Arrow Logic with Arbitrary Intersections - Semantic Scholar

Fundamenta Informaticae 76 (2006) 1–25

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IOS Press

Arrow Logic with Arbitrary Intersections: Applications to Pawlak’s Information Systems Philippe Balbiani∗ Universit´e Paul Sabatier Institut de recherche en informatique de Toulouse 31062 Toulouse Cedex 9, France [email protected]

Dimiter Vakarelov Sofia University Faculty of Mathematics and Computer Science blvd James Bouchier 5, 1126 Sofia, Bulgaria [email protected]

Abstract. We dedicate this paper to the memory of Zdislaw Pawlak, the founder of rough sets methodology in computer science. A great deal of our scientific work was motivated and influenced by Pawlak’s ideas. Keywords: Information systems, Information relations, Information logics.

1. Introduction In order to describe entities referred to as objects in terms of their properties, Pawlak [17] introduced, in 1981, his notion of information system. Soon after that, Orłowska and Pawlak [16] defined the propositional modal logic N IL, non-deterministic information logic, the first information logic. N IL was designed as a logic for reasoning about objects in information systems containing non-deterministic informations. The first completeness theorem for N IL with respect to its standard semantics was given ∗

Address for correspondence: Universit´e Paul Sabatier, Institut de recherche en informatique de Toulouse, 31062 Toulouse Cedex 9, France

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in [19]. After that, a great variety of modal logics related to Pawlak’s information systems have been introduced and studied by several authors: Balbiani [1], Demri [6], Orłowska [13], Vakarelov [20] and others. The book by Demri and Orłowska [8] contains a comprehensive bibliography on the subject (see also [24]). Let us note, however, that all the information logics considered so far are related to information systems containing non-deterministic informations, while the original Pawlak’s systems, introduced in [17], are single-valued (see next section for the relevant definitions). This can be explained by the fact that most of the information relations are interesting only in the non-deterministic case whereas in the case of single-valued systems most of the information relations become trivial or collapse to the strong and weak indiscernibility relations. The aim of this paper is to consider another kind of information logics, which are interesting and meaningful for the original Pawlak’s information systems. The idea, presented in [25], is based on an observation about the relations between arrow structures, used as a semantic basis for arrow logic, and Pawlak’s single-valued information systems. In fact, different systems of arrow logic can be reinterpreted as logics for information systems. Our aim is to consider arrow logics which are naturally related to the original single-valued Pawlak’s information systems. The paper is organized as follows. In section 2, we list the definitions of Pawlak’s information systems, we introduce some standard information relations and we show how they can be used as a semantic basis for information logics. Then we show how the standard information relations degenerate in the case of single-valued information systems. Section 3 is devoted to the definitions concerning arrow logics and single-valued information systems. We discuss there about several kinds of new information relations, typical for arrow logics but not considered yet in information logics. These new information relations compare objects of the information system with respect to different attributes. They have a great expressive power and allow us to code the full information in the system up to an isomorphism. We also present, in section 3, the intended system of arrow logic which can be considered twofold: as a system of arrow logic and as a system of information logic related to the original single-valued Pawlak’s information systems. Section 4 presents the syntax and the semantics of a variant of arrow logic: arrow logic with arbitrary intersections. The axiomatization/completeness and the decidability/complexity issues of this variant are addressed in sections 5 and 6.

2. Pawlak’s information systems 2.1. Information systems We adopt the following general definition of Pawlak’s information systems, named attribute systems in [24] (see also [8]). By an attribute system, A-system for short, we mean any system of the form S = (Ob, At, {V al(a) : a ∈ At}, f ), where • Ob is a nonempty set of objects, • At is a set of attributes, • for each a ∈ At, V al(a) is a set whose elements are called values of the attribute a and • f is a two-argument function, called information function, which assigns to each object x ∈ Ob and to each attribute a ∈ At a subset f (x, a) of V al(a) called information of x according to a.

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Hence, in S, objects are described in terms of attributes. We shall say that S is deterministic if f (x, a) contains at most one element for all x ∈ Ob and for all a ∈ At. S is said to be total if f (x, a) 6= ∅ for all x ∈ Ob and for all a ∈ At. We shall say that S is single-valued if S is both deterministic and total. In this case, for all xS∈ Ob and for all a ∈ At, f (x, a) is a singleton and we may as well consider f as a function ranging over a∈At V al(a). Single-valued information systems have been introduced by Pawlak in [17]. Information systems which are not single-valued are said to be non-deterministic in [16]. The components of A-system S will sometimes be written with subscripts: ObS , AtS , V alS (a) and fS . An example of an attribute is a = LAN GU AGE with V al(a) = {English, F rench, German, Russian}. If, for instance, f (x, a) = {English, Russian} then we say that x knows two languages — English and Russian — and does not know the remaining ones — French and German. If, for instance, f (x, a) = ∅ then we have a very precise information: x speaks no language in V al(a). For more examples, the reader is invited to consult [8]. For single-valued information systems, the information for each object x and each attribute a always contains exactly one value. This is a restrictive condition but it makes the system very simple: in the finite case this is a two-dimensional table which for each object x lists in a horizontal line the value f (x, a) for each attribute a. In the definition given by Pawlak, all components of an information system are supposed to be finite. Nevertheless, for mathematical reasons, we do not systematically assume this. In subsequent sections of this paper, we will be interested in single-valued systems based on a finite number n of attributes. In this case, we will say that n is the dimension of the information system.

2.2. Information relations Information systems contain explicit and implicit informations about objects. Explicit informations are given by the description of objects in terms of attributes whereas implicit informations are given by information relations derived from information systems. Let S be an A-system. We introduce the following binary relations between objects in S: • strong indiscernibility: x ≡S y iff (∀a ∈ AtS )(fS (x, a) = fS (y, a)), • weak indiscernibility: x 'S y iff (∃a ∈ AtS )(fS (x, a) = fS (y, a)), • strong forward inclusion: x ≤S y iff (∀a ∈ AtS )(fS (x, a) ⊆ fS (y, a)), • weak forward inclusion: x S y iff (∃a ∈ AtS )(fS (x, a) ⊆ fS (y, a)), • strong backward inclusion: x ≥S y iff (∀a ∈ AtS )(fS (x, a) ⊇ fS (y, a)), • weak backward inclusion: x S y iff (∃a ∈ AtS )(fS (x, a) ⊇ fS (y, a)), • strong positive similarity: xσS y iff (∀a ∈ AtS )(fS (x, a) ∩ f (S (y, a) 6= ∅) and • weak positive similarity: xΣS y iff (∃a ∈ AtS )(fS (x, a) ∩ f (S (y, a) 6= ∅). Each relation defined above is referred to as an information relation. Next lemma explains why there are not so much interesting information relations in the case of single-valued information systems. Lemma 2.1. Let S be a single-valued information system. Then • the strong relations ≤S , ≥S and σS collapse to the strong indiscernibility relation ≡S and • the weak relations S , S and ΣS collapse to the weak indiscernibility relation 'S .

