothesis we getMω,Ωl−1 0 Γ, (µX .A)′. (4)8 Cut-elimination for the mu-calculus with one variableMoreover, also by the induction hypothesis we get for all (h+1)-positive ∆Mω,Ωl−1 0 ∆, (µX .A)′ =⇒ Mω,Ωh 0 ∆, Γ. (5)Now we plug (4) in (5) and obtain Mω,Ωh 0 Γ as required.We now have all ingredients ready for our main result.Corollary 11. Let Γ be anL -sequent. We haveM Γ =⇒ Mω Γ.Proof. Assume M Γ. By Theorem 8 we get Mω,Ωk Γ for some k. By cut-elimination we obtainMω,Ωk 0 Γ. Then collapsing yields Mω,Ω0 0 Γ which finally gives us Mω Γ by Lemma 4.References[1] David Baelde (2009): Least and greatest fixed points in linear logic. CoRR abs/0910.3383v4. Available athttp://arxiv.org/abs/0910.3383v4.[2] Kai Bru¨nnler & Thomas Studer (2009): Syntactic cut-elimination for common knowledge. Annals of Pureand Applied Logic 160(1), pp. 82–95, doi:10.1016/j.apal.2009.01.014.[3] Kai Bru¨nnler & Thomas Studer (preprint): Syntactic cut-elimination for a fragment of the modal mu-calculus.[4] Wilfried Buchholz (1981): The Ωµ+1-rule. In Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers &Wilfried Sieg, editors: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof TheoreticStudies, Lecture Notes in Mathematics 897, Springer, pp. 189–233, doi:10.1007/BFb0091898.[5] Wilfried Buchholz (2001): Explaining the Gentzen-Takeuti reduction steps: a second-order system. Archivefor Mathematical Logic 40(4), pp. 255–272, doi:10.1007/s001530000064.[6] Wilfried Buchholz & Kurt Schu¨tte (1988): Proof Theory of Impredicative Subsystems of Analysis. Bibliopo-lis.[7] Brian Hill & Francesca Poggiolesi (2010): A Contraction-free and Cut-free Sequent Calculus for Proposi-tional Dynamic Logic. Studia Logica 94(1), pp. 47–72, doi:10.1007/s11225-010-9224-z.[8] Gerhard Ja¨ger, Mathis Kretz & Thomas Studer (2008): Canonical completeness for infinitary µ . Journal ofLogic and Algebraic Programming 76(2), pp. 270–292, doi:10.1016/j.jlap.2008.02.005.[9] Gerhard Ja¨ger & Thomas Studer (2011): A Buchholz rule for modal fixed point logics. Logica Universalis 5,pp. 1–19, doi:10.1007/s11787-010-0022-1.[10] Dexter Kozen (1988): A finite model theorem for the propositional µ–calculus. Studia Logica 47(3), pp.233–241, doi:10.1007/BF00370554.[11] Grigori Mints (to appear): Effective Cut-elimination for a fragment of Modal mu-calculus. Studia Logica .[12] Regimantas Pliuskevicius (1991): Investigation of Finitary Calculus for a Discrete Linear Time Logicby means of Infinitary Calculus. In: Baltic Computer Science, Selected Papers, Springer, pp. 504–528,doi:10.1007/BFb0019366.[13] Gaisi Takeuti (1987): Proof Theory. North-Holland.[14] Alwen Tiu & Alberto Momigliano (2010): Cut Elimination for a Logic with Induction and Co-induction.CoRR abs/1009.6171v1. Available at http://arxiv.org/abs/1009.6171v1.

Cut-elimination for the mu-calculus with one variable

We establish syntactic cut-elimination for the one-variable fragment of the modal mu-calculus. Our method is based on a recent cut-elimination technique by Mints that makes use of Buchholz ’ Ω-rule. 

