On Strictly Positive Modal Logics with S4.3 Frames

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1 On Strictly Positive Modal Logics with S4.3 Frames Stanislav Kikot Birkbeck, University of London, U.K. Agi Kurucz King s College London, U.K. Frank Wolter University of Liverpool, U.K. Michael Zakharyaschev Birkbeck, University of London, U.K. Abstract We investigate the lattice of strictly positive (SP) modal logics that contain the SP-fragment of the propositional modal logic S4.3. We are interested in Kripke (in)completeness of these logics, their computational complexity, as well as the definability of Kripke frames by means of SP-implications. We compare the lattice of these SP-logics with the lattice of normal modal logics above S4.3. We also consider global consequence relations for SP-logics, focusing on definability and Kripke completeness. Keywords: strictly positive modal logic, meet-semilattices with monotone operators, Kripke completeness, definability, decidability. 1 Introduction The lattice NExtS4.3 of propositional normal modal logics containing S4.3 is a rare example of a non-trivial class of well-behaved modal logics. Indeed, all of them are finitely axiomatisable, have the finite model property [6,10], and are decidable in conp [22]. Although Fine [10] complained that the full lattice [of these logics] is one of great complexity, its structure is perfectly understandable (though difficult to depict) if one recalls the fact [7] that every L NExtS4.3 can be axiomatised by Jankov (aka characteristic or frame) formulas for finite linear quasi-ordered frames. In this paper, our concern is the fragment of the modal language that comprises implications σ τ, where σ and τ are strictly positive modal formulas [3]

2 2 On Strictly Positive Modal Logics with S4.3 Frames constructed from variables using,, and the constant. We call such implications SP-implications. A natural algebraic semantics for SP-implications is given by meet-semilattices with monotone operators (SLOs, for short) [21,2,16]. We denote the corresponding syntactic consequence relation by SLO and call any SLO -closed set of SP-implications an SP-logic. The lattice of varieties of SLOs validating the SP-implicational S4 axioms p p and p p (closure SLOs) was studied by Jackson [15] (see also [9]). Jackson showed, in particular, that the SP-fragment P S4 of S4 is complex in the sense that each closure SLO can be embedded into the full complex algebra of some Kripke frame for S4. It follows that the SP-logic given by {p p, p p} is (Kripke) complete in the sense that its SP-implicational SLO -consequences coincide with those over Kripke frames, that is, with P S4. On the other hand, it is shown in [16] that the SP-extension of P S4 with (p q) (p r) (p q r) (wcon) is not complex; yet, it is complete and axiomatises the SP-fragment P S4.3 of S4.3. Both P S4 and P S4.3 are decidable in polynomial time [1,21,16]. Our aim here is to investigate SP-logics above P S4.3 focusing on completeness, definability (of Kripke frames) and computational complexity and compare them with classical modal logics above S4.3. Our first important observation is that, unlike classical modal formulas, a randomly chosen SP-implication will most probably axiomatise an incomplete SP-extension of P S4.3. Such are, for example, p q and p q (p q). In Section 3, we construct infinite sequences of SP-logics sharing the same two-point frames. But the main result of this paper is that we do identify all complete SP-logics above P S4.3. Over S4.3 frames, modal formulas define exactly those frame classes that are closed under cofinal subframes [10,23]. On the other hand, over P S4, SPimplications are capable of defining only FO-definable classes closed under subframes [16] (which implies that frames for any proper extension of P S4.3 are of bounded depth). In Section 4, we show that all FO-definable subframe-closed classes of S4.3-frames are SP-definable. We also give finite axiomatisations of these SP-definable classes by means of SP-analogues of subframe formulas [11,23], and prove that each of the resulting SP-logics is decidable in polynomial time. In Section 5, we investigate the lattice Ext + P S4.3 of SP-logics containing P S4.3 by comparing it with the lattice NExtS4.3. Following [4], we consider two maps: one associates with every modal logic L its SP-fragment π(l), the other one associates with every SP-logic P the modal logic µ(p ) = S4.3 P. We give a frame-theoretic characterisation of π and show that each π 1 (P ) has a greatest element. On the other hand, as follows from Section 3, µ 1 (L) often contains infinitely many incomplete logics. In Section 6, we consider rules ϱ = ι1,...,ιn ι of SP-implications. In modal logics above S4, ϱ is expressed by the formula (ι 1 ι n ) ι. We show that a class of S4.3-frames is modally definable iff it is definable by SP-rules. An SP-logic is complete w.r.t. SP-rules iff it is complex [16]. We show that the