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Proof: An easy routine check which is left to the reader.

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2.3. Information logics Information relations presented in subsection 2.2 are used as a semantic basis for modal logics called information logics. The language of such logics is an extension of the language of classical propositional logic with some modal operators corresponding to some given information relations. For instance the logic N IL from [16] is based on the modal operators [≤], [≥] and [σ] corresponding to the strong information relations ≤, ≥ and σ. The standard semantics of such systems are Kripke structures of the form (W, ≤, ≥, σ) for which there exists an information system S such that W = ObS , ≤=≤S , ≥=≥S and σ = σS . In order to axiomatize and prove the completeness of N IL with respect to its standard semantics, a special method is presented in [19] and consists in the following steps. First, we choose a number of first-order sentences for the relations ≤, ≥ and σ that are true in all information systems. In this way the concrete semantics of the language is replaced by an abstract semantics in the class of all structures (W, ≤, ≥, σ) satisfying the chosen first-order sentences. The structure (W, ≤, ≥, σ) is said to be standard if there is an information system S such that W = ObS , ≤=≤S , ≥=≥S and σ = σS . Second, we prove an abstract characterization theorem stating that each structure in the abstract semantics is isomorphic to some standard structure. This means that the sentences characterizing the relations ≤, ≥ and σ have to be chosen in such a way that the abstract characterization theorem can be proved. Finally, using methods from modal logic (canonical model, bounded morphism, filtration, etc), we axiomatize the logic over its abstract semantics and we prove its completeness as well as some other metalogical theorems concerning decidability and complexity issues. Such a procedure has been applied to a great variety of information logics and the interested reader is invited to consult [8] for a comprehensive bibliography (see also [24]). However, in the list of all such logics, there is no logic related to Pawlak’s original single-valued information systems. This situation is due to lemma 2.1 and to the nature of the information relations considered so far. Starting from the first paper by Orłowska and Pawlak [16], it is a tradition to consider only two kinds of information relations in information systems — the strong and the weak ones. The strong relations are of conjunctive form — “for all attributes” — while the weak relations are of disjunctive form — “for some attribute”. Note also that the information relations considered so far do not combine informations from different attributes like, for instance, the value f (x, a) of object x with respect to attribute a and the value f (y, b) of object y with respect to attribute b. Let us consider, for instance, a single-valued information system with attributes a — the color of eyes — and b — the color of hair. Then the “mixed” information relation Rab defined by • xRab y iff f (x, a) = f (y, b) gives a meaningful information, seeing that xRab y iff x’s eyes and y’s hair are of the same color. Such relations have a great expressive power. By means of them, we can define strong and weak indiscernibility relations for a given set of attributes. Our aim, in this paper, is to present information logics based on such information relations. In fact, such logics already exist — they are the arrow logics introduced in [21, 23, 25]. In the next section, we will shortly present n-dimensional arrow logic and we will show that it corresponds to the n-dimensional single-valued information systems.

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3. Arrow structures 3.1. Arrow logics By an arrow structure of dimension n, n-arrow structure for short, we mean any system of the form S = (Ar, P o, (n)) where: • Ar is a non-empty set of arrows, • P o is a non-empty set of points and • (n) = {1, . . . , n} is the set of all natural numbers between 1 and n. Each i ∈ (n) is considered as a total mapping i: Ar −→ P o called projection. For all x ∈ Ar and for all i ∈ (n), i(x) is called the i-th point of x. Without loss of generality, we assume that Ar ∩ P o = ∅ and that the following axiom is satisfied • (Ax) (∀α ∈ P o)(∃i ∈ (n))(∃x ∈ Ar)(i(x) = α). The meaning of (Ax) is that there is no isolated point. Although it is possible to develop a more general theory of arrows without this axiom, we assume it only because the formulations of some theorems become simpler. Hence, two-dimensional arrow structures are just directed multigraphs without isolated points and, in some sense, n-arrow structures can be considered as a generalization of directed multigraphs. The components of n-arrow structure S will be written with subscripts: ArS and P oS . We shall say that S is normal if it satisfies the following axiom: • (N or) (∀x, y ∈ ArS )((∀i ∈ (n))(i(x) = i(y)) → x = y). Let S be an n-arrow structure. The following binary relations will play an important role in the theory S be the binary relation between arrows in Ar defined by: of n-arrow structure. For all i, j ∈ (n), let Rij S y iff i(x) = j(y). • xRij S y means that the i-th point of x coincides with the j-th point of y. For n = 2, this can Intuitively, xRij be graphically represented as follows: S y: x ←−•−→ y, • xR11 S y: x ←−•←− y, • xR12 S y: x −→•−→ y and • xR21 S y: x −→•←− y. • xR22

Lemma 3.1. Let S be an n-arrow structure. For all x, y, z ∈ ArS and for all i, j, k ∈ (n): S x, • (ρi) xRii S y then yRS x and • (σij) if xRij ji S y and yRS z then xRS z. • (τ ijk) if xRij jk ik

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Moreover, S is normal iff: S y) → x = y). • (N or 0 ) (∀x, y ∈ ArS )((∀i ∈ (n))(xRii

Proof: S , i, j ∈ (n). The proof follows immediately from the definition of the binary relations Rij