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We establish syntactic cut-elimination for the one-variable fragment of the modal mu-calculus. Our method is based on a recent cut-elimination technique by Mints that makes use of Buchholz' Omega-rule

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D. Miller and Z.Ésik (Eds.): Fixed Pointsin Computer Science 2012 (FICS 2012)EPTCS 77, 2012, pp. 47–54, doi:10.4204/EPTCS.77.7Cut-elimination for the mu-calculus with one variableGrigori MintsDept. of PhilosophyStanford UniversityUSAgmints@stanford.eduThomas StuderInst. of Computer Science and Appl. Math.University of BernSwitzerlandtstuder@iam.unibe.chWe establish syntactic cut-elimination for the one-variable fragment of the modal mu-calculus. Ourmethod is based on a recent cut-elimination technique by Mints that makes use of Buchholz’Ω-rule.1 IntroductionThe propositional modalµ-calculus is a well-established modal fixed point logic thatincludes fixedpoints for arbitrary positive formulae. Thus it subsumes many temporal logics (with an always operator),epistemic logics (with a common knowledge operator), and program logics (with an iteration operator).Making use of the finite model property, Kozen [10] introduces a sound and complete infinitarysystem for the modalµ-calculus. In this system greatest fixed points are introduce by means of theω-rule that has a premise for each finite approximation of the greatest fixed point. Jäger et al. [8] showby semanticmethods that the cut rule is admissible in this kind of infinitary systems. So far, however,there is nosyntacticcut-elimination procedure available for the modalµ-calculus. It is our aim in thispaper to present an effective cut-elimination method for the one-variable fragment of theµ-calculus.There are already a few results available on syntactic cut-elimination for modal fixed point logics.Most of them make use of deep inference where rules may not only be applied to outermost connectivesbut also deeply inside formulae. The first result of this kindhas been obtained by Pliuskevicius [12] whopresents a syntactic cut-elimination procedure for lineartime temporal logic. Brünnler and Studer [2]employ nested sequents to develop a cut-elimination procedure for the logic of common knowledge. Hilland Poggiolesi [7] use a similar approach to establish effective ut-elimination for propositional dynamiclogic. A generalization of this method is studied in [3] where it is also shown that it cannot be extendedto fixed points that have a✷-operator in the scope of aµ-operator. Fixed points of this kind occur, forinstance, inCTL in the form of universal path quantifiers.Thus we need a more general approach to obtain syntactic cut-elimination for the modalµ-calculus.A standard proof-theoretic technique to deal with inductive definitions and fixed points is Buchholz’Ω-rule [4, 6]. Jäger and Studer [9] present a formulation of the Ω-rule for non-iterated modal fixedpoint logic and they obtain cut-elimination for positive formulae of this logic. In order to overcome thisrestriction to positive formulae, Mints [11] introduces anΩ-rule that has a wider set of premises, whichenables him to obtain full cut-elimination for non-iterated modal fixed point logic.Mints’ cut-elimination algorithm makes use of, in additionto ideas from [5], a new tool presentedin [11]. It is based on the distinction, see [13], between impl cit and explicit occurrences of formulae ina derivation with cut. If an occurrence of a formula is traceabl to the endsequent of the derivation, thenit is called explicit. If it is traceable to a cut-formula, then it is an implicit occurrence.Implicit and explicit occurrences of greatest fixed points are treated differently in the translationof the