3 Kikot, Kurucz, Wolter, Zakharyaschev 3 only SP-logics in Ext + P S4.3 complete w.r.t. SP-rules are P S5 and the logic of the singleton cluster. These are also the only complete SP-logics for which deciding valid SP-rules is in polynomial time. We make a step towards axiomatising SPrules by giving axiomatisations for the n-element clusters, n > 1. 2 Preliminaries We assume that the reader is familiar with basic notions of modal logic and Kripke semantics [7]. In particular, M, w = ϕ means that ϕ holds at world w in Kripke model M, and F = ϕ says that ϕ is valid in Kripke frame F. We write C = ϕ, for a class C of frames, if ϕ is valid in every F C. We denote SP-implications by ι = (σ τ) and refer to σ, τ as terms. By regarding ι as the equality σ τ = σ, we can naturally evaluate it in SLOs, i.e., structures A = (A,,, ), where (A,, ) is a semilattice with top element and (a b) b = (a b), for any a, b A. We write A = ι to say that σ τ = σ holds in A under any valuation. For a set P {ι} of SP-implications, we write P = SLO ι if A = ι for every SLO A validating all implications in P. We denote by P [A] the set of SP-implications ι for which A = ι. Since equational consequence can be characterised syntactically by Birkhoff s equational calculus [5,13], it is readily seen that P = SLO ι iff P SLO ι, where P SLO ι means that there is a sequence (derivation) ι 0,, ι n = ι, with each ι i being a substitution instance of a member in P or one of the axioms p p, p, p q q p, p q p, or obtained from earlier members of the sequence using one of the rules σ τ τ ϱ, σ ϱ σ τ σ ϱ σ τ ϱ, σ τ σ τ (see also the Reflection Calculus RC of [2,8]). We say that SP-implications ι and ι are P -equivalent, if P {ι} SLO ι and P {ι } SLO ι. For any class C of Kripke frames, the set of modal formulas (with full Booleans) validated by the frames in C is called the modal logic of C and denoted by L[C]; the restriction of L[C] to SP-implications is called the SP-logic of C and denoted by P [C]. For finite C = {F 1,, F n }, we write L[F 1, F n ] and P [F 1,, F n ], respectively. For any set Φ of modal formulas, Fr(Φ) is the class of frames validating Φ. Every Kripke frame F = (W, R) gives rise to a SLO F = (2 W,, W, + ) where + X = {w W R(w, v) for some v X}, for X W (that is, F is the (,, )-type reduct of the full complex algebra of F [12]). As Kripke models over F and valuations in the algebra F are the same thing, we always have P [F] = P [F ]. S4.3 is the modal logic whose rooted Kripke frames are linear quasi-orders F = (W, R), i.e., R is a reflexive and transitive relation on W with xry or yrx, for any x, y W. From now on, we refer to rooted frames for S4.3 as simply

4 4 On Strictly Positive Modal Logics with S4.3 Frames frames. A cluster in F is any set of the form C(x) = {y W xry & yrx}. If W ω and the number of clusters in F is finite, we write F = (m 1,, m n ), 1 m i ω, to say that the ith cluster in F (starting from the root) has m i points. In this case, we also say that F is of depth n and write d(f) = n. A linear order with n points is denoted by L n. We call frame F = (W, R) a subframe of a frame F = (W, R ) and write F F if W W and R is the restriction of R to W. A subframe F of F is proper if W is a proper subset of W. We say that F is a cofinal subframe of F and write F F if F G and, for any x W and y W with xr y, there is z W with yr z. (As we only deal with frames for S4.3, F F iff F is a p-morphic image of F.) The normal extension L of S4.3 with a set Φ of modal formulas is denoted by L = S4.3 Φ; NExtS4.3 = {S4.3 Φ Φ is a set of modal formulas}. The minimal SP-logic P containing P S4.3 and a set Φ of SP-implications is denoted by P = P S4.3 + Φ; Ext + P S4.3 = {P S4.3 + Φ Φ is a set of SP-implications}. Given P and ι, we write P = Kr ι iff ι P [Fr(P )]. It is easy to see that P SLO ι always implies P = Kr ι. We call P complete if the converse also holds for every ι. As shown in [16], P S4.3 is complete and Fr(P S4.3 ) = Fr(S4.3). The picture below gives examples of SLOs for P S4.3 by Hasse diagrams where vertices represent (closed) elements a with a = a and vertices represent elements a with a > a. Note that, for each element a, a is the (unique) smallest element b with b > a. d e a b c a 1 a 2 a n C 1 g g E n 3 Incomplete SP-logics Axiomatising even very simple frames by SP-implications turns out to be a challenging problem because many natural and innocuously looking axiomcandidates will most probably give incomplete SP-logics. For instance, let σ n (p, q) = ( p ( q (p ) ). }{{} used n times, 1 n < ω Consider the following SP-implications (the reader may find it useful to compare them with the SP-implications defined in the next section): ι 2 fun = (p q) (p r) (q r) (p q r), (1) ɛ 1 = p q ( p q), (2) ɛ 2 = (p q) (p r) (q r) ( p (q r) ), (3)

5 Kikot, Kurucz, Wolter, Zakharyaschev 5 ϕ = (p q) (p r) (q r) ( p (q r) ), (4) β n = σ n (p, q) σ n (q, p) (p q), (5) γ n = σ n+1 (p, q) (p q), = σ n+1 (p, q) σ n+1 (q, p). δ n Figure 1 describes inclusions between the SP-logics above P 2 fun = P S4.3 + ι 2 fun axiomatised by these SP-implications. Using results from [15] and Claim 3.1 below, it is not hard to see that all of the depicted inclusions are proper. P [(2)] = P 2 fun + ɛ 1 P [(2), L 2 ] = P 2 fun + ϕ P [L 2 ] = Pfun 2 + ɛ 2 + β 2 Pfun 2 + β 2 Pfun 2 + ɛ 2 Pfun 2 + δ 2 P 2 fun + δ 3 Pfun 2 + γ 2 Pfun 2 + β 3 Pfun 2 + γ 3 Pfun 2 + β 4 Pfun 2 + γ 4 P 2 fun + δ 4 P 2 fun = P S4.3 + ι 2 fun Fig. 1. Complete and incomplete SP-logics above P 2 fun = P S4.3 + ι 2 fun. The only complete SP-logics in the picture are shown by (that they are axiomatised by the indicated SP-implications follows from [15]; see also Theorem 4.4 below). On the other hand, it is not hard to check that, for each SP-logic P in Fig. 1, if P is below P [(2), L 2 ] then the rooted frames for P are L 1, (2), L 2, and if P is below P [L 2 ] but not below P [(2), L 2 ] then the rooted frames for P are L 1, L 2. Therefore, all SP-logics shown by + in the picture are incomplete. Claim 3.1 (i) Pfun 2 +ɛ 2 = SLO ϕ and Pfun 2 +ɛ 2 = SLO β 2, (ii) Pfun 2 +β 2 = SLO ɛ 2, (iii) Pfun 2 + γ n = SLO β n, (iv) Pfun 2 + β n+1 = SLO δ n, (v) Pfun 2 + δ n = SLO β n+1. Proof. Claim (i) can be shown by the SLO A 2 in Fig. 3 (in the appendix), (ii) by A 3, (iii) by C n, (iv) by B n, and (v) by D n. The frames for the incomplete SP-logics above have at most 2 points. Similar incomplete SP-logics can clearly be defined for more complex frames, which makes the lattice Ext + P S4.3 much more involved compared to NExtS Axiomatising frame classes by SP-implications As any SP-implication ι is a Sahlqvist formula, Fr(ι) is FO-definable [20]. As shown in [16], the class of reflexive and transitive frames validating ι is closed under subframes. Using the results from [11,23] on FO-definable subframe logics, we obtain the following theorem (which can be readily proved directly):

