Unification in modal logic Alt1

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Unification in modal logic Alt

1

Philippe Balbiani

1

Institut de recherche en informatique de Toulouse Toulouse University

Tinko Tinchev

2

Department of Mathematical Logic and Applications Sofia University

Abstract

Alt1 is the least modal logic containing the formula ✸x → ✷x. It is determined by

the class of all deterministic frames. The unification problem in Alt1is to determine,

given a formula φ(x1, . . . , xα), whether there exists formulas ψ₁, . . . , ψα such that

φ(ψ1, . . . , ψα) is in Alt₁. In this paper, we show that the unification problem in Alt₁

is in P SP ACE. We also show that there exists an Alt1-unifiable formula that has

no minimal complete set of unifiers. Finally, we study sub-Boolean variants of the unification problem in Alt1.

Keywords: Modal logic Alt1. Computability of unifiability. Unification type.

Sub-Boolean variants.

1 Introduction

Modal logics are essential to the design of logical systems that capture elements of reasoning about knowledge, time, etc. There exists variants of these logics with one or several modalities, with or without the universal modality, etc. The logical problems addressed in their setting usually concern their axioma-tizability, their decidability, etc. Other desirable properties which one should establish whenever possible concern, for example, the admissibility problem and the unifiability problem. About admissibility, an inference rule φ1,...,φn

ψ is

admissible in a modal logic L if for all instances φ′1,...,φ′n

ψ′ of the inference rule,

if φ′

1, . . . , φ′n are in L then ψ′ is in L too [18]. About unifiability, a formula

φis unifiable in a modal logic L if there exists an instance φ′ _{of the formula}

such that φ′ _{is in L [11]. When a modal logic L is axiomatically presented, its}

1 _{Address: Institut de recherche en informatique de Toulouse, Toulouse University, 118 route}

de Narbonne, 31062 Toulouse Cedex 9, FRANCE; balbiani@irit.fr.

2 _{Address: Department of Mathematical Logic and Applications, Sofia University, Blvd}

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admissible inference rules can be added to its axiomatical presentation without changing the set of its theorems. As a result, in order to improve the efficiency of automated theorem provers for modal logics, methods for deciding the ad-missibility of inference rules can be used [8]. The unifiability problem is easily reducible to the admissibility problem, seeing that the formula φ is unifiable in Liff the inference rule ⊥φ is non-admissible in L. In some cases, when L’s

unifi-cation type is finitary, the admissibility problem is reducible to the unifiability problem. Therefore, in order to improve the efficiency of automated theorem provers for modal logics, methods for deciding the unifiability of formulas can be used as well.

Results about unification have been already obtained in many modal logics. Rybakov [17] demonstrated that unification in S4 is decidable. Wolter and Za-kharyaschev [19] showed that unification is undecidable for K4 or K extended with the universal modality. The notion of projectivity has been introduced by Ghilardi [13] to determine the unification type, finitary, of S4 and K4. Je˘r´abek [16] established the unification type, nullary, of K. Within the context of description logics, checking subsumption of concepts is not sufficient and new inference capabilities are required. One of them, unification of concept terms, has been introduced by Baader and Narendran [6] for FL0. Baader and

K¨usters [4] established the EXP T IM E-completeness of unification in FLreg

whereas Baader and Morawska [5] established the N P T IM E-completeness of unification in EL. Much remains to be done, seeing that the computability of unifiability and the unification types are unknown in multifarious modal logics. In this paper, we consider the unification problem in Alt1. Its

section-by-section breakdown is organized as follows. Section 2 defines the syntax, Section 3 introduces the semantics and Section 4 presents unification. In Sec-tion 5, useful Lemmas are proved. They are used in SecSec-tion 6 to prove the soundness/completeness of a nondeterministic algorithm solving unification in polynomial space. In Section 7, it is shown that there exists a unifiable for-mula that has no minimal complete set of unifiers. In Section 8, we study sub-Boolean variants of unification.

2 Syntax

Let AF be a countable set of atomic formulas (denoted x, y, etc). The set F of all formulas (denoted φ, ψ, etc) is inductively defined as follows:

• φ::= x | ⊥ | ¬φ | (φ ∨ ψ) | ✷φ.

We define the other Boolean constructs as usual. The formula ✸φ is obtained as an abbreviation:

• ✸φ ::= ¬✷¬φ.

The modal connective ✷k _{is inductively defined as follows for each k ∈ N:} • ✷0φ::= φ,

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The modal connective ✷<k _{is inductively defined as follows for each k ∈ N:} • ✷<0φ::= ⊤,

• ✷<k+1φ::= ✷<kφ∧ ✷kφ.