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Let W be a nonempty set, n ≥ 2 be a fixed natural number and {Rij : i, j ∈ (n)} be a set of n2 binary relations on W . The relational system W = (W, {Rij : i, j ∈ (n)}) is called an n-dimensional arrow frame, n-arrow frame for short, if the conditions (ρi), (σij) and (τ ijk) from lemma 3.1 are satisfied for all i, j, k ∈ (n) and for all x, y, z ∈ W . If, moreover, W satisfies (N or 0 ) then we shall say that W is S , i, j ∈ (n), for some n-arrow structure S then W = W (S) is called normal. If W = ArS and Rij = Rij a standard n-arrow frame over S. Theorem 3.1. (Abstract characterization theorem) Let W = (W, {Rij : i, j ∈ (n)}) be an n-arrow frame. Then there exists an n-arrow structure S = S(W ) definable in a canonical way such that ArS = S = R . Moreover, if W is a normal n-arrow frame, then S is a normal W and for all i, j ∈ (n), Rij ij n-arrow structure. The truth of the matter is that Theorem 3.2. There is a one-one correspondence, up to isomorphism, between n-dimensional arrow structures and n-dimensional arrow frames. Theorem 3.1 says that each arrow frame is standard whereas theorem 3.2 says that the relations Rij , i, j ∈ (n), code the whole information of an arrow structure up to an isomorphism. More precisely, let S be a given arrow structure and let W (S) be the corresponding standard arrow frame over S. By theorem 3.1, let S(W (S)) be the canonical arrow structure corresponding to W (S) and W (S(W (S)) be the corresponding standard arrow frame over S(W (S)). Theorem 3.2 says that, on one hand, S and S(W (S)) are isomorphic and, on the other hand, W (S) and W (S(W (S))) are isomorphic. See [23] for details. The basic arrow logic of dimension n — BALn — is a propositional modal logic the modal operators of which are [ij], i, j ∈ (n). It is proved in [23] that this logic is complete with respect to the class of all n-dimensional arrow frames with respect to the class of all normal n-dimensional arrow frames. It is also proved there that BALn has the finite model property and is decidable in nondeterministic exponential time.

3.2. Single-valued information systems Now we will show that there is no essential difference between n-dimensional single-valued information systems and n-dimensional arrow structures. Let S = (Ob, At, {V al(a): a ∈ At}, f ) be an ndimensional single-valued information system. We shall assume that it satisfies the following natural condition saying that the system does not contain useless values of attributes: • (∀α ∈

S

a∈At

V al(a))(∃x ∈ Ob)(∃a ∈ At)(f (x, a) = α).

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S Let At = {a1 , . . . , an }, Ar = Ob and P o = a∈At V al(a). We define for all i ∈ (n) and for all x ∈ Ob, i(x) = f (x, ai ). Then, obviously, the obtained system (P o, Ar, (n)) is an n-dimensional arrow structure. So, n-dimensional single-valued information systems can be regarded also as n-dimensional arrow structures. Conversely, let (P o, Ar, (n)) be an n-dimensional arrow structure. Let Ob = Ar and At = (n) = {1, . . . , n}. We define for all i ∈ (n), V al(i) = {α ∈ P o: (∃x ∈ Ar)(i(x) = α}. For the information function, let f (x, i) = i(x). Obviously, the structure defined in this way is an ndimensional single-valued information system. In this system, the objects are the arrows, the attributes are just the natural numbers between 1 and n and the values of the attributes are points. From now on, we will identify n-dimensional single-valued information systems satisfying the condition mentioned above with n-dimensional arrow structures. Now, the relations Rij will be our basic information relations. The intuitive meaning of xRij y is that the value of x with respect to the attribute i is equal to the value of y with respect to the attribute j. In real information systems, two different attributes can have common values and, consequently, such mixed information relations are meaningful and, in fact, widely used. We have mentioned above an example of such a relation: “the color of x’s eyes is equal to the color of y’s hair”. Let A = {i1 , . . . , im } be a nonempty set of attributes. By means of the relations Rii , we may define the strong and the weak indiscernibility relations corresponding to A as follows: • strong indiscernibility: x ≡A y iff for all i ∈ A, xRii y and • weak indiscernibility: x 'A y iff there exists i ∈ A such that xRii y. Since 'A is the union of the binary relations Rii , i ∈ A, then the modality ['A ] corresponding to 'A is definable by the modalities [ii] corresponding to Rii as follows: V • ['A ]p =def i∈A [ii]p. Note, however, that the modality [≡A ] is not definable by the modalities [ii], seeing that ≡A is the intersection of the binary relations Rii , i ∈ A. Hence, its axiomatization presents difficulties. Namely, the rest of the paper is devoted to a solution of these difficulties. It presents a new system of arrow logic which extends basic arrow logic BALn with modalities corresponding to arbitrary intersections of the basic Rij relations, i, j ∈ (n). We also address decidability and complexity issues related to the obtained logical system.

4. Basic concepts 4.1. Syntax Let n be a positive integer such that n ≥ 2 and Φ0 be a countable set of propositional letters, with typical members denoted p, q, etc. We inductively define the set F or(n, Φ0 ) of all formulas over (n, Φ0 ), with typical members denoted φ, ψ, etc, by • φ := p | ⊥ | ¬φ | (φ ∨ ψ) | [α]φ | α! where α ranges over the subsets of (n) × (n). For all subsets α of (n) × (n), the modal operator [α] will correspond to the intersection of the binary relations Rij , (i, j) ∈ α. Hence, [α]φ will be true exactly at those arrows x such that for all arrows y, if xRij y, (i, j) ∈ α, then φ is true at y. In other respects, for all subsets α of (n) × (n), the modal constant α! will be true exactly at those arrows where the binary

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relations Rij , (i, j) ∈ α, are reflexive. See subsection 4.2 for details. Let us adopt the standard rules for omission of the parentheses. We define the other constructs as usual. In particular for all subsets α of (n) × (n), • hαiφ := ¬[α]¬φ. We also define in addition • id := {(i, i): i ∈ (n)}, • for all subsets α of (n) × (n), α−1 := {(j, i): i, j ∈ (n) and (i, j) ∈ α} and • for all subsets α, β of (n) × (n), α; β := {(i, k): i, j, k ∈ (n), (i, j) ∈ α and (j, k) ∈ β}. Lemma 4.1. Let α be a subset of (n) × (n). There exists subsets β, γ of (n) × (n) such that • α ⊆ β; γ, • β ⊆ α; γ −1 and • γ ⊆ β −1 ; α. Proof: Let β = {(i, j): i, j, k ∈ (n) and (i, k) ∈ α} and γ = {(j, k): i, j, k ∈ (n) and (i, k) ∈ α}. The reader u t may easily verify that α ⊆ β; γ, β ⊆ α; γ −1 and γ ⊆ β −1 ; α. For all formulas φ, let k φ k be the length of φ as a string. Let there be given a set Γ of formulas. For all subsets α of (n) × (n), let [α]Γ = {φ: [α]φ is in Γ}. cln (Γ) will denote the least set of formulas such that • Γ ⊆ cln (Γ), • cln (Γ) is closed under subformulas, • cln (Γ) is closed under ¬, • for all subsets α, β of (n) × (n) and for all formulas φ, [α]φ is in cln (Γ) iff [β]φ is in cln (Γ) and • for all subsets α, β of (n) × (n), [α]¬β! is in cln (Γ) whereas clnnor (Γ) will denote the least set of formulas such that • cln (Γ) ⊆ clnnor (Γ) and • clnnor (Γ) is closed under [id]. ˆ will It is important to note that cln (Γ) and clnnor (Γ) are infinite sets of formulas. If Γ is finite then Γ denote the conjunction φ1 ∧ . . . ∧ φN of the formulas φ1 , . . ., φN in Γ.