6 6 On Strictly Positive Modal Logics with S4.3 Frames Theorem 4.1 For every ι / P S4.3, there exists n < ω such that Fr(ι) is closed under subframes and any frame in Fr(ι) is of depth < n. Thus, for any SP-logic P Ext + P S4.3, either P = P S4.3 or Fr(P ) is of bounded depth. Recall from [11,23] that, for any finite frame F, there are modal formulas α(f) and α (F), called the subframe and cofinal subframe formulas for F, respectively, such that, for any frame G, G = α(f) iff F G, G = α (F) iff F G. In fact, any L NExtS4.3 can be represented in the form L = S4.3 α (F 1 ) α (F m ), for some F i. If C is an FO-definable class of frames closed under subframes and N < ω is the maximal depth of frames in C, then there exist frames F 1,, F m of depth N such that L[C] = S4.3 α(l N+1 ) α(f 1 ) α(f m ). (6) Note that S4.3 α(f) = S4.3 α (F) α (F ), where F is obtained from F by adding a single-point cluster on top of F, and that S4.3 α(l n ) = S4.3 α (L n ). Our aim now is (i) to construct SP-analogues κ N (F) of the subframe formulas α(f) such that G = κ N (F) iff G is of depth N and G = α(f); and (ii) to prove that the κ N (F) axiomatise all complete SP-logics. Let F = (n 1,, n f ) be a finite frame with f N < ω. We begin by defining an SP-implication ι N (F) as follows. First, we take the terms τ(f) = ( P1 ( P2 ( P f ) )), (7) where the P i are pairwise disjoint sets of variables with P i = n i, for i f. Next, denote by Σ N (F) the set of all terms of the form ( Q1 ( Q2 ( Q N ) )), where there exist 1 = x 1 < x 2 < < x f+1 = N + 1 such that: for any i and j, if x i j < x i+1 then Q j P i and Q j P i 1, (8) there is i such that Q j = P i 1 for any j with x i j < x i+1. (9) (Note that =, so when some Q j =, the corresponding term in Σ N (F) is P S4 -equivalent to a term of modal depth < N.) It is not hard to see that Σ N (F) (max i P i + 1) N, and so Σ N (F) is polynomial in F. (10) Finally, we set ι N (F) = σ τ(f). σ Σ N (F)

7 Kikot, Kurucz, Wolter, Zakharyaschev 7 For example, ι 1 ((n)) is Q {p 1,...,p n} Q =n 1 Q (p 1 p n ), defining (n 1)-functionality (in particular, ι 1 ((3)) is ι 2 fun in (1)). For every m n, the SP-implication ι m (L n ) is (P S4 -equivalent to) n i=1 [ ( p 1 p 2 [ ( p i 1 [p i+1 ( p n ) ] ) ])] ( p 1 ( p 2 ( p n ) )), defining the property of having depth < n. In particular, ι n (L n ) is ɛ n 1 in (2) (3) for n = 2, 3; ι 2 ((1, 2)) is ϕ in (4), and ι 2 ((2)) is β 2 in (5). In order to define κ N (F), we require the following: Claim 4.2 [15, Lemma 7.7] For any finite set Φ of SP-implications, there is a single SP-implication ι Φ such that P S4 + Φ = P S4 + ι Φ. Now, we set κ N (F) = ι {ι N+1 (L N+1 ),ι N (F)}. Observe that ι n (L n ) is P S4 -equivalent to ι n+1 (L n+1 ), and so κ n (L n ) = ι n (L n ). Theorem 4.3 For any finite S4.3-frame F, any N with d(f) N < ω, and any S4.3-frame G, we have G = κ N (F) iff G is of depth N and G = α(f). Proof. It suffices to show that (i) if F G then G = ι N (F); and (ii) if d(g) N and F G then G = ι N (F). (i) Let F = (n 1,, n f ) and F G. Then there exist clusters (ζ 1 ),, (ζ f ) in G such that (ζ 1,, ζ f ) is a subframe of G and ζ i n i, for i f. We define a model M based on G in the following way. For each i f, take n i distinct points from (ζ i ) and let each validate f j=1 P j \ {p} for each different p P i. All other points in G validate no variables. Then it is easy to see that M, r = ι N (F) for any r in the root cluster of G. (ii) is proved by induction on the pair (d(f), N) for N d(f). To begin with, we claim that if (n) (ζ) for some n < ω, then (ζ) = ι 1 ((n)). Indeed, in this case, ζ < n. Suppose τ((n)) = P 1 for some set P 1 of variables with P 1 = n. Let M be a model on (ζ) such that the left-hand side of ι 1 ((n)) holds at some point in M. Then, for each of the n-many (n 1)-element subsets Q of P 1, Q is in Σ 1 ((n)), and so there is x Q (ζ) with M, x Q = Q. By the pigeonhole principle, there is x (ζ) with M, x = P 1. Now suppose inductively that, for all F and N d(f ), if either d(f ) < f or N < N, then G = ι N (F ) for every G with d(g) N and F G. Let F = (n 1,, n f ), N f, G = (ζ 1,, ζ g ), for some g N, and F G. Suppose M is a model over G such that M, r = σ, for some root r in G. (11) σ Σ N (F)