We adopt the standard rules for omission of the parentheses. Let deg(φ) denote the degree of a formula φ and var(φ) its atom-set. We shall say that a formula φis atom-free iff var(φ) = ∅. Let AF F be the set of all atom-free formulas. In the sequel, we use φ(x1, . . . , xα) to denote a formula whose atomic formulas

form a subset of {x1, . . . , xα}. A substitution is a function σ associating to each

variable x a formula σ(x). We shall say that a substitution σ is closed if for all variables x, σ(x) ∈ AF F . For all formulas φ(x1, . . . , xα), let σ(φ(x1, . . . , xα))

be φ(σ(x1), . . . , σ(xα)). The composition σ ◦ τ of the substitutions σ and τ

associates to each atomic formula x the formula τ (σ(x)). Remark that for all substitutions σ, τ , if τ is closed then σ ◦ τ is closed.

3 Semantics

Our modal language receives a relational semantics and a tuple semantics. 3.1 Relational semantics

A frame is a relational structure of the form F = (W, R) where W is a nonempty set of states (with typical members denoted s, t, etc) and R is a binary relation on W . A model based on a frame F = (W, R) is a relational structure of the form M = (W, R, V ) where V is a function associating to each variable x a set V(x) of states. We inductively define the truth of a formula φ in a model M at state s, in symbols M, s |= φ, as follows:

• M, s |= x iff s ∈ V (x), • M, s (|= ⊥,

• Ms |= ¬φ iff M, s (|= φ,

• M, s |= φ ∨ ψ iff either M, s |= φ, or M, s |= ψ,

• M, s |= ✷φ iff for all states t ∈ W , if sRt then M, t |= φ.

Obviously,

• M, s |= ✸φ iff there exists a state t ∈ W such that sRt and M, t |= φ, • M, s |= ✷kφiff for all states t ∈ W , if sRktthen M, t |= φ,

• M, s |= ✷<kφiff for all states t ∈ W and for all i ∈ N, if sRitand i < k then

M, t |= φ.

Let C be a class of frames. We shall say that a formula φ is C-valid, in symbols C |= φ, if for all frames F = (W, R) in C, for all models M = (W, R, V ) based on F and for all states s ∈ W , M, s |= φ.

3.2 Tuple semantics

For all n ∈ N, an n-valuation is an (n + 1)-tuple (U0, . . . , Un) of subsets of AF .

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• (U₀, . . . , U_n) |= ✷<kφ iff for all i ∈ N, if n ≥ i and i < k then

(U0, . . . , Un−i) |= φ.

We shall say that a formula φ is n-tuple-valid, in symbols |=n φ, iff for all

n-valuations (U0, . . . , Un), (U0, . . . , Un) |= φ.

3.3 Correspondence between the two semantics

In this paper, we will be only interested in the class Cdet of all deterministic

frames, i.e. frames F = (W, R) such that for all states s, t, u ∈ W , if sRt and sRuthen t = u.

Proposition 3.1 Letφ be a formula. The following conditions are equivalent: (i) Cdet|= φ.

(ii) For all n ∈ N, |=nφ.

When the conditions from Proposition 3.1 hold, we shall simply say that φ is valid, in symbols |= φ.

4 Unification

We shall say that a formula φ(x1, . . . , xα) is unifiable iff there exists

ψ1, . . . , ψα ∈ F such that |= φ(ψ1, . . . , ψα). In that case, the substitution

σdefined by σ(x1) = ψ1, . . ., σ(xα) = ψα is called unifier of φ. For instance,

the formula φ = ✷x ∨ ✷y is unifiable. The substitution σ defined by σ(x) = z and σ(y) = ¬z is a unifier of φ. Remark that if a formula possesses a unifier then it possesses a closed unifier. This follows from the fact that for all unifiers σof a formula φ and for all closed substitutions τ , σ ◦ τ is a closed unifier of φ. The unification problem is the decision problem defined as follows:

• given a formula φ(x₁, . . . , x_α), determine whether φ(x₁, . . . , x_α) is unifiable.

We shall say that a substitution σ is equivalent to a substitution τ , in symbols σ≃ τ , if for all variables x, |= σ(x) ↔ τ (x). We shall say that a substitution σ is more general than a substitution τ , in symbols σ , τ , if there exists a substitution υ such that σ ◦ υ ≃ τ . We shall say that a set Σ of unifiers of a unifiable formula φ is complete if for all unifiers σ of φ, there exists a unifier τ _{of φ in Σ such that τ , σ. An important question is the following: when}

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a formula is unifiable, has it a minimal complete set of unifiers? When the answer is “yes”, how large is this set? We shall say that a unifiable formula

• φ is unitary if there exists a minimal complete set of unifiers of φ with

cardinality 1,

• φis finitary if there exists a finite minimal complete set of unifiers of φ but

there exists no with cardinality 1,

• φis infinitary if there exists an infinite minimal complete set of unifiers of φ

but there exists no finite one,

• φis nullary if there exists no minimal complete set of unifiers of φ.