induction rule to the infinitary system. An instance ofthe induction rule that derives a sequent48 Cut-elimination for the mu-calculus with one variableνX.A,B goes to an instance of theω-rule if νX.A is explicit. Otherwise, ifνX.A is traceable to a cut-formula, the induction rule is translated to an instance of theΩ-rule that is preserved until the last stageof cut-elimination. At that stage, called collapsing, theΩ-rule is eliminated completely.In the present paper we show that this method can be extended to a µ-calculus with iterated fixedpoints. Hence we obtain complete syntactic cut-elimination for the one-variable fragment of the modalµ-calculus. Our infinitary system is completely cut-free in the sense that there are not only no cut rulesin the system but also no embedded cuts. Thus our cut-free syst m enjoys the subformula property. Thisis in contrast to the recent cut-elimination results by Baelde [1] and by Tiu and Momigliano [14] for thefinitary systemsµMALL andLinc−, respectively, where theν-introduction rule and the co-induction rulecontain embedded cuts, which results in the loss of the subformula property.2 Syntax and semanticsWe first introduce the languageL . We start with a countable set PROP of atomic propositionspi andtheir negationspi . We useP to denote an arbitrary element of PROP. Moreover, we will use a specialvariableX.Definition 1. Operator forms A,B, . . . are given by the following grammar:A :== pi | pi | X | A∧A | A∨A | ✷A | ✸A | µX.A | νX.A.Formulae Fare defined by:F :== pi | pi | F ∧F | F ∨F | ✷F | ✸F | µX.A | νX.A.The fixed point operatorsµ andν bind the variableX and, therefore, we will talk of free and boundoccurrences ofX. Hence a formula is an operator form without free occurrences of X.The negation of an operator form is inductively defined as follows.1. ¬pi := pi and¬pi := pi2. ¬X := X3. ¬(A∧B) := ¬A∨¬B and¬(A∨B) := ¬A∧¬B4. ¬✷A :=✸¬A and¬✸A :=✷¬A5. ¬µX.A := νX.¬A and¬νX.A := µX.¬ANote that negation is well-defined: the negation of anX-positive operator form is againX-positivesince we have¬X := X. Thus, for example,¬µX.✷(pi ∧X) := νX.¬✷(pi ∧X) := νX.✸¬(pi ∧X) := νX.✸(¬pi ∨¬X) := νX.✸(pi ∨X).For an arbitrary but fixed atomic propositionpi we set⊤ := pi ∨ pi . If A is an operator form, then wewrite A(B) for the result of simultaneously substitutingB for every free occurrence ofX in A. We willalso use finite iterations of operator forms, given as followsA0(B) := B andAk+1(B) := A(Ak(B)).G. Mints & T. Studer 49Γ,P,¬P Γ,µX.A,¬µX.AΓ,A,BΓ,A∨B(∨)Γ,A Γ,BΓ,A∧B(∧)Γ,A✸Γ,✷A,Σ(✷)Γ,A(µX.A)Γ,µX.A(clo)¬A(B),B¬µX.A,B(ind)Γ,A Γ,¬AΓ(cut)Figure 1: SystemM3 System MSystemM derives sequents, that are finite sets of formulae. We denotesequents byΓ,Σ and use thefollowing notation: ifΓ := {A1, . . . ,An}, then✸Γ := {✸A1, . . . ,✸An}, SystemM consists of the axiomsand rules given in Figure 1.4 System MωSystemMω is an infinitary cut-free system for the modalµ-calculus with one variable. It consists of theaxioms and rules given in Figure 2.Γ,P,¬PΓ,A,BΓ,A∨B(∨)Γ,A Γ,BΓ,A∧B(∧)Γ,A✸Γ,✷A,Σ(✷)Γ,A(µX.A)Γ,µX.A(clo)Γ,Ai(⊤) for all natural numbersiΓ,νX.A(ω)Figure 2: SystemMω5 System Mω ,ΩkIn order to embedM into Mω , we need a family of intermediate systemsMω ,Ωk that include additionalrules to derive greatest fixed points that later will be cut away.The languageLΩ extendsL by a new connectiveν ′ to denote those greatest fixed points. Formally,LΩ is given as follows. Operator forms ofLΩ are defined like operator forms ofL with the additionalcase1. If A is an operator form, thenν ′X.A is also an operator form.A formula of LΩ is anLΩ operator form without free occurrence ofX. A formula is agreatest fixedpoint if it has the formνX.A or ν ′X.A.50 Cut-elimination for the mu-calculus with one variableDefinition 