Elementary Unification in Modal Logic KD45

Introduction

Modal logics like S5 or KD45 are essential to the design of logical systems that capture elements of reasoning about knowledge [13,21]. There exists variants of these logics with one or several agents, with or without common knowledge, etc. As in any modal logic, the questions addressed in their setting usually concern their axiomatizability and their decidability. Another desirable question which one should address whenever possible concerns the unifiability of formulas. A formulaϕ(x 1 , . . . , x n ) is unifiable in a modal logic L iff there exists formulas ψ 1 , . . . , ψ n such that ϕ(ψ 1 , . . . , ψ n ) is in L. See [1,11,14,15] for details.

Special acknowledgement is heartly granted to the referees for the feedback we have obtained from them. Their comments and suggestions have greatly helped us to improve the correctness and the readability of our paper. Philippe Balbiani and Tinko Tinchev were partially supported by the programme RILA (contracts 34269VB and DRILA01/2/2015).

Results about the unifiability problem have been already obtained in many modal logics. Rybakov [22,23] demonstrated that the unifiability problem in transitive modal logics like K4 and G is decidable. Wolter and Zakharyaschev [24] showed that the unifiability problem is undecidable for any modal logic between K and K4 extended with the universal modality. The notion of projectivity has been introduced by Ghilardi [15] to determine the unification type, finitary, of transitive modal logics like K4 and G. The unification type, nullary, of modal logics like K, KD and KT has been established in [6,18].

Within the context of description logics, checking subsumption of concepts is not sufficient and new inference capabilities are required. One of them, the unifiability of concept terms, has been introduced by Baader and Narendran [4] for FL 0 . Baader and Küsters [2] established the EXP T IM E-completeness of the unifiability problem in FL reg whereas Baader and Morawska [3] established the NPTIME-completeness of the unifiability problem in EL. Much remains to be done, seeing that the computability of the unifiability problem and the unification types are unknown in multifarious modal logics and description logics.

KD45 is the least modal logic containing the formulas ✷x → ✸x, ✷x → ✷✷x and ✸x → ✷✸x. It is determined by the class of all serial, transitive and Euclidean frames. The elementary unifiability problem in KD45 is to determine, given a parameter-free formula ϕ(x 1 , . . . , x n ), whether there exists parameter-free formulas ψ 1 , . . . , ψ n such that ϕ(ψ 1 , . . . , ψ n ) is in KD45. It is well-known that the elementary unifiability problem in KD45 is NP-complete. Moreover, as proved by Ghilardi and Sacchetti [16], the unifiability problem in KD45 is directed and, consequently, KD45 has either unitary type, or nullary type. See also [7,19]. The directedness of KD45 is a consequence of the characterization by Ghilardi and Sacchetti of the normal extensions of K4 with a directed unifiability problem. This characterization uses advanced notions from algebraic and relational semantics of normal modal logics.

In our paper, we directly show that every KD45-unifiable parameter-free formula has a projective unifier. As an immediate corollary, we conclude that KD45 has unitary type for elementary unification. Section 2 defines the syntax and the semantics of KD45. In Section 3, definitions about the elementary unifiability problem in KD45 are given. Sections 4-6 introduce and study arrows, setarrows and tips which will be our main tools for proving our results. In Section 7, definitions about acceptable agreements as a simplified version of bounded morphisms are given. Section 8 introduces and studies types which are sets of tips. In Sections 9-11, intermediate results about types needed to show that every KD45-unifiable parameter-free formula has a projective unifier are proved.

Let VAR be a countable set of variables (with typical members denoted x, y, etc). Let (x 1 , x 2 , . . .) be an enumeration of VAR without repetitions. The set FOR of all formulas (with typical members denoted ϕ, ψ, etc) is inductively defined as follows:

We write ϕ(x 1 , . . . , x n ) to denote a formula whose variables form a subset of {x 1 , . . . , x n }. The result of the replacement in ϕ(x 1 , . . . , x n ) of variables x 1 , . . . , x n in their places with formulas ψ 1 , . . . , ψ n will be denoted by ϕ(ψ 1 , . . . , ψ n ). We define the other Boolean constructs as usual. We will follow the standard rules for omission of the parentheses. Let ϕ be a formula. We will write ✸ϕ for ¬✷¬ϕ. We will respectively write ϕ ⊥ and ϕ ⊤ for ¬ϕ and ϕ. Let Γ be a finite set of formulas. Considering that ∅ = ⊥ and ∅ = ⊤, we will write ▽Γ for the conjunction of the following formulas:

• ✷ {ϕ : ϕ is a formula in Γ},

• {✸ϕ : ϕ is a formula in Γ}.

