SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC

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1 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC NICK BEZHANISHVILI AND IAN HODKINSON Abstract. We define Salqvist fixed point formulas. By extending te tecnique of Sambin and Vaccaro we sow tat (1) for eac Salqvist fixed point formula ϕ tere exists an LFPformula χ(ϕ), wit no free first-order variable or predicate symbol, suc tat a descriptive µ-frame (an order-topological structure tat admits topological interpretations of least fixed point operators as intersections of clopen pre-fixed points) validates ϕ iff χ(ϕ) is true in tis structure, and (2) every modal fixed point logic axiomatized by a set Φ of Salqvist fixed point formulas is sound and complete wit respect to te class of descriptive µ-frames satisfying {χ(ϕ) : ϕ Φ}. We also give some concrete examples of Salqvist fixed point logics and classes of descriptive µ-frames for wic tese logics are sound and complete. 1. Introduction Modal fixed point logic is an extension of modal logic wit fixed point operators. Modal µ-calculus is a variant of modal fixed point logic obtained by adding to te basic modal logic te least and greatest fixed point operators. Te attractive feature of modal µ-calculus is tat it is very expressive, but still decidable, e.g., [7, Section 5]. Many expressive modal and temporal logics suc as PDL, CTL and CTL are all fragments of te modal µ-calculus, e.g., [7, Section 4.1]. In [18] Kozen defined te syntax and semantics of modal µ-calculus, gave its axiomatization using te so-called fixed point rule (see Definition 2.12), and sowed soundness of tis axiomatization. Walukiewicz [24] proved completeness of Kozen s axiomatization using automata and tableaux. His proof, owever, is complicated and as not been generalized to oter axiomatic systems of µ-calculus. Ambler, Kwiatkowska, Measer [1] proved soundness and completeness of Kozen s axiomatization of modal µ-calculus wit respect to non-standard, order-topological semantics. Tey also extended tis result to all normal fixed point logics logics obtained by adding extra axioms to Kozen s axiomatization of µ-calculus. Later Bonsangue and Kwiatkowska [6] sowed tat in tis semantics te least fixed point can be computed as te intersection of clopen pre-fixed points. Hartonas [15] generalized tese completeness results to te systems of positive (negation-free) modal µ-calculus. Santocanale [21] proved completeness of te flat modal fixed point logic using modal µ-algebras and teir completions. Flat modal fixed point logic is obtained by replacing te fixed point operators by logical connectives; tis as (among oter tings) te effect of severely restricting nesting of fixed point operators. Later Santocanale and Venema [22] gave an alternative proof of te completeness of flat modal fixed point logic by applying coalgebraic metods. Ten Cate and Fontaine [8] used non-standard semantics of modal fixed point logics for proving te finite model property result for te modal fixed point logic axiomatized by te formula µxx. Van 2000 Matematics Subject Classification. 03B45; 03B60; 06F30. Key words and prases. Salqvist s teorem; modal fixed point logic; completeness; correspondence; descriptive frame. 1

2 2 NICK BEZHANISHVILI AND IAN HODKINSON Bentem [2], [3] also investigated tis logic and posed a question weter a version of te Salqvist teorem olds for te systems of µ-calculus. Salqvist s completeness and correspondence teorem is one of te most fundamental results of classical modal logic (see e.g., [5, Sections 3.6 and 5.6]). It states tat every modal logic obtained by adding Salqvist formulas (a large class of formulas of a particular syntactic sape) to te basic modal logic K is sound and complete wit respect to a first-order definable class of Kripke frames. In [20] Sambin and Vaccaro gave an elegant proof of Salqvist s teorem using order-topological metods. An important ingredient of teir proof is te Esakia lemma of [11]. Generalizations of te Salqvist completeness and correspondence (first-order definability) result to larger classes of modal formulas can be found in [13], [14] and [17]. Oter generalizations of te Salqvist correspondence (definability) for modal formulas (in first-order logic wit fixed point operator) were obtained in [2], [3] and [9]. See remarks 5.14 and 5.15 below for more information on tese generalizations. In tis paper we prove a version of Salqvist s teorem for modal fixed point logic. Our language is te modal language extended wit te least fixed point operator µ (we do not ave te greatest fixed point operator ν in our language). Following [1] we consider te order-topological semantics of modal µ-calculus. Descriptive frames are order-topological structures extensively used in modal logic, e.g., [5, Capter 5]. In [1] te autors define, wat we call in tis paper, descriptive µ-frames tose descriptive frames tat admit a topological interpretation of te least fixed point operator. Unlike te classical semantics of fixed point logics, in tis semantics, te least fixed point operator is interpreted as te intersection of not all pre-fixed points, but of all clopen pre-fixed points. We prove tat for tis semantics of modal fixed point logic an analogue of te Esakia lemma still olds (Lemma 4.6). We also define Salqvist fixed point formulas (Definition 5.1) and extend te Sambin Vaccaro metod [20] of proving Salqvist s completeness and correspondence results (Teorems 5.3 and 5.11) from modal logic to modal fixed point logic. More specifically, let LFP denote first-order logic wit te least fixed point operator. (Again te least fixed point operator is interpreted topologically, tat is, as te intersection of clopen pre-fixed points.) We prove tat for every Salqvist fixed point formula ϕ tere exists an LFP-formula χ(ϕ), wit no free first-order variable or predicate symbol, suc tat a descriptive µ-frame validates ϕ iff χ(ϕ) is true in tis structure. Our main result (Teorem 5.13) states tat every modal fixed point logic axiomatized by a set Φ of Salqvist fixed point formulas is sound and complete wit respect to te class of descriptive µ-frames satisfying {χ(ϕ) : ϕ Φ}. We also give some concrete examples of Salqvist fixed point logics and classes of descriptive µ-frames for wic tese logics are sound and complete. Note tat tese results can also be formulated witout mentioning any topology. A general frame is a Kripke frame wit a distinguised set F of admissible subsets of tis frame. A general µ-frame is a general frame in wic all modal µ-formulas are assigned to admissible sets under any assignment of propositional variables to admissible sets, wen te least fixed point operator is interpreted as te intersection of all te admissible pre-fixed points. A descriptive µ-frame can be seen as a general µ-frame were F is te collection of all clopen sets. In tis paper we sow (Teorem 5.13(1)) tat te Salqvist completeness and correspondence results also old for tis general-frame semantics of modal fixed point logic. It needs to be stressed tat our Salqvist completeness and correspondence results apply only to order-topological structures (descriptive µ-frames) and general µ-frames, and do not