2. The levellev(A) of an operator formA is the maximal nesting of fixed point operators inA. Formally we set:1. lev(P) := lev(X) := 0 for all P in PROP2. lev(A∧B) := lev(A∨B) := max(lev(A), lev(B))3. lev(✷A) := lev(✸A) := lev(A)4. lev(µX.A) := lev(νX.A) := lev(ν ′X.A) := lev(A)+1The level of a sequent is the maximum of the levels of its formulae. We say a formula (sequent) isk-positiveif for all ν ′X.A occurring in it we havelev(ν ′X.A)< k.When working inMω ,Ωk , we will use the following notation: the formulaA′ is obtained fromA byreplacing all occurrences ofνX in A with ν ′X.Let k ≥ 0. SystemMω ,Ωk consists of the axioms and rules ofMω (formulated inLΩ) and the addi-tional rules:cut, Ωh, andΩ̃h. Thecut rule is given as followsΓ,A′ Γ,(¬A)′Γ(cut),whereA is a formula withlev(A)≤ k. The rulesΩh andΩ̃h , where 1≤ h≤ k, are informally describedas follows:· · ·Mω ,Ωk−1 0 ∆, (µX.A)′∆, Γ · · ·ΩhΓ, (¬µX.A)′andΓ, (µX.A)′ · · ·Mω ,Ωk−1 0 ∆, (µX.A)′∆, Γ · · ·Ω̃hΓwherelev((¬µX.A)′) = h and∆ ranges overh-positive sequents such that there is a cut-free proof of thesequent∆, (µX.A)′ in Mω ,Ωk−1.Definition 3. We useMω ,Ωk 0 Γ to express that there is a cut-free derivation ofΓ in Mω ,Ωk .In a more formal notation we can state theΩh-rule as follows. If for everyh-positive sequent∆Mω ,Ωk−1 0 ∆, (µX.A)′ =⇒ Mω ,Ωk ∆, Γ,thenMω ,Ωk Γ, (¬µX.A)′,and similarly forΩ̃h.Note that SystemMω ,Ω0 does not includeΩh- or Ω̃h-rules. Hence we immediately get the followinglemma.Lemma 4. Let Γ be anL sequent. We haveMω ,Ω0 0 Γ =⇒ Mω Γ.G. Mints & T. Studer 516 EmbeddingIn this section we present a translation fromM -proofs intoMω ,Ωk -proofs. First we establish an auxiliarylemma.Lemma 5. For all natural numbers h≤ k we have the following.1. If lev(µX.A) = h, thenMω ,Ωk 0 µX.A, ¬µX.A.2. If lev(A) = h, thenMω ,Ωk 0 Γ, A′ =⇒ Mω ,Ωk 0 Γ, A.3. If lev(µX.A) = h, thenMω ,Ωk 0 µX.A, (¬µX.A)′.4. If lev(A) = h, thenMω ,Ωk 0 B,C =⇒ Mω ,Ωk 0 (¬A)(B), A(C).5. If lev(A) = h, thenMω ,Ωk 0 B,C′ =⇒ Mω ,Ωk 0 (¬A)(B), A′(C′).Proof. The five statements are shown simultaneously by induction onh. For space considerations weshow only one particular case of the second statement, whichis shown by induction on the derivation ofΓ, A′ and a case distinction on the last rule. Assume the last rule is an nstance ofΩh with main formulaA′. We haveA′ = (νX.A0)′ with lev(A0) < h. By the premise of theΩh-rule we have for allh-positivesequents∆Mω ,Ωk−1 0 ∆,(µX.¬A0)′ =⇒ Mω ,Ωk 0 ∆,Γ. (1)Trivially we haveMω ,Ωk 0 ⊤,Γ. (2)We also haveMω ,Ωk−1 0 ⊤,(µX.¬A0)′from which we get by the induction hypothesis for the fifth claim of this lemmaMω ,Ωk−1 0 A0(⊤),(¬A0)′((µX.¬A0)′).An application ofclo yieldsMω ,Ωk−1 0 A0(⊤),(µX.¬A0)′.By (1) we getMω ,Ωk 0 A0(⊤),Γ. (3)Note that (2) and (3) are the first two premises of an instance of ω . By further iterating this we obtainfor all iMω ,Ωk 0 Ai0(⊤),Γ.Hence an application ofω yieldsMω ,Ωk 0 νX.A0,Γ.We will need a certain form of the induction rule inMω ,Ωk , which we are going to derive next. Wewrite Σ[(µX.A)′ := B] for the result of simultaneously replacing in every formulain Σ every occurrenceof (µX.A)′ with B.52 Cut-elimination for the mu-calculus with one variableLemma 6. Let A be an operator form withlev(νX.A)≤ k. Let∆,Σ1,Σ2 be h-positive sequents and let Bbe a formula withlev(B)≤ k. Assume thatMω ,Ωk (¬A(B))′, B and Mω ,Ωk (¬A(B))′, B′.Then we have, ifMω ,Ωk−1 0 ∆, Σ1, Σ2thenMω ,Ωk ∆, Σ1[(µX.A)′ := B], Σ2[(µX.A)′ := B′].Lemma 7. Let A be an operator form withlev(νX.A) ≤ k. Further let B be an arbitrary formula withlev(B)≤ k. Assume thatMω ,Ωk (¬A(B))′, B and Mω ,Ωk (¬A(B))′, B′.Then we haveMω ,Ωk (¬µX.A)′, B and Mω ,Ωk (¬µX.A)′, B′.Proof. Let h = lev(νX.A). In view of our assumptions and the previous lemma we know that for allh-positive