8 8 On Strictly Positive Modal Logics with S4.3 Frames Take the term τ(f) from (7) and consider two cases: Case 1: g < f. Then there is j g such that, for all i f, M, x i = P i for some x i (ζ j ). Indeed, suppose otherwise. For any j g, take some i j f with M, x = P ij for any x (ζ j ). However, as g < f, there is σ Σ N (F) with P S4 SLO σ δ for δ = ( P i1 ( P i2 ( P ig ) )), and so M, r = δ by (11), which is a contradiction. Case 2: f g N. As P S4 SLO σ P f for some σ Σ N (F), by (11) there is x with M, x = P f. Let We show that m = max{j M, x = P f for some x (ζ j )}. M, y = P 1 ( P2 ( P f 1 ) ), for some j m, y (ζ j ). (12) Indeed, two cases are possible. (2.1): ζ j < n f for every j with m < j g. Suppose (12) does not hold. Since for any sequence (Q m+1,, Q g ) of (n f 1)-element subsets of P f, we have σ Σ N (F) with P S4 SLO σ δ for the term δ = ( P1 ( P f 1 ( Qm+1 ( Q g ) ))), M, r = δ follows by (11). Therefore, there is j with m < j g such that for each of the n f -many (n f 1)-element subsets Q of P f, we have M, x Q = Q for some x Q (ζ j ), and so M, x = P f for some x (ζ j ) by the pigeonhole principle, contrary to the definition of m. (2.2) There is j such that m < j g and ζ j n f. Let F = (n 1,, n f 1 ). Then F (ζ 1,, ζ m ), and so by IH we have (ζ 1,, ζ m ) = ι m (F ). (13) For every δ Σ m (F ) of the form ( Q 1 ( Q 2 ( Q m ) )), there is some σ Σ N (F) with P S4 SLO σ δ + for the term δ + = ( Q1 ( ( Qm ( Pf ))) ). Thus, M, r = δ + by (11). Therefore, by the definition of m, there is i m such that M, x = Q 1 ( Q 2 ( Q m ) ) for some x (ζ i ), and so M, r = δ for the restriction M of M to (ζ 1,, ζ m ). By (13), we have M, r = τ(f ), and so (12) holds as required. We are now in a position to prove our main theorem on definability, completeness (axiomatisability) and computational complexity: Theorem 4.4 Let C be any first-order definable class of S4.3-frames closed under subframes and different from Fr(P S4.3 ), and let N < ω be the maximal

9 Kikot, Kurucz, Wolter, Zakharyaschev 9 depth of frames in C. Then there is a finite set F C of frames of depth N such that the following conditions hold for the SP-logic { P S4.3 + {κ N (F) F F C }, if F C, P C = P S4.3 + κ N+1 (L N+1 ), otherwise: (i) C = Fr(P C ). (ii) P C is Kripke complete, and so P [C] = P C. (iii) P C is decidable in PTime. Proof. Let F C = {F 1,, F m }, where the frames F i are supplied by (6). Then, for any frame G of depth N, we have G C iff F G for any F F C. (i) is a straightforward consequence of Claim 4.2 and Theorem 4.3. (ii) By an F C -normal form we mean any term of the following form: ( P 1 ( P 2 ( P k ) )) where k N and each P i is a finite set of variables such that there is no F F C with F ( P 1, P 2,, P k ). (We do not require that the P i are disjoint.) For a set Σ of terms, let var(σ) be the set of propositional variables in Σ. We write var(σ) for var ( {σ} ). Claim For every term σ, there is a set N σ of F C -normal forms such that var(n σ ) = var(σ), N σ is polynomial in the size of σ, and ( P S4.3 + ι N+1 (L N+1 ) + {ι N (F) F F C } SLO σ ). ϱ N σ ϱ Proof. We proceed via a series of steps (a) (c). (a) As shown in [16, Claim 48.1]), by wcon, σ is P S4.3 -equivalent to a conjunction of terms of the form ( Q 1 ( Q 2 ( Q k ) )), where k < ω and each Q i is a finite set of variables. Each such term describes a full linear branch in the term tree of σ, and so (p1) each k is polynomial in the size σ of σ; (p2) each Q i of each term is polynomial in σ ; and (p3) the overall number of terms we obtain this way is polynomial in σ. (b) By ι N+1 (L N+1 ), we can take k N in (a). By (10) and (p1) (p3), the number of terms we thus obtain is polynomial in σ, so we are done if F C =. (c) Let F C. We claim that each term in (b) is P S4 + {ι N (F) F F C }- equivalent to a conjunction of F C -normal forms. Indeed, observe first that if ι N (F) = (σ τ) then P S4 SLO τ σ always holds. Now suppose χ = ( Q 1 ( Q 2 ( Q k ) )) for k N and finite sets Q i of variables with F ( Q 1, Q 2,, Q k ) for some F F C. Clearly, we may assume that d(f) = k. Let F = (n 1,, n k ) and n i Q i for i k. For i k, let P i = {p i 1,, p i n i } be the variables in