3 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC 3 imply tat every Salqvist modal fixed point logic is sound and complete wit respect to Kripke frames. Te paper is organized as follows: in Section 2 we recall a duality between modal µ- algebras and descriptive µ-frames and also te completeness of normal modal fixed point logics wit respect to modal µ-algebras and descriptive µ-frames. In Section 3 we compare different kinds of order-topological semantics of modal µ-formulas. A modal fixed point analogue of te Esakia lemma is proved in Section 4. In Section 5 we define Salqvist fixed point formulas and prove te Salqvist completeness and correspondence results for modal fixed point logic. In Section 6 we discuss a few examples of Salqvist fixed point logics and teir frame correspondents and conclude te paper wit some remarks in Section 7. Acknowledgment Te work of te first autor was partially supported by te EP- SRC grant EP/F032102/1. Te autors are very grateful to Clemens Kupke and Dimitar Vakarelov for many interesting discussions on te subject of te paper. 2. Preliminaries In tis section we set up te scene: we introduce te basic definitions of a modal µ-algebra and descriptive µ-frame, discuss a duality between tem and te consequences of tis duality for te completeness of axiomatic systems of modal fixed point logic Classical fixed points. Let (L, ) be a complete lattice and f : L L a monotone map, tat is, for eac a, b L wit a b we ave f(a) f(b). Ten te celebrated Knaster Tarski teorem states tat f as a least and greatest fixed point LF P (f) and GF P (f), respectively. Moreover, tese fixed points can be computed as follows: LF P (f) = {a L : f(a) a} and GF P (f) = {a L : a f(a)}. Tere is anoter way of computing LF P (f). In particular, for an ordinal α we let f 0 (0) = 0, f α (0) = f(f β (0)) if α = β + 1, and f α (0) = β<α f β (0), if α is a limit ordinal. Ten LF P (f) = f α (0), for some ordinal α suc tat f α+1 (0) = f α (0). We briefly recall te syntax and Kripke semantics for te modal µ-calculus. Te language of modal µ-calculus consists of a countably infinite set of propositional variables (x, y, p, q, x 0, x 1, etc), constants and, connectives,,, modal operators and, µxϕ(x, x 1,..., x n ) for all formulas ϕ(x, x 1,..., x n ), were x occurs under te scope of an even number of negations. Formulas of modal µ-calculus will be called modal µ-formulas. A formula tat does not contain any µ-operators will be called a modal formula. A Kripke frame is a pair (W, R), were W is a non-empty set and R W 2 a binary relation. Given a Kripke frame (W, R), an assignment is a map from te propositional variables to te powerset P(W ) of W. Te satisfiability and validity of a modal formula in a Kripke model and frame, respectively, are defined in a standard way (see, e.g., [5]). For eac modal formula ϕ we denote by [ϕ] te set of points satisfying ϕ under te assignment.

4 4 NICK BEZHANISHVILI AND IAN HODKINSON A propositional variable x is bound in a modal µ-formula ϕ if it occurs in te scope of some µx. A variable is free if it is not bound. We say tat a modal µ formula ϕ(x, x 1,... x n ) is positive in x if all te free occurrences of te variable x are under te scope of an even number of negations. A modal µ-formula ϕ(x, x 1,... x n ) is called negative in x if all te free occurrences of te variable x are under te scope of an odd number of negations. Let (W, R) be a Kripke frame. For eac modal µ-formula ϕ and an assignment, we define te semantics [ϕ] of ϕ by induction on te complexity of ϕ. If ϕ is a propositional variable, a constant, or is of te form ψ χ, ψ χ, ψ, ψ or ψ, ten te semantics of ϕ is defined in a standard way. Now assume tat ϕ(x, x 1,..., x n ) is a modal µ-formula positive in x. Ten by te induction ypotesis, te semantics of ϕ is already defined for eac assignment. Let be a fixed assignment. Ten ϕ and give rise to a map f ϕ, : P(W ) P(W ) defined by f ϕ, (U) = [ϕ] U x, were U x (x) = U and U x (y) = (y) for eac variable y x. It is well known tat if ϕ is positive in x, ten f ϕ, is monotone wit respect to te inclusion order. It is also well known tat (P(X), ) is a complete lattice were meets and joins are te set-teoretic intersections and unions, respectively. Tus, by te Knaster Tarski teorem f ϕ, as a least fixed point and [µxϕ] is defined to be te least fixed point of f ϕ, Modal algebras and descriptive frames. Given a Kripke frame (W, R) we let R 0 = {(w, w) : w W } and for eac d 0 we let R d denote te dt iteration of R. Tat is, for w, v W we ave wr d v iff tere exists u W suc tat wr d 1 u and urv. For eac w W and d ω we let R d (w) = {v W : wr d v}. We will write R(w) instead of R 1 (w). Also for eac U W we let [R]U = {v W : R(v) U} and R U = {v W : R(v) U }. Recall tat a Stone space is a compact and Hausdorff topological space wit a basis of clopen sets. A descriptive frame is a pair (W, R) suc tat W is a Stone space and R a binary relation on W suc tat R(w) is a closed set for eac w W and te collection Clop(W ) of all clopen subsets of W is closed under te operations [R] and R. Te latter condition is equivalent to R C Clop(W ) for eac C Clop(W ). We also note tat Clop(W ) is a Boolean algebra wit te operations,, \, and constants W and. We denote by Cl(W ) and Op(W ) te collections of all closed and all open subsets of W, respectively. We also note tat Cl(W ) and Op(W ) are complete lattices (see, e.g., [23]). For Cl(W ) te meet is te intersection and te join te closure of te union and for Op(W ) te meet is te interior of te intersection and te join te union. Te next lemma is well known e.g., [11] or [19]. It will be used in te subsequent sections. Lemma 2.1. Let (W, R) be a descriptive frame. Ten (1) R F Cl(W ) for eac F Cl(W ) and R U Op(W ) for eac U Op(W ), (2) [R]F Cl(W ) for eac F Cl(W ) and [R]U Op(W ) for eac U Op(W ), (3) R d (w) Cl(W ) for eac w W and d 0. Recall tat a modal algebra is a pair B = (B, ), were B is a Boolean algebra and a unary operation on B satisfying for eac a, b B, (1) 0 = 0 and (2) (a b) = a b. Now we will briefly spell out te constructions establising a duality between modal algebras and descriptive frames. For eac descriptive frame F = (W, R) te algebra Clop(F) = (Clop(W ), R ) is a modal algebra. For eac modal algebra B = (B, ) we consider te set W B of all ultrafilters of B and define a topology on W B by declaring te set {â : a B}, were â = {w W B : a w}, as a basis of te topology. We define a relation R B on W B by wr B v iff a w for eac a v (for w, v W B ). Ten (W B, R B )

5 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC 5 is a descriptive frame and tis correspondence is (up to isomorpism) one-to-one. Tat is, B = (Clop(W B ), R B ) and F = (W Clop(F), R Clop(F) ) Modal µ-algebras and descriptive µ-frames. Definition 2.2. (1) Let B = (B, ) be a modal algebra. A map from propositional variables to B is called an algebra assignment. We define a (possibly partial) semantics for modal µ-formulas by te following inductive definition. [ ] = 0 [ ] = 1 [x] = (x), were x is a propositional variable, [ϕ ψ] = [ϕ] [ψ], [ϕ ψ] = [ϕ] [ψ], [ ϕ] = [ϕ], [ ϕ] = [ϕ], [ϕ] = [ϕ], For a B we denote by a x a new algebra assignment suc tat a x(x) = a and a x(y) = (y) for eac propositional variable y x. If ϕ(x, x 1,..., x n ) is positive in x, ten [µxϕ(x, x 1,..., x n )] = {a B : [ϕ(x, x 1,..., x n )] a x a}, if tis meet exists; oterwise, te semantics for µxϕ(x, x 1,..., x n ) is undefined. (2) A modal algebra (B, ) is called a modal µ-algebra if [ϕ] is defined for any modal µ-formula ϕ and any algebra assignment. Notation: To simplify te notations instead of [ϕ(x 1,..., x n )] wit (x i ) = a i, 1 i n, we will simply write ϕ(a 1,..., a n ). Recall tat a modal algebra (B, ) is called complete if B is a complete Boolean algebra; tat is, for eac subset S of B te meet S and te join S exist. It is straigtforward to see tat every complete modal algebra is a modal µ-algebra. Lemma 2.3. Let B = (B, ) be a modal µ-algebra and an algebra assignment. Ten for eac modal µ-formula ϕ positive in x, [µxϕ] is te least fixed point of te map (a [ϕ] a x ) for a B. Proof. Te result follows from te definition of [µxϕ] and te standard argument of te proof of te Knaster Tarski teorem. Definition 2.4. Let (W, R) be a descriptive frame, F P(W ) and an arbitrary assignment, tat is, a map from te propositional variables to P(W ). We define te semantics for modal µ-formulas by te following inductive definition. [ ] F =, [ ] F = W, [x] F = (x), were x is a propositional variable, [ϕ ψ ] F = [ϕ]f [ψ ]F,