A model is a function V : VAR −→ 2 N associating to each variable x a set V (x) of nonnegative integers. We inductively define the truth of a formula ϕ in model V at nonnegative integer s, in symbols V, s |= ϕ, as follows:

As a result, V, s |= ✸ϕ iff there exists a positive integer t such that

• for all ϕ ∈ Γ, there exists a positive integer t such that V, t |= ϕ.

We shall say that a model V is uniform iff for all variables x, either V (x) = ∅, or V (x) = N. We shall say that a formula ϕ is satisfiable iff there exists a model V such that V, 0 |= ϕ. We shall say that a formula ϕ is valid, in symbols |= ϕ, iff for all models V , V, 0 |= ϕ. The following result is well-known and can be proved by using the canonical model construction, the technique of the generated subframe and the bounded morphism lemma [10]. Proposition 1. For all formulas ϕ, |= ϕ iff ϕ ∈ KD45.

Proof. Left to the reader.

Unification

A substitution is a function σ : VAR −→ FOR associating to each variable a formula. We shall say that a substitution σ is closed iff for all variables x, σ(x) is a variable-free formula. For all formulas

The composition σ • τ of the substitutions σ and τ is the substitution associating to each variable x the formula τ (σ(x)). We shall say that a substitution σ is equivalent to a substitution τ , in symbols σ ≃ τ , iff |= σ(x) ↔ τ (x) for all variables x. We shall say that a substitution σ is more general than a substitution τ , in symbols σ τ , iff there exists a substitution υ such that σ • υ ≃ τ . Note that the notation τ σ is also used in many papers. We shall say that a formula ϕ is unifiable iff there exists a substitution σ such that |= σ(ϕ). In that case, σ is a unifier of ϕ. We shall say that a unifiable formula ϕ is projective iff there exists a unifier Proof. Remark that every variable-free formula is either KD45-equivalent to ⊥, or KD45-equivalent to ⊤. Hence, by Proposition 2, in order to determine if a given formula ϕ(x 1 , . . . , x n ) is unifiable, it suffices to nondeterministically choose (ψ 1 , . . . , ψ n ) in {⊥, ⊤} n such that |= ϕ(ψ 1 , . . . , ψ n ). Since the validity of a given variable-free formula can be checked in polynomial time, therefore the elementary unifiability problem in KD45 is in NP. As for the NP-hardness of the elementary unifiability problem in KD45, it follows from Proposition 3.

Proposition 5. Let ϕ be a unifiable formula. If ϕ is projective then ϕ possesses a most general unifier.

Proof. Suppose ϕ is projective. Let σ be a unifier of ϕ such that |= ϕ ∧ ✷ϕ → (σ(x) ↔ x) for all variables x. Let τ be a unifier of ϕ and x be a variable. Hence, |= τ (ϕ) and

From now on, let us fix n ∈ N.

Formulas of the form ϕ(x 1 , . . . , x n ) will be called n-formulas. From now on, they will be denoted ϕ( x).

Arrows

We define A n = {⊥, ⊤} n . Elements of A n are n-tuples of bits. They will be called narrows. They will be denoted α, β, etc. Remark that Card(A n ) = 2 n . For all n-arrows α = (α 1 , . . . , α n ), we will write α( x) for the associated n-formula

The following result says that the n-formula associated to an n-arrow is always satisfiable.

Lemma 6. Let α be an n-arrow. There exists a model

Proof. Left to the reader.

Remark that for all n-arrows α = (α 1 , . . . , α n ) and for all n-tuples ψ of formulas,

As a result, Lemma 7. Let ψ be an n-tuple of formulas, V be a model and s be a nonnegative integer.

. . , n} be such that either α i = ⊥ and β i = ⊤, or α i = ⊤ and β i = ⊥. Without loss of generality, assume α i = ⊥ and

For all n-tuples ψ of formulas, for all models V and for all nonnegative integers s, let α[ ψ, V, s] be the n-arrow such that for all i ∈ {1, . . . , n}, Proof. By Lemmas 7 and 8.