sequents∆Mω ,Ωk−1 0 ∆, (µX.A)′ =⇒ Mω ,Ωk ∆, B.Hence by an application of theΩh-rule we concludeMω ,Ωk (¬µX.A)′, B. Similarly, we can deriveMω ,Ωk (¬µX.A)′, B′.Theorem 8. Let Γ be a sequent ofL . AssumeM Γ and assume further for any sequent∆ occurringin that proof we havelev(∆)≤ k. Then we haveMω ,Ωk Γ.Proof. An operationσ on sequents is called ’-operation ifσ(Γ,A1, . . . ,An) = Γ,A′1, . . . ,A′n. The result ofapplyingσ to a sequentΓ is denotedΓσ .To establish the theorem, we show by induction on the depth ofeM -proof that for all ’-operationsσ , we haveMω ,Ωk Γσ . We distinguish the following cases for the last rule.1. Γ is an axiom different fromΓ0,µX.A,¬µX.A. ThenΓσ is an axiom ofMω ,Ωk , too.2. Γ is Γ0,µX.A,¬µX.A. ThenΓσ follows either by the first or the third claim of Lemma 5 dependingon whether¬µX.A is replaced byσ or not.3. The last rule is an instance of∧, ∨, ✷ or clo. We can apply the same rule inMω ,Ωk .4. The last rule is a cutΓ,A Γ,¬AΓ.We extend the current ’-operationσ to a ’-operationτ such that(Γ,A)τ = Γσ ,A′ and(Γ,¬A)τ =Γσ ,(¬A)′ By the induction hypothesis for the ’-operationτ we obtainMω ,Ωk Γσ ,A′ as well asMω ,Ωk Γσ ,(¬A)′. With an instance of cut we getMω ,Ωk Γσ .5. The last rule is an instance of the induction rule. Then theendsequent has the form¬µX.A, Bwhich isνX.¬A, B. There are two possible cases.(a) The principal occurrence ofνX.¬A is not changed byσ . By the induction hypothesis we canderive(¬A(B))′, Bσ and(¬A(B))′, B′. We obtain our claim by the following proof.G. Mints & T. Studer 53· · ·I.H.(¬A(B))′, BσI.H.(¬A(B))′, B′⊤,B′L. 5(¬A)(⊤),(A(B))′cut(¬A)(⊤),B′...(¬A)i(⊤),B′L. 5(¬A)i+1(⊤),(A(B))′cut(¬A)i+1(⊤),Bσ · · ·ωνX.¬A, Bσ(b) The principal occurrence ofνX.¬A is changed byσ . Let τ1,τ2 be ’-operations such that(¬A(B),B)τ1 = (¬A(B))′,Band(¬A(B),B)τ2 = (¬A(B))′,B′.By the induction hypothesis forτ1 andτ2 we obtainMω ,Ωk (¬A(B))′, B and Mω ,Ωk (¬A(B))′, B′.We apply Lemma 7 and concludeMω ,Ωk (¬µX.A)′, Bσ .7 Cut eliminationWe eliminate instances ofcut in the standard way, see for instance [5, 11], by pushing themup thederivation. When an instance ofcut with cut formulae(µX.A)′ and(¬µX.A)′ meets the instance ofΩhthat introduces(¬µX.A)′, this pair of inferences is replaced byΩ̃h.Lemma 9 (Cut-elimination). If Mω ,Ωk Γ, thenMω ,Ωk 0 Γ.The cut-elimination process terminates in a formally cut-free derivation that may contain instancesof Ω̃h-rules. Now we show that these instances ofΩ̃h also can be eliminated.Lemma 10(Collapsing). Let Γ be an(h+1)-positive sequent. IfMω ,Ωk 0 Γ, thenMω ,Ωh 0 Γ.Proof. By transfinite induction on the derivation inMω ,Ωk . The only interesting case is when the last ruleis an instance of̃Ωl for h< l ≤ k as followsΓ, (µX.A)′ · · ·Mω ,Ωl−1 0 ∆, (µX.A)′∆, Γ · · ·Ω̃lΓNote thatΓ,(µX.A)′ is l -positive. Thus by the induction hypothesis we getMω ,Ωl−1 0 Γ, (µX.A)′. (4)54 Cut-elimination for the mu-calculus with one variableMoreover, also by the induction hypothesis we get for all(h+1)-positive∆Mω ,Ωl−1 0 ∆, (µX.A)′ =⇒ Mω ,Ωh 0 ∆, Γ. (5)Now we plug (4) in (5) and obtainMω ,Ωh 0 Γ as required.We now have all ingredients ready for our main result.Corollary 11. Let Γ be anL -sequent. We haveM Γ =⇒ Mω Γ.Proof. AssumeM Γ. By Theorem 8 we getMω ,Ωk Γ for somek. By cut-elimination we obtainMω ,Ωk 0 Γ. Then collapsing yieldsMω ,Ω0 0 Γ which finally gives usMω Γ by Lemma 4.References[1] David Baelde (2009):Least and greatest fixed points in linear logic. CoRRabs/0910.3383v4. Available athttp://arxiv.org/abs/0910.3383v4.[2] Kai Brünnler & Thomas Studer (2009):Syntactic cut-elimination for common knowledge. 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Cut-elimination for the mu-calculus with one variable

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