6 6 NICK BEZHANISHVILI AND IAN HODKINSON [ϕ ψ ] F = [ϕ]f [ψ ]F, [ ϕ] F = W \ [ϕ]f, [ ϕ] F = R [ϕ]f, [ϕ] F = [R][ϕ]F, We denote by U x a new assignment suc tat U x (x) = U and U x (y) = (y) for eac propositional variable y x and U P(W ). Let ϕ(x, x 1,..., x n ) be positive in x, ten [µxϕ(x, x 1,..., x n ))] F = {U F : [ϕ(x, x 1,..., x n )] F U x We assume tat = W. U}. Let (W, R) be a descriptive frame. We call a map from te propositional variables to P(W ) a set-teoretic assignment. If maps eac propositional variable to Cl(W ), ten is called a closed assignment, and if maps eac propositional variable to Clop(W ), ten is called a clopen assignment. Let be any assignment. Ten [ ] F is called te clopen semantics if F = Clop(W ), [ ] F is called te closed semantics if F = Cl(W ) and [ ] F is called te classical or set-teoretic semantics if F = P(W ). Notation: To simplify te notations instead of [ϕ(x 1,..., x n )] F wit (x i) = U i, 1 i n, we will simply write ϕ(u 1,..., U n ) F. Moreover, we will skip te index F if it is clear from te context or is irrelevant (e.g., wen ϕ is a modal formula). A set C suc tat ϕ(c, (x 1 ),..., (x n )) C is called a pre-fixed point. Lemma 2.5. Let (W, R) be a descriptive frame, F P(W ) and an arbitrary assignment. Ten for eac modal µ-formula ϕ(x, x 1..., x n ) positive in x, ϕ(, (p 1 ),..., (p n )) F is monotone. Tat is, for U, V W, U V implies ϕ(u, (p 1 ),..., (p n )) F ϕ(v, (p 1 ),..., (p n )) F. Proof. We will prove te lemma by induction on te complexity of ϕ. As agreed above we will skip te index F. Our induction ypotesis is: 1) if ϕ(x, x 1..., x n ) is positive in x, ten ϕ(, (p 1 ),..., (p n )) is monotone and 2) if ϕ(x, x 1..., x n ) is negative in x, ten ϕ(, (p 1 ),..., (p n )) is anti-tone. Te cases ϕ =, ϕ =, ϕ is a propositional variable, ϕ = ψ χ, ϕ = ψ χ, ϕ = ψ, ϕ = ψ and ϕ = ψ are proved as in standard modal logic (see, e.g., [5]). Now let ϕ = µyψ(y, x, x 1,..., x n ) be positive in x and inductively assume te result for ψ. Ten, by te induction ypotesis, for eac U, V W wit U V and C F we ave ψ(c, U, (p 1 ),..., (p n )) ψ(c, V, (p 1 ),... (p n )). So if ψ(c, V, (p 1 ),..., (p n )) C, ten ψ(c, U, (p 1 ),..., (p n )) C. Terefore, te set {C F : ϕ(c, U, (p 1 ),..., (p n )) C} contains te set {C F : ϕ(c, V, (p 1 ),..., (p n )) C}. But tis means tat µyψ(y, U, (p 1 ),..., (p n )) = {C F : ψ(c, U, (p 1 ),..., (p n )) C} {C F : ψ(c, V, (p 1 ),..., (p n )) C} = µyψ(y, V, (p 1 ),..., (p n )). Terefore, we obtained tat ϕ(u, (p 1 ),..., (p n )) ϕ(v, (p 1 ),..., (p n )). Te case of ψ negative in x is similar. Definition 2.6. A descriptive frame (W, R) is called a descriptive µ-frame if for eac clopen assignment and for eac modal µ-formula ϕ, te set [ϕ] Clop(W ) is clopen. Obviously, eac finite descriptive frame is a descriptive µ-frame. We will see more examples of descriptive µ-frames later in tis section and in te following section.

7 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC 7 Now we will discuss a duality between modal µ-algebras and descriptive µ-frames. Tis duality was first obtained in [1] and later improved in [6]. A generalization of tis duality to positive modal µ-algebras can be found in [15]. Lemma 2.7. Let (W, R) be a descriptive µ-frame. Ten te modal algebra (Clop(W ), R ) is a modal µ-algebra. Proof. In order to sow tat (Clop(W ), R ) is a modal µ-algebra, we need to prove tat [ϕ] exists for eac modal µ-formula ϕ and eac algebra assignment. Note tat in tis case an algebra assignment for (Clop(W ), R ) is te same as a clopen assignment for (W, R). So we will not distinguis tem. We prove te lemma by induction on te complexity of ϕ. Our induction ypotesis is: for any clopen assignment, [ϕ] is defined and [ϕ] = [ϕ] Clop(W ), were [ϕ] Clop(W ) is te clopen semantics of ϕ in te descriptive µ-frame (W, R) wit te clopen assignment. Te cases ϕ =, ϕ =, ϕ is a propositional variable, ϕ = ψ χ, ϕ = ψ χ, ϕ = ψ, ϕ = ψ and ϕ = ψ are proved as in te duality teorem for modal algebras and descriptive frames. Now assume ϕ(x, x 1,..., x n ) is a modal µ-formula positive in x for wic te induction ypotesis olds. We consider any clopen assignment. By te induction ypotesis, for eac C Clop(W ), we ave [ϕ] C x = [ϕ] Clop(W ). We will denote tis set by C x ϕ(c, (x 1 ),..., (x n )). Let C = {C Clop(W ) : ϕ(c, (x 1 ),..., (x n )) C}. Since (W, R) is a descriptive µ-frame, C is clopen. We will sow tat C = C. Tat C exists will be an obvious consequence of tis. Let G = C. So G Clop(W ). Ten G is a lower bound of C. On te oter and, for eac C C we ave G C. By monotonicity, ϕ(g, (x 1 ),..., (x n )) ϕ(c, (x 1 ),..., (x n )) C. Tus, ϕ(g, (x 1 ),..., (x n )) C = = G. Terefore, G belongs to C. So G is a lower bound tat belongs to te set, wic means tat G = C. Tus, [µxϕ] is defined and is equal to [ϕ] Clop(W ). Tis C x completes te induction, and so (Clop(W ), R ) is a modal µ-algebra. [ϕ] Clop(W ) Let (W, R) be a descriptive µ-frame, a clopen assignment and ϕ be a modal µ-formula positive in x. Let (C [ϕ] Clop(W ) ) be te map from Clop(W ) to Clop(W ) sending eac C x clopen set C to [ϕ] Clop(W ). It is easy to see tat tis map is well defined and, by Lemma 2.5, C x monotone. Corollary 2.8. Let (W, R) be a descriptive µ-frame and a clopen assignment. Ten for eac modal µ-formula ϕ positive in x, [µxϕ] Clop(W ) is te least fixed point of te map (C [ϕ] Clop(W ) ) for C Clop(W ). C x Proof. Te result follows immediately from te proof of Lemma 2.7. It follows from te proof tat G = [µxϕ] Clop(W ) is a fixed point of (C [ϕ] Clop(W ) ) and by te definition of G, C x it is contained in every (pre-)fixed point. Lemma 2.9. Let B = (B, ) be a modal µ-algebra. frame (W B, R B ) is a descriptive µ-frame. Ten te corresponding descriptive