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4.2. Semantics A Kripke model is a triple • M = (W, {Rij : i, j ∈ (n)}, V ) where (W, {Rij : i, j ∈ (n)}) is an n-dimensional arrow frame and V is a function assigning to each proposition letter p a subset V (p) of W . Given a Kripke model M = (W, {Rij : i, j ∈ (n)}, V ) and a subset α of (n) × (n), let R(α) be the binary relation on W such that for all arrows x, y in W , xR(α)y iff for all i, j ∈ (n), if (i, j) ∈ α then xRij y. Hence, it follows immediately that for all x, y, z ∈ W and for all subsets α, β of (n) × (n), • xR(id)x, • if xR(α)y then yR(α−1 )x and • if xR(α)y and yR(β)z then xR(α; β)z. Moreover, xR(α ∪ β)y iff xR(α)y and xR(β)y. Let us remark that a Kripke model M = (W, {Rij : i, j ∈ (n)}, V ) is normal iff for all arrows x, y in W , if xR(id)y then x = y. Lemma 4.2. Let M = (W, {Rij : i, j ∈ (n)}, V ) be a Kripke model, x, y be arrows in W and α be a subset of (n) × (n). xR(α)y iff for all subsets β, γ of (n) × (n), • if β ⊆ α; γ −1 then for all arrows z in W , if zR(γ)y then xR(β)z and • if γ ⊆ β −1 ; α then for all arrows z in W , if xR(β)z then zR(γ)y. Proof: The proof is left to the reader.

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We inductively define the notion of a formula φ being true in a Kripke model M = (W, {Rij : i, j ∈ (n)}, V ) at a arrow x in W , in symbols M, x |= φ, by • M, x |= p iff x is in V (p), • M, x 6|= ⊥, • M, x |= ¬φ iff M, x 6|= φ, • M, x |= φ ∨ ψ iff M, x |= φ or M, x |= ψ, • M, x |= [α]φ iff for all arrows y in W , if xR(α)y then M, y |= φ and • M, x |= α! iff xR(α)x. A formula φ is said to be valid in a Kripke model M = (W, {Rij : i, j ∈ (n)}, V ), in symbols M |= φ, if φ is true at all arrows in W . We shall say that a formula φ is valid, in symbols |= φ, if φ is valid in all Kripke models. A formula φ is said to be normal-valid, in symbols |=nor φ, if φ is valid in all normal Kripke models.

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Lemma 4.3. For all subsets α, β of (n) × (n) and for all formulas φ, • |= [id]φ → φ, • |= φ → [α]hα−1 iφ, • |= [α; β]φ → [α][β]φ, • |= hα ∪ βiφ → hαiφ ∧ hβiφ, • |= id!, • |= α! → α−1 !, • |= α! ∧ β! → (α; β)!, • |= (α ∪ β)! ↔ α! ∧ β!, • |= α! → ([α]φ → φ) and • |= hαiβ! → (α; β; α−1 )!. Proof: The proof is left to the reader.

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Lemma 4.4. For all formulas φ, • |=nor hidiφ → φ. Proof: The proof is left to the reader.

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We shall say that a set Γ of formulas is satisfiable in a Kripke model M = (W, {Rij : i, j ∈ (n)}, V ), in symbols sat(M, Γ), if all formulas in Γ are true at some arrow in W . A set Γ of formulas is said to be satisfiable, in symbols sat(Γ), if Γ is satisfiable in some Kripke model. We shall say that a set Γ of formulas is normal-satisfiable, in symbols satnor (Γ), if Γ is satisfiable in some normal Kripke model. Lemma 4.5. For all finite set Γ of formulas, ˆ and • sat(Γ) iff 6|= ¬Γ ˆ • satnor (Γ) iff 6|=nor ¬Γ. Proof: The proof is left to the reader.

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5. Axiomatization and completeness 5.1. Axiomatization Let Ln be the least normal logic in our language that contains the formulas • [id]p → p, • p → [α]hα−1 ip, • [α; β]p → [α][β]p, • hα ∪ βip → hαip ∧ hβip, • id!, • α! → α−1 !, • α! ∧ β! → (α; β)!, • (α ∪ β)! ↔ α! ∧ β!, • α! → ([α]p → p) and • hαiβ! → (α; β; α−1 )! and Lnor n be the least normal logic in our language that contains Ln and the formula • hidip → p. Consider a normal logic L in our language containing Ln . A formula φ is said to be L-deducible from a set Γ of formulas, in symbols Γ `L φ, if there exists formulas φ1 , . . ., φN in Γ such that φ1 ∧. . .∧φN → φ is in L. We use the notation `L φ when Γ = ∅. Proposition 5.1. (Soundness of Ln ) Let φ be a formula such that `Ln φ. |= φ. Proof: By lemma 4.3.

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Proposition 5.2. (Soundness of Lnor φ. |=nor φ. n ) Let φ be a formula such that `Lnor n Proof: By lemmas 4.3 and 4.4.

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5.2. Completeness Now we present the proof of the completeness of Ln and the proof of the completeness of Lnor n . These proofs use the techniques of the subordination model.

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5.2.1.

Subordination model

Consider a normal logic L in our language containing Ln . We shall say that a set Γ of formulas is Lconsistent if Γ 6`L ⊥. A set Γ of formulas is said to be maximal if for all formulas φ, φ is in Γ or ¬φ is in Γ. Lemma 5.1. (Lindenbaum’s lemma) Let Γ be a L-consistent set of formulas. There exists a maximal L-consistent set Γ0 of formulas such that Γ ⊆ Γ0 . Proof: The proof uses general techniques that can be found in most elementary texts. See [4], [5] or [10].