10 10 On Strictly Positive Modal Logics with S4.3 Frames ι N (F) and Q i = {q1, i, q Q i i }. We take the following substitution instance of ι N (F): for any i k and j n i 1, we substitute qj i for pi j, and qi n i q Q i i for p i n i. Then, by (8) and (9), χ is equivalent to a conjunction of terms of the form ( R 1 ( R 2 ( R l ) )) for some l N such that (n1) for every i = 1,, l there is some j i k with R i Q ji ; (n2) there is j k such that R i < Q j, for every i with j i = j. If a term ϑ of this form is not an F C -normal form, then again there is some F F C such that d(f ) = l and F ( R 1, R 2,, R l ). Thus, we can continue with applying ι N (F ) to ϑ. By (n1) and (n2), sooner or later the procedure stops and we obtain a set of F C -normal forms. In fact, by (10) and (p1) (p3), both the number of necessary steps and the number of terms we obtain in each step are polynomial in σ. Claim For every finite set Σ {ξ} of F C -normal forms such that var(σ) var(ξ), if P S4 SLO Σ ξ then G = Σ ξ, for some G C. Proof. Suppose ξ = ( P 1 ( P 2 ( P k ) )) where k N, and each P i is a finite set of variables such that there is no F F C with F ( P 1, P 2,, P k ). Then ( P 1, P 2,, P k ) C. By P S4, we may assume that neither P i P i+1 nor P i+1 P i for any i = 1,, k. Let M be the following model over ( P 1, P 2,, P k ) : for each i = 1,, k, let each point in ( P i ) validate k j=1 P j \ {p} for each different p P i. Let χ be any F C -normal form such that var(χ) var(ξ). Then χ is of the form ( R 1 ( R 2 ( R l ) )) for some l N, R i k j=1 P j for i l. If P S4 SLO χ ξ then there is no subsequence (R i1,, R i k ) of (R 1,, R l ) with P j R ij for j k. Thus, it is not hard to check that M, r = χ for any root r. As we clearly have M, r = ξ, it follows that M, r = Σ ξ for any set Σ of F C -normal forms with var(σ) = var(ξ) and P S4 SLO Σ ξ. Now the proof of (ii) can be completed as follows. Suppose P C = Kr σ τ. It is not hard to show [15, Lemma 5.1] that σ τ is P S4 -equivalent to σ τ and we may also assume that var(σ) = var(τ). Thus by Claim 4.4.1, we have P C = Kr χ χ N σ ϑ N τ ϑ. and so P C = Kr χ N σ χ ϑ, for every ϑ N τ. Again, it is not hard to show [15, Lemma 5.1] that we may assume that var(n σ ) var(ϑ). Therefore, by (i) and Claim 4.4.2, P S4 SLO χ N σ χ ϑ, for every ϑ N τ, and so P S4 SLO χ χ N σ ϑ N τ ϑ. By the completeness of P S4 and Claim 4.4.1, we obtain that P C SLO σ τ. (iii) follows from Claims and 4.4.2, and the tractability of P S4.

11 Kikot, Kurucz, Wolter, Zakharyaschev 11 5 Modal logics vs. SP-logics The set NExtS4.3 ordered by forms a distributive lattice, a pseudo-boolean algebra, to be more precise [19]. On the other hand, as follows from [15, Proposition 5.11], the lattice Ext + P S4.3 is not even modular because it contains the pentagon N 5 lattice; see Fig. 1. Following [4], we compare these two lattices of logics using the map π : NExtS4.3 Ext + P S4.3, which gives the SP-fragment π(l) of any modal logic L, and the map µ: Ext + P S4.3 NExtS4.3, which gives the modal extension µ(p ) = S4.3 P of any SP-logic P. We call L a modal companion of π(l), and P an SP-companion of µ(p ). It was observed in [4] that µ and π are monotone and form a Galois connection: µ(p ) L iff P π(l). In this section, we obtain answers to some open questions from [4] in the context of NExtS4.3 and Ext + P S4.3. First, we give a characterisation of those SP-logics that have modal companions (which actually holds for all multi-modal SP-logics). Theorem 5.1 An SP-logic P has a modal companion iff P is complete. Proof. ( ) If π(l) = P and ι / P, then ι / S4.3 P, and so, by Sahlqvist completeness, there is a Kripke frame for P refuting ι, as required. ( ) It is readily seen that, if P is complete, then S4.3 P is its modal companion. Next, we give a frame-theoretic characterisation of modal companions of any complete SP-logic in Ext + P S4.3, which consists of two parts. The first part describes the set π 1 (P S4.3 ) of modal companions of P S4.3 and shows that it comprises exactly the logics in NExtS4.3 that have frames of unbounded depth. We say that such logics are of slice ω. Theorem 5.2 For any modal logic L NExtS4.3, we have π(l) = P S4.3 iff L Grz.3 = S4.3 α((2)). Thus, Grz.3 is the greatest companion of P S4.3. Proof. ( ) If L Grz.3, then α(l n ) L, for some n < ω. By Theorem 4.3, it follows that κ n (L n ) L, contrary to π(l) = P S4.3. ( ) Suppose there is ι π(l) \ P S4.3. By Theorem 4.1, there is n < ω such that L n = ι, which is impossible since L n = Grz.3. In other words, π 1 (P S4.3 ) comprises S4.3 and all those (infinitely many) logics in NExtS4.3 whose classes of frames are not FO-definable. By a standard Löwenheim Skolem Tarski argument, if C is FO-definable, then both L[C] and P [C] are determined by the countable frames in C. So the remaining logics can be classified according to the depth of their countable frames. (This classification was suggested in [14,18].) We say that a modal logic L is of slice n (0 < n < ω) if L n = L but L n+1 = L. Within NExtS4.3, the logics L of slice n form the infinite interval S4.3 α(l n+1 ) = L[(ω,, ω)] L L[L }{{} n ] = S4.3 α(l n+1 ) α ( (2) ). n In particular, the logics L of slice 1 form the (infinite) interval S5 L L[L 1 ]. (It is shown in [18] that all logics of finite slices above S4 are locally finite.)