8 8 NICK BEZHANISHVILI AND IAN HODKINSON Proof. We need to sow tat for eac modal µ-formula ϕ and eac clopen assignment, te set [ϕ] Clop(W B) is clopen. We prove tis by induction on te complexity of ϕ. By te definition of a modal µ-algebra and duality we ave B = (Clop(W B ), R B ). Terefore, as in te proof of Lemma 2.7, we will identify algebra assignments for (Clop(W B ), R B ) wit clopen assignments for (W B, R B ). Our induction ypotesis is: for any clopen assignment we ave [ϕ] Clop(W B) = [ϕ], were [ϕ] is te semantics of ϕ in te algebra (Clop(W B ), R B ). As in te proof of Lemma 2.7, te cases ϕ =, ϕ =, ϕ is a propositional variable, ϕ = ψ χ, ϕ = ψ χ, ϕ = ψ, ϕ = ψ and ϕ = ψ are proved as in te duality teorem for modal algebras and descriptive frames. Now let ϕ(x, x 1,..., x n ) be a modal µ-formula positive in x and let be any clopen assignment. By te assumed induction ypotesis for ϕ, for eac C Clop(W B ), we ave [ϕ] Clop(W B) = [ϕ] C x C x. As in te proof of Lemma 2.7, we denote tis set by ϕ(c, (x 1 ),..., (x n )). We also denote te set {C Clop(W B ) : ϕ(c, (x 1 ),..., (x n )) C} by C. Since B is a modal µ-algebra, (Clop(W B ), R B ) is also a modal µ-algebra. Terefore, D = C exists and is a clopen set. Tus, D is contained in C. Moreover, te same argument as in te proof of Lemma 2.7 sows tat ϕ(d, (x 1 ),... (x n )) D. So D C and ence C D. Terefore, [ϕ] Clop(W B) = C = D is clopen. Tis completes te induction, and tus (W B, R B ) is a descriptive µ-frame. As every complete modal algebra is a modal µ-algebra, it follows from Lemma 2.9 tat a descriptive frame dual to a complete modal algebra is a descriptive µ-frame. descriptive µ-frames of tis kind will be eavily used in te next section. Remark From now on, we will identify clopen assignments of a descriptive frame (W, R) wit algebra assignments of (Clop(W ), R ). It is easy to see tat te correspondence between descriptive µ-frames and modal µ- algebras is (up to te isomorpism) one-to-one. Putting everyting togeter we obtain te following teorem. Teorem ([1]) Te correspondence between modal algebras and descriptive frames restricts to a one-to-one correspondence between modal µ-algebras and descriptive µ-frames. We note tat te duality result of [1] is a bit different tan ours since in [1] descriptive µ-frames are defined as tose descriptive frames were meets of clopen pre-fixed points are clopen. It was later observed in [6] tat tese meets are in fact te intersections of clopen pre-fixed points. In [15] te duality is obtained for distributive modal µ-lattices (algebraic models of negation-free µ-calculus). Our duality result can be seen as a restricted case of [15] wen te distributive µ-lattice is a (Boolean) modal µ-algebra. In [1] and [15] te above correspondence between modal µ-algebras and descriptive µ-frames is also extended to a dual equivalence of te corresponding categories Axiomatic systems of modal fixed point logic. Next we briefly discuss te connection of modal µ-algebras and descriptive µ-frames wit te axiomatic systems of µ-calculus. If ϕ and ψ are formulas and x a variable, we will denote by ϕ[ψ/x] te formula obtained by replacing in ϕ eac occurrence of x by ψ. Definition 2.12.

9 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC 9 (1) [18] Te axiomatization of Kozen s system K µ consists of te following axioms and rules propositional tautologies, If ϕ and ϕ ψ, ten ψ (Modus Ponens), If ϕ, ten ϕ[ψ/x] (Substitution), If ϕ, ten ϕ (Necessitation), (x y) (x y) (K-axiom), ϕ[µxϕ/x] µxϕ (Fixed Point axiom), If ϕ[ψ/x] ψ, ten µxϕ ψ (Fixed Point rule), were x is not a bound variable of ϕ and no free variable of ψ is bound in ϕ. (2) [1, 8] Let Φ be a set of modal µ-formulas. We write K µ + Φ for te smallest set of formulas wic contains bot K µ and Φ and is closed under te Modus Ponens, Substitution, Necessitation and Fixed Point rules. We say tat K µ +Φ is te extension of K µ by Φ. We also call K µ + Φ a normal modal fixed point logic. Let L = K µ + Φ be a normal modal fixed point logic. A modal µ-algebra (B, ) is called an L-algebra if it validates all te formulas in Φ. A descriptive µ-frame (W, R) is called an L-frame if (W, R) validates all te formulas in Φ wit respect to clopen assignments. Teorem [1, 8] Let L be a normal modal fixed point logic. Ten (1) L is sound and complete wit respect to te class of modal µ-l-algebras. (2) L is sound and complete wit respect to te class of descriptive µ-l-frames. We note tat [1] prove tis result using te Lindenbaum-Tarski algebra and canonical model constructions, wile [8] give an alternative proof using te so-called replacement map and translations. 3. A comparison of different semantics of fixed point operators In tis section we investigate te connections between different kinds of semantics of modal µ-formulas. Te results and examples discussed ere are not directly relevant for te Salqvist completeness and correspondence teorem proved in Section 5. Tus, te reader interested only in te Salqvist teorem for modal fixed point logic can skip tis section. In te previous section we introduced various (e.g. clopen, closed, set-teoretic) semantics for modal µ-formulas. An obvious question is: ow different are all tese semantics? In tis section we will address tis question. We will first consider classes of descriptive µ-frames for wic te semantics coincide. After tat we will give examples of descriptive µ-frames for wic te semantics differ. Recall tat a modal algebra is called locally finite if its every finitely generated subalgebra is finite. Let (W, R) be a descriptive frame and a clopen assignment. Let also B = (Clop(W ), R ). Ten for eac modal µ-formula ϕ wose only free variables are x 1,..., x n we associate a modal subalgebra of B generated by te elements (x 1 ),..., (x n ) and denote it by B ϕ. Teorem 3.1. Let (B, ) be a locally finite modal algebra and (W, R) its dual descriptive frame. Ten for eac formula ϕ, clopen assignment, and F suc tat Clop(W ) F P(W ), we ave [ϕ] F = [ϕ]p(w ) B ϕ. Consequently, (W, R) is a descriptive µ-frame and (B, ) is a modal µ-algebra.