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Lemma 5.2. (Diamond lemma) Let Γ be a maximal L-consistent set of formulas, α be a subset of (n) × (n) and φ be a formula such that [α]φ is not in Γ. There exists a maximal L-consistent set ∆ of formulas such that [α]Γ ⊆ ∆ and φ is not in ∆. Proof: Let ∆ = [α]Γ ∪ {¬φ}. The reader may easily verify that ∆ is L-consistent. By lemma 5.1 there exists a maximal L-consistent set ∆0 of formulas such that ∆ ⊆ ∆0 . Since [α]Γ ⊆ ∆, then [α]Γ ⊆ ∆0 . Since u t ¬φ is in ∆, then φ is not in ∆0 . A L-subordination model is a triple • M = (W, R, S) where W is a non-empty set of arrows, R is a function assigning to each subset α of (n) × (n) a binary relation R(α) on W such that • for all arrows x in W , xR(id)x, • for all subsets α of (n) × (n) and for all arrows x, y in W , if xR(α)y then yR(α−1 )x, • for all subsets α, β of (n) × (n) and for all arrows x, y, z in W , if xR(α)y and yR(β)z then xR(α; β)z, • R(∅) = W × W and • for all subsets α, β of (n) × (n), R(α ∪ β) = R(α) ∩ R(β) and S is a function assigning to each arrow in W a maximal L-consistent set of formulas such that • for all subsets α of (n) × (n), for all formulas φ and for all arrows x in W , if [α]φ is in S(x) then for all arrows y in W , if xR(α)y then φ is in S(y) and • for all subsets α of (n) × (n) and for all arrows x in W , α! is in S(x) iff xR(α)x. We shall say that a L-subordination model M = (W, R, S) is perfect if • for all subsets α of (n) × (n), for all formulas φ and for all arrows x in W , if [α]φ is not in S(x) then there exists a arrow y in W such that xR(α)y and φ is not in S(y).

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13

Lemma 5.3. Let Γ be a maximal L-consistent set of formulas. The structure • MΓ = (WΓ , RΓ , SΓ ) where • WΓ = {0}, • RΓ is the function assigning to each subset α of (n) × (n) a binary relation RΓ (α) on WΓ such that – 0RΓ (α)0 iff α! is in Γ and • SΓ is the function assigning to each arrow in WΓ a maximal L-consistent set of formulas such that – SΓ (0) = Γ is a finite L-subordination model. MΓ is said to be the initial model of Γ. Proof: The proof is left to the reader.

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Lemma 5.4. Let M = (W, R, S) be a finite L-subordination model, α be a subset of (n) × (n), φ be a formula and x be a arrow in W such that [α]φ is not in S(x). Let ] be a positive integer such that ] is not in W and Γ be a maximal L-consistent set of formulas such that [α]S(x) ⊆ Γ and φ is not in Γ. The structure • M0 = (W 0 , R0 , S 0 ) where • W 0 = W ∪ {]}, • R0 is the function assigning to each subset β of (n) × (n) a binary relation R0 (β) on W 0 such that – for all arrows y, z in W , yR0 (β)z iff yR(β)z, – for all arrows y in W , yR0 (β)] iff there exists subsets β 0 , β 00 of (n) × (n) such that yR(β 0 )x, β 00 ! is in Γ and β ⊆ β 0 ; α; β 00 , – for all arrows z in W , ]R0 (β)z iff there exists subsets β 0 , β 00 of (n) × (n) such that β 0 ! is in Γ, xR(β 00 )z and β ⊆ β 0 ; α−1 ; β 00 and – ]R0 (β)] iff β! is in Γ and • S 0 is the function assigning to each arrow in W 0 a maximal L-consistent set of formulas such that – for all arrows y in W , S 0 (y) = S(y) and – S 0 (]) = Γ is a finite L-subordination model. We shall say that M0 is a local completion of M with respect to (α, φ, x). Proof: The proof is left to the reader.

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5.2.2.

Completeness of Ln

Theorem 5.1. (Completeness of Ln ) Let Γ be a Ln -consistent set of formulas. sat(Γ). Proof: By lemma 5.1 there exists a maximal Ln -consistent set Γ0 of formulas such that Γ ⊆ Γ0 . Let (α1 , φ1 , x1 ), (α2 , φ2 , x2 ), . . . be a list of 2(n)×(n) ×F or(n, Φ0 )×IN such that each item appears infinitely often. We inductively define the sequence M1 = (W1 , R1 , S1 ), M2 = (W2 , R2 , S2 ), . . . of finite Ln -subordination models by • M1 is, by lemma 5.3, the initial model of Γ0 and • for all positive integers N , if xN is a arrow in WN such that [αN ]φN is not in SN (xN ) then MN +1 is, by lemma 5.4, a local completion of MN with respect to (αN , φN , xN ) else MN +1 is MN . The reader may easily verify that the structure • M = (W, R, S) where • W = W1 ∪ W2 ∪ . . ., • R is the function assigning to each subset α of (n) × (n) a binary relation R(α) on W such that – for all arrows x, y in W , xR(α)y iff there exists a positive integer N such that x is in WN , y is in WN and xRN (α)y and • S is the function assigning to each arrow in W a maximal Ln -consistent set of formulas such that – for all arrows x in W , if there exists a positive integer N such that x is in WN then S(x) = SN (x) is a perfect Ln -subordination model. The reader may easily verify that the structure 0 : i, j ∈ (n)}, V 0 ) • M0 = (W 0 , {Rij

where • W0 = W, 0 is the binary relation on W 0 such that • for all i, j ∈ (n), Rij 0 y iff xR({(i, j)})y – for all arrows x, y in W 0 , xRij

and • V 0 is the function assigning to each proposition letter p a subset V 0 (p) of W 0 such that – for all arrows x in W 0 , x is in V 0 (p) iff p is in S(x) is a Kripke model such that for all arrows x in W 0 and for all formulas φ, M0 , x |= φ iff φ is in S(x). Since Γ ⊆ Γ0 and S(0) = Γ0 , then sat(Γ). u t

P. Balbiani and D. Vakarelov / Arrow logic with arbitrary intersections

5.2.3.

15

Completeness of Lnor n

nor Theorem 5.2. (Completeness of Lnor n ) Let Γ be a Ln -consistent set of formulas. satnor (Γ).

Proof: 0 0 By lemma 5.1 there exists a maximal Lnor n -consistent set Γ of formulas such that Γ ⊆ Γ . Let (α1 , φ1 , x1 ), (n)×(n) (α2 , φ2 , x2 ), . . . be a list of 2 × F or(n, Φ0 ) × IN such that each item appears infinitely often. We inductively define the sequence M1 = (W1 , R1 , S1 ), M2 = (W2 , R2 , S2 ), . . . of finite Lnor n subordination models by • M1 is, by lemma 5.3, the initial model of Γ0 and • for all positive integers N , if xN is a arrow in WN such that [αN ]φN is not in SN (xN ) then MN +1 is, by lemma 5.4, a local completion of MN with respect to (αN , φN , xN ) else MN +1 is MN . The reader may easily verify that the structure • M = (W, R, S) where • W = W1 ∪ W2 ∪ . . ., • R is the function assigning to each subset α of (n) × (n) a binary relation R(α) on W such that – for all arrows x, y in W , xR(α)y iff there exists a positive integer N such that x is in WN , y is in WN and xRN (α)y and • S is the function assigning to each arrow in W a maximal Lnor n -consistent set of formulas such that – for all arrows x in W , if there exists a positive integer N such that x is in WN then S(x) = SN (x) is a perfect Lnor n -subordination model. Let us remark that R(id) is an equivalence relation on W . In the sequel for all arrows x in W , the set of all arrows in W equivalent to x modulo R(id) will be denoted | x | and the quotient set of W modulo R(id) will be denoted W|R(id) . The reader may easily verify that the structure 0 : i, j ∈ (n)}, V 0 ) • M0 = (W 0 , {Rij