12 12 On Strictly Positive Modal Logics with S4.3 Frames We now characterise the modal companions of complete SP-logics properly extending P S4.3. Given the class Fr ω (P ) of all countable rooted frames for P, denote by Fr ω (P ) its smallest subclass whose closure under subframes gives Fr ω (P ). (It is not hard to see that Fr ω (P ) always exists.) Theorem 5.3 For any complete SP-logic P P S4.3, π 1 (P ) = {L NExtS4.3 Fr ω (P ) Fr ω (L) Fr ω (P )}. Thus, L[ Fr ω (P )] is the greatest modal companion of P. Proof. Suppose Fr ω (P ) Fr ω (L) Fr ω (P ) and show that π(l) = P. If ι L, then Fr ω (L) = ι, and so Fr ω (P ) = ι. By Theorem 4.1, it follows that Fr ω (P ) = ι and ι P. The implication ι P ι L is trivial. Next, we have to prove that every modal companion L of P belongs to the specified interval. It suffices to show that if L = S4.3 P α (F) is a modal companion of P, then Fr ω (P ) = α (F). Suppose otherwise and take some G Fr ω (P ) such that F G. By the construction of Fr ω (P ), we can always find a finite frame G such that F G G and G H, for any H in Fr ω (P ) different from G. But then α(g ) L, and so κ n (G ) P by Theorem 4.3, where n is the slice of P, which is impossible because G Fr ω (P ). Intuitively, L[ Fr ω (P )] saturates S4.3 P with all those formulas α (F) that do not derive any α(g) / S4.3 P. We illustrate this by a few examples. Example 5.4 (1) The SP-logic P [(ω, ω)] = P S4.3 +κ 3 (L 3 ) has only one modal companion S4.3 α(l 3 ) since (ω, ω) = α (F), for any F of depth 2. (2) The SP-logic P [(2, 1)] = P S4.3 + κ 3 ((1, 2)) + κ 3 ((3)) has two modal companions: S4.3 α(l 3 ) α ( (1, 2) ) α ( (3) ) = L[(2), (2, 1)] and its extension with α ( (2) ), i.e., L[(2, 1)]. Note that S4 α ( (2) ) = S4 p p = S4.1. (3) The companions L of P [(ω, 1)] = P S4.3 + κ 3 ((1, 2)) form the infinite interval L[(ω), (ω, 1)] = S4.3 α(l 3 ) α ( (1, 2) ) L S4.3 α(l 3 ) α ( (2) ) = L[(ω, 1)]. As follows from our results above, π maps NExtS4.3 onto the complete SP-logics in Ext + P S4.3. On the other hand, for a complete SP-logic P, there may exist Kripke incomplete logics P such that Fr(P ) = Fr(P ). All these logics form the set µ 1 (S4.3 P ). Thus, for every L NExtS4.3, the set µ 1 (L) has π(l) as its greatest element; see Fig. 4 in the appendix. We do not know whether µ 1 (L) always has a least element, whether it has non-finitely axiomatisable SP-logics, and whether there is a continuum of them. As we saw above, any SP-logic different from P S4.3 belongs to some finite slice. In Fig. 2, we show slice 1 and a part of slice 2 (to minimise clutter, we only give the frames or SLOs determining modal and SP-logics; as before, indicates incomplete SP-logics). The structure of slice 1 was detailed in [15]. As shown in [16], it has two incomplete SP-logics: P [S] and P [E 1 ], where S

13 Kikot, Kurucz, Wolter, Zakharyaschev 13 is a SLO with two elements a such that a = =, and E 1 is on p. 4. Note that µ ( P [S] ) is inconsistent, while µ 1( L[(1)] ) = { P [(1)], P [E 1 ] }. Slice 2 is much more involved. Its sublattice of Kripke complete logics comprises the SP-logics of the form P [C], where C is a finite set of frames of depth 2 at least one of which is of depth 2. As shown in Section 3, modal logics L[C] of slice 2 typically have infinitely many SP-companions in µ 1 (L); see Fig. 1. On the other hand, Example 5.4 shows SP-logics with multiple modal companions. NExtS4.3 Ext + P S4.3 µ S (1) (2) (3) (ω) π (1, 1) (2), (1, 1) (3), (1, 1) (2, 1) (ω), (1, 1) π 1 (P [(2, 1)]) (2), (2, 1) (3, 1) (2), (3, 1) (3), (3, 1) π 1 (P [(3, 1)]) (ω, 1) µ π (1) π (2) E 1 µ 1 (L[(1)]) (3) (ω) (1, 1) (2), (1, 1) (3), (1, 1) (ω), (1, 1) π (3, 1) π µ (ω, 1) µ 1 (L[(1, 1)]) µ 1 (L[(2), (1, 1)]) Fig. 2. Modal and SP-logics of slices 1 and 2. 6 SP-rules An SP-rule, ϱ, takes the form ι1,...,ιn ι, where ι 1,, ι n, ι are SP-implications. We say that ϱ is valid in a SLO A and write A = ϱ if A satisfies the quasiequation (ι 1 ι n) ι, where (σ τ) = (σ τ = σ). We identify the rule ι with ι. Given a set R of SP-rules and an SP-rule ϱ, we set R = SLO ι if A = ϱ for any SLO A validating every rule in R. We denote by R[A] the set of SP-rules that are valid in A and, for a frame F, set R[F] = R[F ]. For a class C of SLOs (or frames), we define the SP-rule logic R[C] of C as the intersection of all R[A] with A C. Clearly, there is a one-to-one correspondence between SP-rule logics and quasi-varieties of SLOs. The minimal SP-rule logic R containing P S4.3 and a set Φ of SP-rules is denoted by P S4.3 + Φ. We call R globally complete in case ϱ P S4.3 + Φ iff ϱ R[F] for any frame F validating all rules in R. If Φ is a set of SP-implications, then P S4.3 + Φ is complete whenever it is globally complete. However, the converse only holds for two non-trivial SP-logics containing P S4.3 (see appendix for the proof): Theorem 6.1 A non-trivial P Ext + P S4.3 is globally complete iff P is complex iff P {P S5, P [(1)]}.