10 10 NICK BEZHANISHVILI AND IAN HODKINSON Proof. Since (B, ) is locally finite and (B, ) is isomorpic to B = (Clop(W ), R ), for formula ψ we ave tat B ψ is finite. We now prove by induction on te complexity of any subformula ψ of ϕ tat for any clopen assignment and any F wit Clop(W ) F P(W ) we ave: (1) [ψ ] F = [ψ ]P(W ) B ψ. If ψ =, ψ =, ψ is a propositional variable, ψ = χ 1 χ 2, ψ = χ 1 χ 2, ψ = χ, ψ = χ or ψ = χ, ten (1) easily follows from te induction ypotesis. Now let ψ = µxχ, were χ(x, x 1,..., x n ) is a modal µ-formula positive in x. Let g be any clopen assignment and F suc tat Clop(W ) F P(W ). For eac l ω we let: (2) S 0 = and S l+1 = [χ] P(W ). Claim 3.2. S l B ψ g, for eac l ω. Proof. Since B ψ g is a modal subalgebra of B, obviously S 0 = B ψ g. Now we assume tat S l B ψ g and prove tat S l+1 B ψ g. Since S l B ψ g and g is a clopen assignment, te assignment g S l x is also clopen. Terefore, by our assumption, (1) olds for g S l x and χ. So [χ] P(W ) B χ. Now S g S l x g S l l B ψ g yields tat te subalgebra of B generated by te x elements S l, g(x 1 ),..., g(x n ) is equal to te subalgebra of B generated by te elements g(x 1 ),..., g(x n ). Tus, B χ g S l x and te proof of te claim. = B ψ g. So S l+1 = [χ] P(W ) g S l x g S l x B ψ g, wic completes te induction It follows from Lemma 2.5, tat S l S l+1 for all l. Terefore, as B ψ g is finite, tere is m ω suc tat S l = S m for all l > m. Let U F be suc tat [χ] F U. By induction gx U on l, we sow tat S l U, for all l ω. Obviously, S 0 U. Now assume S l U. Ten by (2), (1) and Lemma 2.5, we ave S l+1 = [χ] P(W ) = [χ] F [χ] F U. So S g S l x g S l g U l U, x x for all l ω. By (1) and (2), [χ] F = [χ] P(W ) = S gx Sm gx Sm m+1 = S m. As B ψ g Clop(W ) F we obtain tat S m F. Terefore, S m is a pre-fixed point tat is contained in every pre-fixed point. So S m = [µxχ] F g. As F was arbitrary, we also ave S m = [µxχ] P(W g ), wic togeter wit te fact tat S m B ψ g completes te induction. Finally, as Clop(W ) F, we deduce tat [ϕ] Clop(W ) Clop(W ). So (W, R) is a descriptive µ-frame and by Lemma 2.7, (B, ) is a modal µ-algebra. Tis finises te proof of te teorem. Next we will sow tat for descriptive µ-frames corresponding to complete modal algebras closed and clopen semantics coincide. For tis we will first recall a topological caracterization of te Stone spaces dual to complete Boolean algebras. Teorem 3.3. (see e.g. [23]) Let B be a Boolean algebra and W its dual Stone space. Ten B is complete iff for eac closed subset F W, te interior of F is clopen iff for eac open subset U W, te closure of U is clopen. Stone spaces satisfying te condition of Teorem 3.3 are called extremally disconnected [23]. Lemma 3.4. Let W be a non-empty set wit te discrete topology and let R be a binary relation on W. Ten

11 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC 11 (1) Te Stone-Čec compactification β(w ) of W is extremally disconnected. (2) Te Boolean algebra Clop(β(W )) is isomorpic to te Boolean algebra P(W ). (3) Let (W B, R B ) be te dual space of B = (P(W ), R ). Ten W B is (up to isomorpism) te Stone-Čec compactification of W, W is te subset of W B consisting of all te isolated points and R B W 2 = R. Proof. Te proofs of (1) and (2) can be found in [23]. Te proof of (3) can be easily derived from (2) using te duality of descriptive frames and modal algebras. Teorem 3.5. Let (W, R) be a descriptive µ-frame dual to a complete modal algebra. Ten for eac modal µ-formula ϕ and eac clopen assignment, we ave [ϕ] Clop(W ) = [ϕ] Cl(W ). Proof. We will prove te teorem by induction on te complexity of ϕ. Our inductive ypotesis is: For any clopen assignment and any subformula ψ of ϕ, we ave (3) [ψ ] Clop(W ) = [ψ ] Cl(W ). If ψ is a constant, propositional variable or of te form ψ = χ 1 χ 2, ψ = χ 1 χ 2, ψ = χ, ψ = χ, ψ = χ, ten (3) easily follows from te induction ypotesis. Now let ψ = µxχ, were χ(x, x 1,..., x n ) is a modal µ-formula positive in x. Ten [µxχ] Cl(W ) = {F Cl(W ) : [χ] Cl(W ) F } (by definition) F x {U Clop(W ) : [χ] Cl(W ) U x = {U Clop(W ) : [χ] Clop(W ) U x = [µxχ] Clop(W ). So it remains to prove tat U} (as Clop(W ) Cl(W )) U} (by (3)) [µxχ] Clop(W ) [µxχ] Cl(W ). Let F = {F Cl(W ) : [χ] Cl(W ) F }. We also let G = [µxχ] Cl(W ) F x and D = Int(G), were Int(G) is te interior of G. Since W corresponds to a complete algebra, by Teorem 3.3, D is clopen. So, by (3), [χ] Cl(W ) D x by Lemma 2.5, [χ] Cl(W ) D x obtain [χ] Clop(W ) D x = [χ] Clop(W ) D x. Obviously for eac F F we ave D F. So, [χ] Cl(W ) F. Terefore, [χ] Cl(W ) F = G and, by (3), we F x D x G. But since (W, R) is a descriptive µ-frame, is a clopen assignment and D is a clopen, [χ] Clop(W ) is clopen. Hence, [χ] Clop(W ) D x D x pre-fixed point contained in G, wic implies tat [µxχ] Clop(W ) finises te induction and te proof of te teorem. Int(G) = D. Tus, D is a clopen G = [µxχ] Cl(W ). Tis It is still an open problem weter Lemma 3.5 olds for descriptive µ-frames not corresponding to complete or locally finite algebras. Next we will give an example of a descriptive µ-frame, a closed assignment and a modal µ-formula ϕ for wic closed and clopen semantics differ. Example 3.6. Let Z be te set of integers wit te discrete topology. We define a relation R on Z by zry iff y = z + 1 or y = z 1 for z, y Z. Ten A = (P(Z), R ) is a complete modal algebra and terefore it is a modal µ-algebra. Let (W, R ) be its dual descriptive frame. By Lemma 3.4, te subframe consisting of all principal ultrafilters in W will be