For instance, the formula x is unitary: the substitution σ defined by σ(x) = ⊤ constitutes a minimal complete set of unifiers of it. We do not know whether there exists finitary, or infinitary formulas. We will show in Section 7 that the formula x → ✷x is nullary.

5 Unification problem: lemmas

Let ψ(x) be an arbitrary formula with at most one atomic formula. Lemma 5.1 For allk ∈ N, the following conditions are equivalent:

(i) ψ(x) is unifiable;

(ii) there exists φ ∈ AF F such that |= ψ(φ);

(iii) there exists φ ∈ AF F such that |= ✷k_{⊥ → ψ(φ) and |= ✸}k_{⊤ → ψ(φ).}

Remark that Lemma 5.1 still holds when one considers a formula ψ(x1, . . . , xα) with more than one atomic formula. In this case, simply

re-place the “there exists φ . . .” by “there exists φ1, . . . , φα . . .”. Concerning the

remainder of this Section and Section 6, the same remark is on as well. Hence, without loss of generality, we will always consider in the remainder of this Sec-tion and in SecSec-tion 6 that ψ is a formula with at most one atomic formula. In this case, for all n ∈ N, an n-valuation is comparable to an (n + 1)-tuple of bits. Let k ∈ N be such that deg(ψ(x)) ≤ k. For all φ ∈ AF F and for all n ∈ N, if k ≤ nthen let Vk(φ, n, i) = “if |=n−k+iφthen 1 else 0“ for each i ∈ N such

that i ≤ k.

Lemma 5.2 For allφ∈ AF F and for all n ∈ N, if k ≤ n then the following

conditions are equivalent:

(i) |=nψ(φ);

(ii) (Vk(φ, n, 0), . . . , Vk(φ, n, k)) |= ψ(x).

Lemma 5.3 For allφ∈ AF F , the following conditions are equivalent: (i) |= ✸k_{⊤ → ψ(φ);}

(ii) for all n ∈ N, if k ≤ n then (Vk(φ, n, 0), . . . , Vk(φ, n, k)) |= ψ(x).

For all φ ∈ AF F and for all n ∈ N, if k ≤ n then let Vk(φ, n) =

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N _{is such that k ≤ n}. The atom-free formulas φ}′ _{and φ}′′ _{are said to be}

k-equivalent, in symbols φ′_≡

kφ′′, iff fk(φ′) = fk(φ′′).

Proposition 5.4 ≡k is an equivalence relation on AF F possessing finitely

many equivalence classes.

Proof. By definitions of ≡k and fk, knowing that for all φ ∈ AF F , fk(φ) is a

nonempty set of (k + 1)-tuples of bits. ✷

Lemma 5.5 For allφ′_{, φ}′′_{∈ AF F , if φ}′ _≡

k φ′′ then the following conditions

are equivalent:

(i) |= ✸k_{⊤ → ψ(φ}′_);

(ii) |= ✸k_{⊤ → ψ(φ}′′_).

For all φ ∈ AF F and for all n ∈ N, let ak(φ, n) = Vk(φ, n · (k + 1) + k).

For all φ ∈ AF F , let gk(φ) = {(ak(φ, n), ak(φ, n + 1)): n ∈ N}. We shall say

that the atom-free formulas φ′ _{and φ}′′_{are k-congruent, in symbols φ}′ ∼₌ k φ′′,

iff gk(φ′) = gk(φ′′).

Proposition 5.6 ∼=k is an equivalence relation on AF F possessing finitely

many equivalence classes.

Proof. By definitions of ∼=k and gk, knowing that for all φ ∈ AF F , gk(φ) is a

nonempty set of pairs of (k + 1)-tuples of bits. ✷

Proposition 5.7 For allφ′_{, φ}′′_{∈ AF F , if φ}′_∼

=kφ′′ thenφ′≡kφ′′.