14 14 On Strictly Positive Modal Logics with S4.3 Frames It follows that the axiomatisations of SP-logics given above do not provide axiomatisations of the corresponding SP-rule logics, except for two cases. This is in sharp contrast to normal modal logics containing S4.3 where the introduction of rules does not extend the expressive power of formulas as any rule ϱ = ι1,...,ιn ι can be equivalently expressed by the formula (ι 1 ι n ) ι. We next observe that SP-rules have sufficient expressive power to define any modally definable class of S4.3-frames: Theorem 6.2 For every finite S4.3-frame F, there is an SP-rule ϱ(f) such that, for any S4.3-frame G, we have G = ϱ(f) iff F is a p-morphic image of G. Proof. Let F = (W, R) be a finite S4.3-frame with root r. For each x W, take a variable p x and the following set F of SP-implications: p x p y if R(x, y), p x p y q if R(x, y), p x p y q for x y, p y for y in the final cluster of F. Let F ϱ(f) = p r q It is straightforward to show that ϱ(f) is as required. Corollary 6.3 A class of frames is modally definable iff it is SP-rule definable. We now return to the axiomatisation problem for SP-rule logics. The SLOs on p. 4 show that the rules from Theorem 6.2 do not axiomatise globally complete SP-rule logics. Here, we give an axiomatisation of R[(2)], which can be easily generalised to any R[(n)] with n > 2. Consider the rules ϱ 1 = (a c i), (a (c i c j ) b), 1 i j 3 a b ϱ 2 = (c 1 b), (c 2 b), (a c 1 ), (a c 2 ), (a (c 1 c 2 ) b) a b ϱ 3 = (a c 1), (a c 2 ), (c 1 b), (a (c 1 c 2 ) b), (a (a c 2 ) b) a b It is easy to see that a cluster (n) validates all the ϱ i iff n 2. Thus, restricted to the set of clusters, each rule ϱ i defines the intended class of frames. Let R (2) = P S5 + {ϱ 1, ϱ 2, ϱ 3 }. The proof of the following theorem is given in the appendix: Theorem 6.4 (i) R (2) = R[(2)]. (ii) P S5 + Φ R[(2)], for any proper subset Φ of {ϱ 1, ϱ 2, ϱ 3 }. The computational complexity of deciding R[C] has been analysed in [17]. In contrast to SP-implications, deciding SP-rules is often conp-hard. In fact, if C is a non-empty class of S4.3-frames of the form Fr(P ) for an SP-logic P Ext + P S4.3, then R[C] is in PTime iff C is the class of all clusters or a singleton cluster. Otherwise, R[C] is conp-complete.

15 Kikot, Kurucz, Wolter, Zakharyaschev 15 References [1] Baader, F., S. Brandt and C. Lutz, Pushing the EL envelope, in: Procs. of IJCAI (2005), pp [2] Beklemishev, L., Calibrating provability logic: from modal logic to reflection calculus, in: Advances in Modal Logic, vol. 9, College Publications, 2012 pp [3] Beklemishev, L., Positive provability logic for uniform reflection principles, Annals of Pure and Applied Logic 165 (2014), pp [4] Beklemishev, L., A note on strictly positive logics and word rewriting systems (2015), corr abs/ [5] Birkhoff, G., On the structure of abstract algebras, Proc. Cambridge Phil. Soc. 31 (1935), pp [6] Bull, R., That all normal extensions of S4.3 have the finite model property, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 12 (1966), pp [7] Chagrov, A. and M. Zakharyaschev, Modal Logic, Oxford Logic Guides 35, [8] Dashkov, E., On the positive fragment of the polymodal provability logic GLP, Mathematical Notes 91 (2012), pp [9] Davey, B., M. Jackson, J. Pitkethly and M. Talukder, Natural dualities for three classes of relational structures, Algebra Universalis (2007), pp [10] Fine, K., The logics containing S4.3, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 17 (1971), pp [11] Fine, K., Logics containing K4, part II, Journal of Symbolic Logic 50 (1985), pp [12] Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic 44 (1989), pp [13] Grätzer, G., Universal Algebra, Springer, 1979, 2nd edition. [14] Hosoi, T., On intermediate logics, Journal of the Faculty of Science, University of Tokyo 14 (1967), pp [15] Jackson, M., Semilattices with closure, Algebra Universalis 52 (2004), pp [16] Kikot, S., A. Kurucz, Y. Tanaka, F. Wolter and M. Zakharyaschev, Kripke completeness of strictly positive modal logics over meet-semilattices with operators, CoRR abs/ (2017). URL [17] Kurucz, A., F. Wolter and M. Zakharyaschev, Islands of tractability for relational constraints: towards dichotomy results for the description logic EL, in: Advances in Modal Logic, vol. 8 (2010), pp [18] Maksimova, L., Modal logics of finite slices, Algebra and Logic 14 (1975), pp [19] Maksimova, L. and V. Rybakov, Lattices of modal logics, Algebra and Logic 13 (1974), pp [20] Sahlqvist, H., Completeness and correspondence in the first and second order semantics for modal logic, in: Procs. of the 3rd Scandinavian Logic Symposium, 1975 pp [21] Sofronie-Stokkermans, V., Locality and subsumption testing in EL and some of its extensions, in: Advances in Modal Logic, vol. 7 (2008), pp [22] Spaan, E., Complexity of Modal Logics, Ph.D. thesis, Department of Mathematics and Computer Science, University of Amsterdam (1993). [23] Zakharyaschev, M., Canonical formulas for K4. Part II: Cofinal subframe logics, Journal of Symbolic Logic 61 (1996), pp