12 12 NICK BEZHANISHVILI AND IAN HODKINSON isomorpic to (Z, R) and every singleton consisting of a principal ultrafilter will be clopen in W. We will denote tis subspace wit te restricted order by (Z, R ). (In fact, topologically, as mentioned in Lemma 3.4, W is te Stone Čec compactification of Z wit te discrete topology.) Let M = W \ Z denote te closed set of all non-principal ultrafilters of A. For eac z Z we let F z = {U Z : z U}. Obviously, F z is a principal ultrafilter of A and eac principal ultrafilter of A is of te form F z for some z Z. Claim 3.7. (1) For eac principal ultrafilter F z Z and non-principal ultrafilter F M we ave (F z R F ) and (F R F z ). (2) For eac non-principal ultrafilter F, tere exists a non-principal ultrafilter F suc tat F R F. (3) R M = M. Proof. (1) Since F is a non-principal ultrafilter, it contains all cofinite subsets of Z. Let V = Z \ R ({z}). Ten V is cofinite and terefore V F. Moreover, (z + 1) / F and (z 1) / F. Tus, z / R (V ) and so R (V ) / F z. Tis implies tat (F z R F ). On te oter and, {z} F z. But R ({z}) = {z + 1, z 1} / F. Terefore, (F R F z ). (2) Let F M. We consider te set S = { R U : U F }. We generate a filter by S and ten extend it to a maximal filter. Te filter generated by S is proper. To see tis, assume R U 1 R U n S. Ten U 1,..., U n F and since n i=1 U i, tere exists z Z suc tat z n i=1 U i. But ten (z + 1) n i=1 R U i. Now we extend tis filter to a maximal filter F. By te definition, R U belongs to F for eac U F. So we ave F R F. By (1) F must be non-principal. (Alternatively, we could take te filter F = {u + 1 : u F } and sow tat it satisfies condition (2) of te claim.) (3) Follows directly from (2) and (1). Next we define a closed (not clopen) assignment on W by (p) = {F 0 } M. Consider te formula ϕ(x, p) = p x. Ten, using te claim, it is easy to see tat te only closed pre-fixed points of ϕ(x, (p)) are te wole space W and te set E M = {F z : z is even or negative even} M. However, te only clopen pre-fixed point of ϕ(x, (p)) is te wole space W. Terefore, [µxϕ] Clop(W ) = W [µxϕ] Cl(W ) = E M. It is also easy to see tat ), for F E M = ϕ(e M, (p)). Tus, E M is a closed fixed point of te map (F [ϕ] Clop(W ) F x Cl(W ). Tis implies tat [µxϕ] Clop(W ) is not te least closed fixed point of (F [ϕ] Clop(W ) for F Cl(W ). Example 3.8. We note tat if, in te previous example, we consider te clopen assignment (p) = {F 0 }, ten [µxϕ] P(W ) = E = {F z : z is even or negative even}. Tus every pre-fixed point of ϕ(x, (p)) contains E. Moreover, it is easy to see tat E is an open set and tus, by Teorem 3.3, te closure of E, wic we denote by E, is a clopen set. It is not ard to see tat E is a pre-fixed point of ϕ(x, (p)). Terefore, E is te least clopen pre-fixed point of ϕ(x, (p)). So E = [µxϕ] Clop(W ) [µxϕ] P(W ). In Example 3.8 we ave tat [µxϕ] Clop(W ) is te closure of [µxϕ] P(W ). Te next example sows not only tat tis is not te case in general, but also tat te closure of [µxϕ] P(W ) may not be even a fixed point of ϕ(x, (p)). F x ),

13 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC 13 Example 3.9. We will give an example of a descriptive µ-frame (W, R), a clopen assignment and a modal formula ϕ(x, p) suc tat te closure of [µxϕ] P(W ) is not a fixed point of ϕ(x, (p)). Let Z be te set of integers wit te discrete topology. Let W = β(z) be te Stone Čec compactification of Z. We define a relation R on W by zry iff (z, y Z and y = z + 1 or y = z 1 or z W and y β(z) \ Z). It is easy to ceck tat (W, R) is a descriptive frame. Moreover, by Lemma 3.4, (W, R) is a descriptive µ-frame. Now we define a clopen assignment (p) = {0}. Consider te formula ϕ(x, p) = p x. Ten [µx(p x)] P(W ) is equal to te set of all even and negative even numbers. Te closure of tis set contains a proper subset of β(z) \ Z and, as is easy to ceck, is not a fixed point of ϕ(x, (p)). Note tat in tis case we ave [µx(p x)] Clop(W ) = [µx(p x)] Cl(W ) = W. Remark We can combine Examples 3.6 and 3.9 by taking te disjoint union of te frames defined in tese examples. Tis will give us an example of a (single) descriptive µ-frame (W, R), a closed assignment and a modal µ-formula ϕ suc tat all te tree semantics of ϕ differ and, moreover, neiter closed nor clopen semantics of ϕ is te closure of te set-teoretic semantics of ϕ. We skip te details. 4. Te intersection lemma In tis section we address two issues. We prove te analogue of te Esakia Sambin Vaccaro lemma, wic will play an essential role in Section 5 in proving Salqvist s completeness and correspondence results for modal fixed point logic. We also discuss weter clopen semantics gives rise to fixed points for closed and set-teoretic assignments. We use te analogue of te Esakia Sambin Vaccaro lemma in proving tat te clopen semantics gives a fixed point for closed assignments. We also sow tat in general te clopen semantics does not provide a fixed point for set-teoretic assignments. Note tat te only fact in tis section tat will be used in te proof of te Salqvist teorem for modal fixed point logic (Section 5) is Lemma 4.6. In more detail, let (W, R) be a descriptive µ-frame, a clopen assignment, and ϕ(x, x 1,..., x n ) a modal µ-formula positive in x. Ten, by Corollary 2.8, [µxϕ] Clop(W ) is te least fixed point of te map (C [ϕ] Clop(W ) ), for C Clop(W ). On te oter and, Example 3.6 sows C x tat tere exist a descriptive µ-frame (W, R), a closed assignment and a modal formula ϕ positive in x suc tat [µxϕ] Clop(W ) is not te least fixed point of te map (F [ϕ] Clop(W ) ), F x for F Cl(W ). Te next question we are going to address is weter [µxϕ] Clop(W ) is a (not necessarily least) fixed point for te maps (F [ϕ] Clop(W ) ) and (U [ϕ] Clop(W ) ) for F x gx U F Cl(W ), U P(W ), a closed assignment, and set-teoretic assignment g, respectively. In fact, we will prove tat for a closed assignment, [µxϕ] Clop(W ) is a fixed point of te map (F [ϕ] Clop(W ) ) for F Cl(W ). We will also sow tat tere exist a descriptive µ-frame F x (W, R) and a set-teoretic assignment g suc tat [µxϕ] Clop(W g ) is not a fixed point of te map (U [ϕ] Clop(W ) ) for U P(W ). gx U Definition 4.1. We call a modal µ-formula ϕ positive if it does not contain any negation. ϕ is called negative if ϕ is positive. Remark 4.2. We note tat ϕ is positive implies tat ϕ is positive in eac variable, but not vice versa.