The following result will be useful when we study the most general unifiers of unifiable n-formulas.

Lemma 10. Let V be a model. For all n-arrows α, there exists a model V ′ such that V ′ , 0 |= α( x) and for all variables x and for all positive integers s, s ∈

Proof. Left to the reader.

Setarrows

Let S n = 2 A n \ {∅}. Elements of S n are nonempty sets of n-arrows. They will be called n-setarrows. They will be denoted a, b, etc. Remark that Card(S n ) = 2 2 n − 1. For all n-setarrows a = {α 0 , . . . , α k }, we will write a( x) the associated n-formula

The following result says that the n-formula associated to an n-setarrow is always satisfiable.

Lemma 11. Let a be an n-setarrow. There exists a model V such that V, 0 |= a( x).

Proof. Left to the reader.

Remark that for all n-setarrows a = {α 0 , . . . , α k } and for all n-tuples ψ of formulas, a( ψ) = ▽{ α 0 ( ψ), . . . , α k ( ψ)}. As a result, The following result will be useful when we study the most general unifiers of unifiable n-formulas. It can be proved by induction on ϕ( x).

Lemma 15. Let ψ be an n-tuple of formulas. Let

Let P n = A n × S n . Elements of P n are couples consisting of an n-arrow component and an n-setarrow component. They will be called n-tips. They will be denoted p, q, etc. Remark that Card(P n ) = 2 n ×(2 2 n −1). For all n-tips p = (α, a), we will write p( x) the associated n-formula

The following result says that the n-formula associated to an n-tip is always satisfiable.

Lemma 16. Let p be an n-tip. There exists a model

Proof. Left to the reader.

Remark that for all n-tips p = (α, a) and for all n-tuples ψ of formulas, p( ψ) = α( ψ) ∧ a( ψ). As a result, Lemma 17. Let ψ be an n-tuple of formulas and V be a model.

Proof. By Lemmas 7 and 12.

For all n-tuples ψ of formulas and for all models

: s is a positive integer} and Lemma 18. Let ψ be an n-tuple of formulas and V be a model.

Acceptable agreements

In this section, we give definitions of acceptable agreements as a simplified version of bounded morphisms. We shall say that a function f : N −→ N associating to each nonnegative integer a nonnegative integer is acceptable iff for all positive integers s, f (s) is a positive integer and f −1 (s) contains a positive integer. We shall say that a function f : N −→ N associating to each nonnegative integer a nonnegative integer is an n-agreement between models V and V ′ iff for all i ∈ {1, . . . , n} and for all nonnegative integers s, s ∈

Lemma 20. Let f be an acceptable n-agreement between models

Proof. By induction on ϕ( x).

We shall say that a function f : N −→ N associating to each nonnegative integer a nonnegative integer is an ω-agreement between models V and V ′ iff for all variables x and for all nonnegative integers s, s ∈

Lemma 21. Let f be an acceptable ω-agreement between models

Proof. By induction on ϕ.

Types

Let T n = 2 A n ×S n . Elements of T n are sets of n-tips. They will be called n-types. They will n n − be denoted T , U, etc. Remark that Card(T n ) = 2 2 ×(2 2 1) . We shall say that an n-type T is complete for an n-setarrow a iff for all n-arrows α, (α, a) ∈ T . We shall say that an n-type T is empty for an n-setarrow a iff for all n-arrows α, if α ∈ a then (α, a) ∈ T . We shall say that an n-type T is full for an n-setarrow a iff for all n-arrows α, if α ∈ a then (α, a) ∈ T . We shall say that an n-type T is saturated iff for all n-arrows α, β and for all n-setarrows a, if (α, a) ∈ T and β ∈ a then (β, a) ∈ T . The following result will be of crucial importance in the remaining sections of our paper.

Proposition 22. Let T be a saturated n-type. For all n-setarrows a, exactly one of the following conditions holds: (i) T is complete for a; (ii) T is not complete for a and T is empty for a; (iii) T is not complete for a and T is full for a.

Proof. Left to the reader.