where • W 0 = W|R(id) , 0 is the binary relation on W 0 such that • for all i, j ∈ (n), Rij 0 | y | iff xR({(i, j)})y – for all arrows | x |, | y | in W 0 , | x | Rij

and

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• V 0 is the function assigning to each proposition letter p a subset V 0 (p) of W 0 such that – for all arrows | x | in W 0 , | x | is in V 0 (p) iff p is in S(x) is a normal Kripke model such that for all arrows | x | in W 0 and for all formulas φ, M0 , | x ||= φ iff φ is in S(x). Since Γ ⊆ Γ0 and S(0) = Γ0 , then satnor (Γ). u t

6. Decidability and complexity 6.1. Decidability Now we present the proof of the decidability of Ln and the proof of the decidability of Lnor n . These proofs use the techniques of the filtration. 6.1.1.

Non-standard models

A non-standard model is a quadruple • M = (W, R, D, V ) where • W is a set of arrows, • for all subsets α of (n) × (n), R(α) is a binary relation on W and D(α) is a subset of W such that – for all arrows x in W , xR(id)x, – for all subsets α of (n) × (n) and for all arrows x, y in W , if xR(α)y then yR(α−1 )x, – for all subsets α, β of (n) × (n) and for all arrows x, y, z in W , if xR(α)y and yR(β)z then xR(α; β)z, – for all subsets α, β of (n) × (n) and for all arrows x, y in W , if xR(α ∪ β)y then xR(α)y and xR(β)y, – for all arrows x in W , x is in D(id), – for all subsets α of (n) × (n) and for all arrows x in W , if x is in D(α) then x is in D(α−1 ), – for all subsets α, β of (n) × (n) and for all arrows x in W , if x is in D(α) and x is in D(β) then x is in D(α; β), – for all subsets α, β of (n) × (n) and for all arrows x in W , x is in D(α ∪ β) iff x is in D(α) and x is in D(β), – for all subsets α of (n) × (n) and for all arrows x in W , if x is in D(α) then xR(α)x and – for all subsets α, β of (n) × (n) and for all arrows x, y in W , if xR(α)y and y is in D(β) then x is in D(α; β; α−1 ) and • V is a function assigning to each proposition letter p a subset V (p) of W . A non-standard model M = (W, R, D, V ) is said to be normal if for all arrows x, y in W , if xR(id)y then x = y. We inductively define the notion of a formula φ being true in a non-standard model

P. Balbiani and D. Vakarelov / Arrow logic with arbitrary intersections

17

M = (W, R, D, V ) at a arrow x in W , in symbols M, x |= φ, by • M, x |= p iff x is in V (p), • M, x 6|= ⊥, • M, x |= ¬φ iff M, x 6|= φ, • M, x |= φ ∨ ψ iff M, x |= φ or M, x |= ψ, • M, x |= [α]φ iff for all arrows y in W , if xR(α)y then M, y |= φ and • M, x |= α! iff x is in D(α). We shall say that a formula φ is valid in a non-standard model M = (W, R, D, V ), in symbols M |= φ, if φ is true at all arrows in W . A formula φ is said to be ns-valid, in symbols |=ns φ, if φ is valid in all non-standard models. We shall say that a formula φ is normal-ns-valid, in symbols |=ns nor φ, if φ is valid in all normal non-standard models. A set Γ of formulas is said to be satisfiable in a non-standard model M = (W, R, D, V ), in symbols sat(M, Γ), if all formulas in Γ are true at some arrow in W . We shall say that a set Γ of formulas is ns-satisfiable, in symbols satns (Γ), if Γ is satisfiable in some non-standard model. A set Γ of formulas is said to be normal-ns-satisfiable, in symbols satns nor (Γ), if Γ is satisfiable in some normal non-standard model. Proposition 6.1. (Non-standard soundness of Ln ) Let φ be a formula such that `Ln φ. |=ns φ. Proof: The proof is left to the reader.

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Proposition 6.2. (Non-standard soundness of Lnor φ. |=ns n ) Let φ be a formula such that `Lnor nor φ. n Proof: The proof is left to the reader. 6.1.2.

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Completeness with respect to non-standard models

Theorem 6.1. (Non-standard completeness of Ln ) Let Γ be a Ln -consistent set of formulas. satns (Γ). Proof: By lemma 5.1 there exists a maximal Ln -consistent set Γ0 of formulas such that Γ ⊆ Γ0 . The reader may easily verify that the structure • M = (W, R, D, V ) where • W is the set of all maximal Ln -consistent sets of formulas, • for all subsets α of (n) × (n), R(α) is the binary relation on W such that for all arrows x, y in W , xR(α)y iff [α]x ⊆ y, • for all subsets α of (n) × (n), D(α) is the subset of W such that for all arrows x in W , x is in D(α) iff α! is in x and

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• V is the function assigning to each proposition letter p a subset V (p) of W such that for all arrows x in W , x is in V (p) iff p is in x is a non-standard model such that for all arrows x in W and for all formulas φ, M, x |= φ iff φ is in x. Since Γ ⊆ Γ0 , then satns (Γ). u t nor Theorem 6.2. (Non-standard completeness of Lnor n ) Let Γ be a Ln -consistent set of formulas. ns satnor (Γ).

Proof: The proof is similar to the proof of theorem 6.1. 6.1.3.