14 14 NICK BEZHANISHVILI AND IAN HODKINSON Lemma 4.3. Let (W, R) be a descriptive frame, a closed assignment and ϕ(x, x 1,..., x n ) a positive modal µ-formula. Ten te set [ϕ] Clop(W ) is closed. Consequently, te map (F [ϕ] Clop(W ) ) mapping eac closed set F to [ϕ] Clop(W ) is well defined and monotone. F x F x Proof. We will prove te result by induction on te complexity of ϕ. If ϕ is a constant or propositional variable, ten as is closed, [ϕ] Clop(W ) is obviously closed. Te cases ϕ = ψ χ and ϕ = ψ χ are trivial since finite unions and intersections of closed sets are closed. Te cases ϕ = ψ and ϕ = ψ follow directly from Lemma 2.1(1),(2). Finally, te case ϕ = µxψ is also easy since any intersection of clopen sets is closed. Terefore, (F [ϕ] Clop(W ) ) is a well-defined map from Cl(W ) to Cl(W ). Monotonicity of tis map F x follows from Lemma 2.5. Remark 4.4. Since Cl(W ) is a complete lattice, by te Knaster Tarski teorem, te map (F [ϕ] Clop(W ) ) will ave te least fixed point. As te meet in Cl(W ) coincides wit te F x intersection, te least point will be te intersection of all closed pre-fixed points. However, as was sown in Example 3.6, tis least fixed point may be different from [µxϕ] Clop(W ). Next we prove an auxiliary lemma wic is an extension of te so-called intersection lemma of Esakia Sambin Vaccaro [11], [20] to te modal µ-case. Tis lemma will be an essential ingredient in te proof of te Salqvist completeness result in Section 5. We will be concerned only wit te clopen semantics. So we will skip te sup index Clop(W ) everywere. We first recall Esakia s lemma. Let W be any set. A set F P(W ) is called downward directed if for eac F, F F, tere exists F F suc tat F F F. Lemma 4.5. [11](Esakia) Let (W, R) be a descriptive frame and F Cl(W ) a downward directed set. Ten R {F : F F} = { R F : F F} Next we prove a modal µ-analogue of te Intersection Lemma of [20]. Lemma 4.6. Let (W, R) be a descriptive frame. 1 Let also F, F 1,..., F n W be closed sets and let A Clop(W ) be a downward directed set suc tat A = F. Ten for eac positive modal µ-formula ϕ(x, x 1,..., x n ) we ave ϕ(f, F 1,..., F n ) = {ϕ(u, F 1,..., F n ) : U A}. Proof. We will prove te lemma by induction on te complexity of ϕ. Te modal cases are already proved in [20]. We briefly recall tese proofs to make te paper self contained. If ϕ = or ϕ =, ten te lemma is obvious. If ϕ is a propositional variable, ten te lemma is again obvious since every closed set is te intersection of te clopen sets containing it. First let ϕ = ψ χ. Ten ϕ(f, F 1,..., F n ) = ψ(f, F 1,..., F n ) χ(f, F 1,..., F n ) = {ψ(u, F 1,..., F n ) : U A} {χ(u, F 1,..., F n ) : U A} (ind) = {ψ(u, F 1,..., F n ) χ(u, F 1,..., F n ) : U A} = {(ψ χ)(u, F 1,..., F n ) : U A} = {ϕ(u, F 1,..., F n ) : U A}. 1 Note tat we do not require tat (W, R) is a descriptive µ-frame.

15 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC 15 Now let ϕ = ψ χ. Since ϕ is positive we ave tat ϕ(f, F 1,..., F n ) ϕ(u, F 1,..., F n ) for eac U A. Tus, ϕ(f, F 1,..., F n ) {ϕ(u, F 1,..., F n ) : U A}. Now suppose w / ϕ(f, F 1,..., F n ). Ten, by te induction ypotesis, w / {ψ(c, F 1,..., F n ) : C A} {χ(d, F 1,..., F n ) : D A}. So w / {ψ(c, F 1,..., F n ) : C A} and w / {χ(d, F 1,..., F n ) : D A}. Terefore, tere exists C, D A suc tat w / ψ(c, F 1,..., F n ) and w / χ(d, F 1,..., F n ). Since A is downward directed, tere exists E A suc tat E C D. As bot ψ and χ are positive, by Lemma 2.5, we ave w / ψ(e, F 1,..., F n ) and w / χ(e, F 1,..., F n ). Tus, w / {ψ(e, F 1,..., F n ) χ(e, F 1,..., F n ) : E A} and terefore, w / {ϕ(u, F 1,..., F n ) : U A}. Tis means tat {ϕ(u, F 1,..., F n ) : U A} ϕ(f, F 1,..., F n ). So ϕ(f, F 1,..., F n ) = {ϕ(u, F 1,..., F n ) : U A}. Now suppose ϕ = ψ. We will need to use te following fact, wic easily follows from Lemma 2.5: if A is downward directed, ten {ψ(u, F 1,..., F n ) : U A} is also downward directed. So ϕ(f, F 1,..., F n ) = R ψ(f, F 1,..., F n ) = R {ψ(u, F 1,..., F n ) : U A} (ind yp) = { R ψ(u, F 1,..., F n ) : U A} (Esakia s lemma) = {ϕ(u, F 1,..., F n ) : U A}. Now assume ϕ = ψ. We recall tat R commutes wit all unions. Ten ϕ(f, F 1,..., F n ) = [R]ψ(F, F 1,..., F n ), = W \ R {W \ ψ(u, F 1,..., F n ) : U A} (ind yp) = W \ { R (W \ ψ(u, F 1,..., F n )) : U A} = {[R]ψ(U, F 1,..., F n ) : U A} = {ϕ(u, F 1,..., F n ) : U A}. Finally, let ϕ = µxψ(x, y, x 1,..., x n ). Ten we need to sow µxψ(x, F, F 1,..., F n ) = {µxψ(x, U, F 1,..., F n ) : U A}. By Lemma 2.5, for eac U A we ave µxψ(x, F, F 1,..., F n ) µxψ(x, U, F 1,..., F n ). Terefore, µxψ(x, F, F 1,..., F n ) {µxψ(x, U, F 1,..., F n ) : U A}. Now suppose w {µxψ(x, C, F 1,..., F n ) : C A}. Ten we ave tat w µxψ(x, C, F 1,..., F n ) for eac C A. So for eac C A and eac V Clop(W ) wit ψ(v, C, F 1,..., F n ) V we ave w V. Assume U Clop(W ) is suc tat ψ(u, F, F 1,..., F n ) U. By te induction ypotesis we ave ψ(u, F, F 1,..., F n ) = {ψ(u, C, F 1,... F n ) : C A}. Tus {ψ(u, C, F 1,..., F n ) : C A} U. By Lemma 4.3, eac ψ(u, C, F 1,..., F n ) is a closed set. Terefore, as U is open, by compactness, tere exist finitely many C 1,..., C k A suc tat k i=1 ψ(u, C i, F 1,..., F n ) U. As A is downward directed, tere exists C A suc tat C k i=1 C i. Ten, by Lemma 2.5, ψ(u, C, F 1,..., F n ) U. But ten w U. Tus, w {U Clop(W ) : ψ(u, F, F 1,..., F n ) U} = µxψ(x, F, F 1,..., F n ), wic finises te proof of te lemma. Corollary 4.7. Let (W, R) be a descriptive frame, F 1,..., F n, G 1,..., G k W closed sets and ϕ(x 1,..., x n, y 1,..., y k ) a positive modal µ-formula. Ten (1) ϕ(f 1,..., F n, G 1,..., G k ) = {ϕ(c 1,..., C n, G 1,..., G k ) : F i C i Clop(W ), 1 i n}. (2) ϕ(f 1,..., F n, G 1,..., G k ) = {ϕ(c 1,..., C n, G 1,..., G k ) : F i C i A i, 1 i n}, were A i Clop(W ) is downward directed and A i = F i, for eac 1 i n.