16 16 On Strictly Positive Modal Logics with S4.3 Frames Appendix A 1 A 2 n n n B n C n D n Fig. 3. SLOs validating P 2 fun. L[ C] L[C] = L π 1 (P ) µ π µ 1 (L) P = P [C] Fig. 4. Maps π and µ.

17 Kikot, Kurucz, Wolter, Zakharyaschev 17 Proof of Theorem 6.1 The first equivalence is shown in [16]. For the second one, it is known from [16] that P S5 and P [(1)] are complex. Conversely, let P {P S5, P [(1)]}. Suppose first that (1, 1) Fr(P ). Then the SLO C 1 on p. 4 validates P but there is no frame F Fr(P S4.3 ) such that C 1 is embeddable into F. Thus, P is not complex. Now suppose (1, 1) Fr(P ). Then P P S5 and there exists a minimal n > 2 such that (n) Fr(P ). Then the SLO E n on p. 4 validates P but there is no frame F Fr(P ) such that E n is embeddable into F. Proof of Theorem 6.4 (ii) Take E 3 on p. 4 and A 3 and A 4 shown below: A 3 A 4 Then E 3, A 3, A 4 all validate P S5 and E 3 = ϱ 1, but E 3 = ϱ 2, ϱ 3 ; A 3 = ϱ 2, but A 3 = ϱ 1, ϱ 3 ; A 4 = ϱ 3, but A 4 = ϱ 1, ϱ 2. For (i), it suffices to provide an embedding of any SLO A with A = R (2) into F, for a union F of two-element clusters. Recall that a filter F in A is a subset of A containing and such that a F and a b imply b F and a, b F imply a b F. For a filter F in A, we set F = { a a F } and define F = (W, R) using a set X of pairs of filters in A. Let (F 1, F 2 ) X if F 1, F 2 are filters in A such that F 1 F 2 and F 2 F 1 ; if F 1 F or F 2 F for a filter F, then F 1 F or F 2 F. For any w = (F 1, F 2 ) X, take fresh 1 w and 2 w and set W = {1 w, 2 w w X }, R = {(1 w, 2 w ), (1 w, 1 w ), (2 w, 2 w ), (2 w, 1 w ) w X }. By definition, F = (W, R) is a union of two-element clusters. We now embed A into F. We show that f(a) = {1 F1,F 2 a F 1 } {2 F1,F 2 a F 2 } is an embedding. We first show that if a b, then f(a) f(b). We may assume that a b. Let F 0 be a maximal filter containing a such that b F 0. We show that there exists a pair (F 1, F 2 ) X with a F 1 and b F 1. Let M be the set of all maximal filters G in A such that G F 0 and F 0 G. Observe that there exists a filter G M containing a set X A iff (a 1 a n ) F 0 for all a 1,, a n X. It follows that, for every filter F in A with F 0 F, there exists G M with F G. We now make a case

18 18 On Strictly Positive Modal Logics with S4.3 Frames distinction according to the cardinality of M. M = 1. Let M = {G}. Then G F 0 and (F 0, G) X. We have a F 0 and b F 0, as required. M = 2. Let M = {G 0, G 1 }. We distinguish two cases: Case 1: G 0, G 1 F 0. Since neither G 0 G 1 nor G 1 G 0, we obtain G 0 F 0 and G 1 F 0. Thus, b G 0 G 1. Then we find c 1 G 0 and c 2 G 1 and a F 0 such that a (c 1 c 2) b. Then, using the condition that filters are closed under, we find c 1 G 0, c 2 G 1, and a F 0 such that c 1 b, c 2 b, a c 1, a c 2, and a (c 1 c 2 ) b. By rule ϱ 2, a b, and we have derived a contradiction as b F 0 follows. Case 2: G 1 F 0. Then G 0 F 0 as there exists at least one filter G M with G F 0. Assume first that G 0 F 0. Then b G 0. Similarly to Case 1 we thus find c 1 G 0, c 2 G 1, and a F 0 such that a c 1, a c 2, c 1 b, a (c 1 c 2 ) b, and a (a c 2 ) b. But then, by rule ϱ 3, a b, and we have derived a contradiction. Assume now that G 0 = F 0. Then (F 0, G 1 ) X. We are done as a F 0 and b F 0. M 3. Let G 0, G 1, G 2 M. Then we find a F 0, c 1 G 0, c 2 G 1, and c 3 G 2 such that a c i for 1 i 3 and a (c i c j ) b for 1 i j 3. But then a b, by rule ϱ 1, and we have derived a contradiction. We next show that f( a) = f(a) for all a A. Suppose first 1 F1,F 2 f(a). Then a F 1 or a F 2. Then a F 1. Then 1 F1,F 2 f( a), as required. Conversely, suppose 1 F1,F 2 f( a). Then a F 1. Then there exists a filter F with a F and F 1 F and F F 1. By definition, F 1 F or F 2 F. Thus 1 F1,F 2 f(a). Finally, f(a 1 a 2 ) = f(a 1 ) f(a 2 ) can we proved in a straightforward way.