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Finite model property

Proposition 6.3. Ln has the finite model property with respect to the class of all non-standard models. Proof: Let Γ be a finite Ln -consistent set of formulas. By theorem 6.1 there exists a non-standard model M = (W, R, D, V ) and a arrow x in W such that for all formulas φ in Γ, M, x |= φ. Let ≡Γ be the equivalence relation on W such that • for all arrows x, y in W , x ≡Γ y iff for all formulas φ in cln (Γ), M, x |= φ iff M, y |= φ. In the sequel for all arrows x in W , the set of all arrows in W equivalent to x modulo ≡Γ , i.e. the equivalence class of x modulo ≡Γ , will be denoted | x | and the quotient set of W modulo ≡Γ will be denoted W|≡Γ . Although cln (Γ) is an infinite set of formulas, W|≡Γ is a finite set of equivalence classes modulo ≡Γ . Moreover Card(W|≡Γ ) = O(2kφk ). The reader may easily verify that the structure • M0 = (W 0 , R0 , D0 , V 0 ) where • W 0 = W|≡Γ , • for all subsets α of (n) × (n), R0 (α) is the binary relation on W 0 such that – for all arrows | x |, | y | in W 0 , | x | R0 (α) | y | iff for all subsets β, γ of (n) × (n) and for all formulas [δ]φ in cln (Γ), if β ⊆ α; γ −1 and M, x |= [β]φ then M, y |= [γ −1 ]φ and if γ ⊆ β −1 ; α and M, y |= [γ −1 ]φ then M, x |= [β]φ, • for all subsets α of (n) × (n), D 0 (α) is the subset of W 0 such that – for all arrows | x | in W 0 , | x | is in D 0 (α) iff M, x |= α! and • V 0 is the function assigning to each proposition letter p a subset V 0 (p) of W 0 such that – for all arrows | x | in W 0 , | x | is in V 0 (p) iff p is in cln (Γ) and x is in V (p)

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19

is a finite non-standard model such that for all arrows | x | in W 0 and for all formulas φ in cln (Γ), M0 , | x ||= φ iff M, x |= φ. u t Proposition 6.4. Lnor n has the finite model property with respect to the class of all normal non-standard models. Proof: The proof is similar to the proof of proposition 6.3.

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Theorem 6.3. Ln is decidable. Proof: By proposition 6.1, theorem 6.1 and proposition 6.3.

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Theorem 6.4. Lnor n is decidable. Proof: By proposition 6.2, theorem 6.2 and proposition 6.4.

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6.2. Complexity It is immediate from the previous section that membership in {Γ: Γ is a finite set of formulas such that sat(Γ)} is in N EXP T IM E and membership in {Γ: Γ is a finite set of formulas such that satnor (Γ)} is in N EXP T IM E. Now we present the proof that membership in {Γ: Γ is a finite set of formulas such that sat(Γ)} is in EXP T IM E and membership in {Γ: Γ is a finite set of formulas such that satnor (Γ)} is in EXP T IM E. These proofs use the techniques of the collapsing models. 6.2.1.

Complexity of formulas in Ln

We prove an exponential-time upper bound for deciding the satisfiability of formulas in Ln . Let there be given a finite set Γ of formulas. An atom for Γ is a maximal Boolean-consistent subset x of cln (Γ) such that [id]x ⊆ x and • for all subsets α, β, γ of (n) × (n) and for all formulas φ, if [α; γ]φ is in x or [β; γ]φ is in x then [(α ∪ β); γ]φ is in x, • id! is in x, • for all subsets α of (n) × (n), if α! is in x then α−1 ! is in x, • for all subsets α, β of (n) × (n), if α! is in x and β! is in x then (α; β)! is in x, • for all subsets α, β of (n) × (n), (α ∪ β)! is in x iff α! is in x and β! is in x, • for all subsets α of (n) × (n), if α! is in x then for all subsets β of (n) × (n) and for all formulas φ, if [α; β]φ is in x then [β]φ is in x and • for all subsets α, β of (n) × (n), if hαiβ! is in x then (α; β; α−1 )! is in x. Denote the set of atoms for Γ by At(Γ). Let us remark that the size of At(Γ) is at most exponential in Σ{k φ k: φ is a formula in Γ}.

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Lemma 6.1. The structure • MΓ = (WΓ , RΓ , DΓ , VΓ ) where • WΓ = At(Γ), • for all subsets α of (n) × (n), RΓ (α) is the binary relation on WΓ such that – for all atoms x, y for Γ, xRΓ (α)y iff for all subsets β, γ of (n) × (n) and for all formulas φ, if β ⊆ α; γ −1 and [β]φ is in x then [γ −1 ]φ is in y and if γ ⊆ β −1 ; α and [γ −1 ]φ is in y then [β]φ is in x, • for all subsets α of (n) × (n), DΓ (α) is the subset of WΓ such that – for all atoms x for Γ, x is in DΓ (α) iff α! is in x and • VΓ is the function assigning to each proposition letter p a subset VΓ (p) of WΓ such that – for all atoms x for Γ, x is in VΓ (p) iff p is in x is a non-standard model. MΓ is said to be the initial model of Γ. Proof: The proof is left to the reader.

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From now on we assume that there exists Ln -consistent atoms for Γ. Lemma 6.2. Let M = (W, R, D, V ) be a submodel of the initial model of Γ. A arrow x in W is said to be bad if there exists a subset α of (n) × (n) and a formula φ such that ¬[α]φ is in x and for all arrows y in W , if xR(α)y then φ is in y. The structure • M0 = (W 0 , R0 , D0 , V 0 ) where • W 0 = W \ {x: x is a bad arrow in W }, • for all subsets α of (n) × (n), RΓ0 (α) is the binary relation on W 0 such that – for all arrows x, y in W 0 , xR0 (α)y iff xR(α)y, • for all subsets α of (n) × (n), DΓ0 (α) is the subset of W 0 such that – for all arrows x in W 0 , x is in D 0 (α) iff x is in D(α) and • V 0 is the function assigning to each proposition letter p a subset V 0 (p) of W 0 such that – for all arrows x in W 0 , x is in V 0 (p) iff x is in V (p) is a non-standard model. We shall say that M0 is the restriction of M with respect to Γ.

P. Balbiani and D. Vakarelov / Arrow logic with arbitrary intersections

Proof: The proof is left to the reader.

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We inductively define the sequence M1 = (W1 , R1 , D1 , V1 ), M2 = (W2 , R2 , D2 , V2 ), . . . of nonstandard models by • M1 is, by lemma 6.1, the initial model of Γ and • for all positive integers N , MN +1 is, by lemma 6.2, the restriction of MN with respect to Γ. It follows from the finiteness of At(Γ) that there is a positive integer N0 for which the construction closes up, i.e. for all positive integers N , if N ≥ N0 then MN = MN0 . Lemma 6.3. (Diamond lemma) Let x be a arrow in WN0 . For all subsets α of (n) × (n) and for all formulas φ, if ¬[α]φ is in x then there exists a arrow y in W such that xR(α)y and φ is not in y. Proof: The proof is left to the reader.

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Lemma 6.4. (Truth lemma) Let x be a arrow in WN0 . For all formulas φ in cln (Γ), φ is in x iff MN0 , x |= φ. Proof: The proof is by induction on the structure of φ.