16 16 NICK BEZHANISHVILI AND IAN HODKINSON Proof. Te result follows from Lemma 4.6 by a trivial induction. Next we will apply Lemma 4.6 to sow tat for eac descriptive frame (W, R), positive modal formula ϕ, and a closed assignment, te set [µxϕ] Clop(W ) is a fixed point of te map (F [ϕ] Clop(W ) ) for F Cl(W ). F x Lemma 4.8. Let (W, R) be a descriptive µ-frame and ϕ(x, x 1,..., x n ) a positive modal µ- formula. Let G = µxϕ(x, F 1,..., F n ), were F 1,... F n W be closed sets. Ten ϕ(g, F 1,..., F n ) = G, tat is, G is a fixed point of te map (F [ϕ] Clop(W ) ) for F Cl(W ). F x Proof. We first sow tat G is a pre-fixed point, tat is, ϕ(g, F 1,..., F n ) G. Let V be an arbitrary clopen pre-fixed point: tat is, ϕ(v, F 1,..., F n ) V. Ten, by te definition of G, we ave G V. By Lemma 2.5 we obtain ϕ(g, F 1,..., F n ) ϕ(v, F 1,..., F n ) V. Terefore, ϕ(g, F 1,..., F n ) {V Clop(W ) : ϕ(v, F 1,..., F n ) V } = G. Conversely, as G is te intersection of closed sets, G is closed. Terefore, by Corollary 4.7, we ave (4) ϕ(g, F 1,..., F n ) = {ϕ(u, U 1,..., U n ) : G U Clop(W ), F i U i Clop(W ), 1 i n}. Let U and U 1,..., U n be arbitrary clopen sets wit G U and F i U i for 1 i n. We sow tat G ϕ(u, U 1,..., U n ). Te fact tat G U means tat {V Clop(W ) : ϕ(v, F 1,..., F n ) V } U. Terefore, te same argument as in te proof of Lemma 4.6 sows tat tere exists a clopen set V U suc tat ϕ(v, F 1,..., F n ) V. By Corollary 4.7, ϕ(v, F 1,..., F n ) = {ϕ(v, C 1,..., C n ) : C i Clop(W ), F i C i U i, 1 i n}. But ten a similar argument as in te proof of Lemma 4.6 sows tat tere exist clopen sets C 1,..., C n suc tat F i C i U i for 1 i n and ϕ(v, C 1,..., C n) V. By monotonicity we ave ϕ(ϕ(v, C 1,..., C n), F 1,..., F n ) ϕ(v, F 1,..., F n ) ϕ(v, C 1,..., C n). Since (W, R) is a descriptive µ-frame, ϕ(v, C 1,..., C n) is a clopen set. Tus ϕ(v, C 1,..., C n) is a clopen pre-fixed point of ϕ(, F 1,..., F n ). Tis means tat G ϕ(v, C 1,..., C n). But since ϕ is monotone and C i U i for 1 i n we ave ϕ(v, C 1,..., C n) ϕ(u, U 1,..., U n ). Tus, as U, U 1,..., U n were arbitrary, we obtain by (4) tat G {ϕ(u, U 1,..., U n ) : G U, F i U i, 1 i n} = ϕ(g, F 1,..., F n ). Next we will see tat an analogue of Lemma 4.8 does not old for set-teoretic assignments. Example 4.9. We will give an example of a descriptive µ-frame (W, R), a set-teoretic (neiter clopen nor closed) assignment, and a formula ϕ(x, p) suc tat {C Clop(W ) : ϕ(c, (p)) C} is no longer a fixed point of ϕ(x, (p)). Let N be te set of natural numbers wit te discrete topology. Let W = β(n) be te Stone Čec compactification of N. Let M = β(n)\n. We define a relation R on W by zry iff z W and y M. It is easy to ceck tat (W, R) is a descriptive frame. Moreover, by Lemma 3.4, (W, R) is a descriptive µ-frame. Now define an assignment (p) = E, were E is te set of all even numbers. Obviously, is neiter clopen nor closed. Consider a formula ϕ(x, p) = p x. Ten te clopen semantics of ϕ(x, (p)) is equal to E, te closure of E. To see tis, note tat every clopen containing E must contain E. Tus, E {C Clop(W ) : ϕ(c, (p)) C}. On te oter and, W is extremally disconnected. So E is clopen. Also note tat (W \ E) M. Tus, R (W \ E) = W. Ten [R]E = W \ R (W \ E) = W \ W =. Terefore, ϕ(e, (p)) = E [R]E = E E. So E is a clopen pre-fixed point of ϕ(x, (p)) and we [µxϕ] Clop(W )

17 SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC 17 ave tat {C Clop(W ) : ϕ(c, (p)) C} E. Finally, note tat, as computed above, ϕ(e, (p)) = E E. Tus, E is not a fixed point of ϕ(x, (p)). 5. Salqvist fixed point formulas In tis section we extend te proof of te Salqvist completeness and correspondence results of [20] from modal logic to modal µ-calculus Completeness. For eac m ω we let 0 x = x and m+1 x = ( m x). Definition 5.1. A formula ϕ(x 1,..., x n ) is called a Salqvist fixed point formula if it is obtained from formulas of te form m x i (m ω, i n) and positive formulas (in te language wit te µ-operator) by applying te operations and. Remark 5.2. We note tat wen considering te language witout fixed point operators, te above definition of te Salqvist formula is different from te standard definition (see e.g., [5]), but is equivalent to it. Any Salqvist formula of [5] is equivalent to a conjunction of Salqvist formulas in te aforementioned sense. Teorem 5.3. Let (W, R) be a descriptive frame, 2 w W and ϕ(x 1,..., x l ) a Salqvist fixed point formula. If w [ϕ] Clop(W ) f, for eac clopen assignment f, ten w [ϕ] Clop(W ), for eac set-teoretic assignment. Proof. Since ϕ(x 1,..., x l ) is a Salqvist fixed point formula, tere exists a formula α(p 1,..., p n, q 1,..., q m ) using only and suc tat all listed propositional variables occur and no propositional variable occurs twice in α, and tere exist positive formulas π 1,..., π m and formulas ψ 1,..., ψ n, were eac ψ i is of te form d i s i, for some d i ω and s i {x 1,..., x l } suc tat (5) ϕ(x 1,... x l ) = α(ψ 1 /p 1,..., ψ n /p n, π 1 /q 1,..., π m /q m ). Let be an assignment suc tat w / [ϕ] Clop(W ). For eac subformula β of α we define a world w β W by induction suc tat (6) w β / [β(ψ 1 /p 1,..., ψ n /p n, π 1 /q 1,..., π m /q m )] Clop(W ). As te basic step of te induction we put w α = w. Now assume β is a subformula of α and w β is already defined and satisfies (6). Tere are tree possible cases: (1) β is atomic. Ten tere is noting to define. (2) β = γ. Ten we ave w β / [γ(ψ 1,..., ψ n, π 1,..., π m )] Clop(W ). So tere exists v W suc tat w β Rv and v / [γ(ψ 1,..., ψ n, π 1,..., π m )] Clop(W ). Ten we put w γ = v. (3) β = γ δ. Ten we put w γ = w δ = w β. We obviously ave w γ / [γ(ψ 1,..., ψ n, π 1,..., π m )] Clop(W ) and w δ / [δ(ψ 1,..., ψ n, π 1,..., π m )] Clop(W ). By te construction, for eac atomic subformula p i (i n) and q j (j m) of α we ave (7) w pi [ d i s i ] Clop(W ) and w qj / [π j ] Clop(W ) 2 Note again tat we do not require tat (W, R) is a descriptive µ-frame.