We shall say that an n-type T is closed iff for all n-setarrows a, there exists an n-arrow γ such that if T is not complete for a then either T is empty for a and (γ, {γ}) ∈ T , or T is full for a and (γ, a) ∈ T . We shall say that an n-type is perfect iff it is saturated and closed. Let T be a perfect n-type. Hence, T is saturated and closed. Thus, by Proposition 22, for all n-setarrows a, exactly one of the following conditions holds: (i) T is complete for a; (ii) T is not complete for a and T is empty for a; (iii) T is not complete for a and T is full for a. Since T is closed, therefore for all n-setarrows a, let γ T,a be an n-arrow such that if T is not complete for a then either T is empty for a and (γ T,a , {γ T,a }) ∈ T , or T is full for a and (γ T,a , a) ∈ T . For all n-tips p = (α, a), let δ T,p be the n-arrow such that if p ∈ T then δ T,p = γ T,a else δ T,p = α. Let ψ[T ]( x) be the n-tuple of n-formulas such that for all i ∈ {1, . . . , n},

The aim of this section is to demonstrate that

Proof. Suppose p ∈ T . Let β be the n-arrow component of p and b be the n-setarrow

Proof.

i . Since T is empty for b and α ∈ b, therefore

Since s is an arbitrary positive integer, therefore

, therefore by Lemma 17, p = q. Since q ∈ T , therefore p ∈ T . Case "q ∈ T ": Let a be the n-setarrow component of q. Since q ∈ T , therefore by Lemma 29,

, therefore by Lemma 7, γ T,a is the n-arrow component of p. Since q ∈ T , therefore T is not complete for a. Since T is saturated, therefore by Proposition 22, either T is empty for a, or T is full for a. In the former case, (γ T,a , {γ T,a }) ∈ T . Moreover, by Lemma 30, |= a( x) → {γ T,a

, therefore by Lemma 12, {γ T,a } is the n-setarrow component of p. Since γ T,a is the n-arrow component of p and (γ T,a , {γ T,a }) ∈ T , therefore p ∈ T . In the latter case, (γ T,a , a) ∈ T . Moreover, by Lemma 31, |= a

About most general unifiers

Let ϕ( x) be an n-formula. Let T be the n-type

Lemma 34. T is saturated.

Proof. Let α, β be n-arrows and a be an n-setarrow such that (α, a) ∈ T and β ∈ a. Hence, |= α ( x)∧a ( x) → ϕ( x)∧✷ϕ( x). Let V be a model such that V, 0 |= β ( x) and V, 0 |= a( x). By Lemma 10, let V ′ be a model such that V ′ , 0 |= α ( x) and for all variables x and for all positive integers s, s ∈

Moreover, recall that β ∈ a. Let s β be a positive integer such that

Lemma 37. T is closed.

Proof. By Lemma 34, T is saturated. Hence, by Proposition 22, for all n-setarrows a, exactly one of the following conditions holds: (i) T is complete for a; (ii) T is not complete for a and T is empty for a; (iii) T is not complete for a and T is full for a. By Lemma 36, let γ be an n-arrow such that (γ, {γ}) ∈ T . For all n-setarrows a, let γ T,a be an arbitrary n-arrow if condition (i) holds, the n-arrow γ if condition (ii) holds and an arbitrary n-arrow in a if condition (iii) holds. The reader may easily verify that for all n-setarrows a, if T is not complete for a then either T is empty for a and (γ T,a , {γ T,a }) ∈ T , or T is full for a and (γ T,a , a) ∈ T .

Lemma 38. T is perfect.

Proof. By Lemmas 34 and 37.

By Lemma 37, T is closed. Hence, for all n-setarrows a, let γ T,a be an n-arrow such that if T is not complete for a then either T is empty for a and (γ T,a , {γ T,a }) ∈ T , or T is full for a and (γ T,a , a) ∈ T . For all n-tips p = (α, a), let δ T,p be the n-arrow such that if p ∈ T then δ T,p = γ T,a else δ T,p = α. Proof. Let V be a model. Since

Case "q ∈ T ": Hence, by Lemma 17, V, 0 |= ψ i [T ]( x) ↔ δ T,q i . Let α be the n-arrow component of q. Since q ∈ T , therefore δ T,q

Case "q ∈ T ": Hence, |= q( x) → ϕ( x) ∧ ✷ϕ( x). Let V ′ be a model such that V ′ , 0 |= q( x) and V ′ , 0 |= ϕ( x) ∧ ✷ϕ( x). Since V, 0 |= q( x), therefore by Lemma 15, V, 0 |= ϕ( x) ∧ ✷ϕ( x): a contradiction. Since V is an arbitrary model such that

From all this, it follows that