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Lemma 6.5. Let x be an atom for Γ. For all positive integers N , • if x is not in WN then `Ln ¬ˆ x and

W W • for all formulas φ in cln (Γ), `Ln x ˆ → [α]( {ˆ y : y is a arrow in WN such that φ is in y} ∨ {ˆ y: y is a arrow in WN such that xRN (α)y and φ is not in y}). Proof: The proof is by induction on N .

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Proposition 6.5. sat(Γ) iff there exists a arrow x in WN0 such that Γ ⊆ x. Proof: Suppose that there exists a arrow x in WN0 such that Γ ⊆ x. By lemma 6.4 satns (Γ). By theorem 5.1 and proposition 6.1 sat(Γ). ˆ Remark that `Ln Γ ↔ W{ˆ Suppose that sat(Γ). By proposition 5.1 6`Ln ¬Γ. x: x is an atom for Γ such that Γ ⊆ x}. Hence there exists an atom x for Γ such that Γ ⊆ x and 6`Ln ¬ˆ x. By lemma 6.5 x is a arrow in WN0 . u t Theorem 6.5. Membership in {Γ: Γ is a finite set of a formulas such that sat(Γ)} is in EXP T IM E. Proof: It suffices to remark that M1 can be constructed in time an exponential in Σ{k φ k: φ is a formula in Γ} and for all positive integer N , MN +1 can be constructed in time a polynomial in the size of WN . u t Let us remark that membership in {Γ: Γ is a finite set of a formulas such that sat(Γ)} is not known to be EXP T IM E-hard.

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6.2.2.

Complexity of formulas in Lnor n

Lnor n , as it turns out, is very similar to Ln with respect to complexity. Let there be given a finite set Γ of formulas. A normal atom for Γ is a maximal Boolean-consistent subset x of clnnor (Γ) such that [id]x ⊆ x, • for all subsets α, β, γ of (n) × (n) and for all formulas φ, if [α; γ]φ is in x or [β; γ]φ is in x then [(α ∪ β); γ]φ is in x, • id! is in x, • for all subsets α of (n) × (n), if α! is in x then α−1 ! is in x, • for all subsets α, β of (n) × (n), if α! is in x and β! is in x then (α; β)! is in x, • for all subsets α, β of (n) × (n), (α ∪ β)! is in x iff α! is in x and β! is in x, • for all subsets α of (n) × (n), if α! is in x then for all subsets β of (n) × (n) and for all formulas φ, if [α; β]φ is in x then [β]φ is in x, • for all subsets α, β of (n) × (n), if hαiβ! is in x then (α; β; α−1 )! is in x and • x is closed under [id]. Denote the set of normal atoms for Γ by Atnor (Γ). Let us remark that the size of Atnor (Γ) is at most exponential in Σ{k φ k: φ is a formula in Γ}. Lemma 6.6. The structure • MΓ = (WΓ , RΓ , DΓ , VΓ ) where • WΓ = Atnor (Γ), • for all subsets α of (n) × (n), RΓ (α) is the binary relation on WΓ such that – for all atoms x, y for Γ, xRΓ (α)y iff for all subsets β, γ of (n) × (n) and for all formulas φ, if β ⊆ α; γ −1 and [β]φ is in x then [γ −1 ]φ is in y and if γ ⊆ β −1 ; α and [γ −1 ]φ is in y then [β]φ is in x, • for all subsets α of (n) × (n), DΓ (α) is the subset of WΓ such that – for all atoms x for Γ, x is in DΓ (α) iff α! is in x and • VΓ is the function assigning to each proposition letter p a subset VΓ (p) of WΓ such that – for all atoms x for Γ, x is in VΓ (p) iff p is in x is a normal non-standard model.

P. Balbiani and D. Vakarelov / Arrow logic with arbitrary intersections

Proof: The proof is left to the reader.

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u t

From now on we assume that there exists Lnor n -consistent atoms for Γ. Following the line of reasoning suggested in the proofs of lemmas 6.2 to 6.5 and proposition 6.5 the reader may easily obtain the Theorem 6.6. Membership in {Γ: Γ is a finite set of a formulas such that satnor (Γ)} is in EXP T IM E. Let us remark that membership in {Γ: Γ is a finite set of a formulas such that satnor (Γ)} is not known to be EXP T IM E-hard.

7. Conclusion In this paper we have introduced two extensions of the basic arrow logic of dimension n: Ln and Lnor n . Both of them can be reinterpreted as information logics for single-valued information systems with n attributes. Hence the modalities of our propositional language can be considered as modal operators corresponding to information relations in single-valued information systems. The modality [≡], corresponding to the information relation of strong indiscernibility, is definable in our propositional language. New modalities of “mixed type”, concerning different attributes and not considered previously in information logics, are also definable in our propositional language. We have completely axiomatized Ln and Lnor n and have established the complexity of the membership problems in these logics. Is it possible to generalize the results of this paper for non-deterministic Pawlak’s information systems? A possible answer to this question could be the hyper arrow logic of dimension n introduced in [25]. This logic corresponds to hyper arrow structures of dimension n which differ from arrow structures in the following way. Their projection functions i, i = 1, . . . , n, are now assumed to be functions assigning to each arrow x a (possibly empty) set i(x) of points. Hyper arrow structures are just nondeterministic Pawlak’s information systems with exactly n attributes. The basic modalities of hyper arrow logic are based on the following relations between arrows: • x ≤ij y iff i(x) ⊆ j(y), • xΣij y iff i(x) ∩ j(y) 6= ∅ and • xN y iff i(x) ∪ j(y) 6= P o. By means of such binary relations we can define the usual strong information relations in the following way: • strong inclusion: x ≤ y iff for all i ∈ (n), x ≤ii y, • strong positive similarity: xσy iff for all i ∈ (n), xΣii y, • strong negative similarity: xνy iff for all i ∈ (n), xNii y and • strong indiscernibility: x ≡ y iff for all i ∈ (n), x ≤ii y and y ≤ii x. Weak versions of these information relations, denoted respectively by , Σ, N and ' in [25], are defined analogously. The complete axiomatization and the decidability of the logic based on arbitrary intersections of the basic relations ≤ ij, Σij and Nij , however, is open.

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Acknowledgement Special acknowledgement is heartly granted to the anonymous referee who made several helpful comments for improving the readability of an earlier version of the paper. Our research has been partly supported by the COST action “Theory and Applications of Relational Structures as Knowledge Instruments” and the Bulgarian Ministry of Science and Education (contract NIP-1510).

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