Proof. Let φ′_{, φ}′′_{∈ AF F . Suppose φ}′_∼

=k φ′′and φ′(≡kφ′′. Hence, gk(φ′) =

gk(φ′′) and fk(φ′) (= fk(φ′′). Thus, either there exists n′ ∈ N such that k ≤

n′ _{and V}

k(φ′, n′) (∈ fk(φ′′), or there exists n′′ ∈ N such that k ≤ n′′ and

V_k_(φ′′_{, n}′′_{) (∈ f}

k(φ′). Without loss of generality, assume there exists n′ ∈ N

such that k ≤ n′ _{and V}

k(φ′, n′) (∈ fk(φ′′). By the division algorithm, let

m, l ∈ Nbe such that n′_{= m · (k + 1) + l and l < k + 1.}

Case m = 0. Since k ≤ n′_{, n}′ _{= m · (k + 1) + l and l < k + 1, therefore}

n′ _{= k. Hence, V}

k(φ′, n′) = ak(φ′,0). Since gk(φ′) = gk(φ′′), therefore let

n′′_{∈ N be such that (a}

k(φ′,0), ak(φ′,1)) = (ak(φ′′, n′′), ak(φ′′, n′′+ 1)). Since

V_k_(φ′, n′_{) = a}

k(φ′,0), therefore Vk(φ′, n′) = Vk(φ′′, n′′· (k + 1) + k).

Case m (= 0. Since gk(φ′) = gk(φ′′), therefore let n′′ ∈ N be such that

(ak(φ′, m −1), ak(φ′, m)) = (ak(φ′′, n′′), ak(φ′′, n′′+ 1)). Hence, Vk(φ′,(m −

1) · (k + 1) + k, i) = Vk(φ′′, n′′· (k + 1) + k, i) and Vk(φ′, m· (k + 1) + k, i) =

Vk(φ′′,(n′′+ 1) · (k + 1) + k, i) for each i ∈ N such that i ≤ k. Since either

n′ _{= m · (k + 1) + l and i ≤ k − (l + 1) and V}

k(φ′, m· (k + 1) + l, i) =

Vk(φ′,(m − 1) · (k + 1) + k, i + (l + 1)), or k − l ≤ i and Vk(φ′, m· (k + 1) + l, i) =

Vk(φ′, m· (k + 1) + k, i − (k − l)) for each i ∈ N such that i ≤ k, therefore either

i ≤ k −(l + 1) and Vk(φ′, n′, i) = Vk(φ′′, n′′· (k + 1) + k, i + (l + 1)), or k − l ≤ i

and Vk(φ′, n′, i) = Vk(φ′′,(n′′+ 1) · (k + 1) + k, i − (k − l)) for each i ∈ N such

that i ≤ k. Thus, Vk(φ′, n′, i) = Vk(φ′′,(n′′+ 1) · (k + 1) + l, i) for each i ∈ N

such that i ≤ k. Consequently, Vk(φ′, n′) = Vk(φ′′,(n′′+ 1) · (k + 1) + l).

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Lemma 5.8 For allφ′_{, φ}′′_{∈ AF F , if φ}′ ∼₌

k φ′′ then the following conditions

are equivalent:

(i) |= ✸k_{⊤ → ψ(φ}′_);

(ii) |= ✸k_{⊤ → ψ(φ}′′_).

We shall say that a nonempty set B of pairs of (k + 1)-tuples of bits is modally definable iff there exists φ ∈ AF F such that B = gk(φ). For all

nonempty sets B of pairs of (k + 1)-tuples of bits, let ⊲Bbe the domino relation

on B defined as follows:

• (b′₁, b′′₁) ⊲_B(b₂′, b′′₂) iff b′′₁ = b′₂.

We shall say that a path in the directed graph (B, ⊲B) is weakly Hamiltonian

iff it visits each vertex at least once. Let 1k+1 be the (k + 1)-tuple of 1 and

0k+1be the (k + 1)-tuple of 0.

Proposition 5.9 For all nonempty sets B of pairs of (k + 1)-tuples of bits,

the following conditions are equivalent:

(i) B is modally definable;

(ii) the directed graph (B, ⊲B) contains a weakly Hamiltonian path either

end-ing with(1k+1, 1k+1), or ending with (0k+1, 0k+1).

Proof. Let B be a nonempty set of pairs of (k + 1)-tuples of bits.

If. Suppose the directed graph (B, ⊲B) contains a weakly Hamiltonian path

either ending with (1k+1, 1k+1), or ending with (0k+1, 0k+1). Let s ∈ N

and (b′

0, b′′0), . . . , (b′s, b′′s) ∈ B be such that ((b′0, b′′0), . . . , (b′s, b′′s)) is a weakly

Hamiltonian path either ending with (1k+1, 1k+1), or ending with (0k+1, 0k+1).

Let (β0, . . . , βs·(k+1)+k) be the sequence of bits determined by the sequence

(b′

0, , . . . , b′s) of (k + 1)-tuples of bits.

Case (b′

s, b′′s) = (1k+1, 1k+1). Let φ = !{✸i✷⊥: i ∈ N is such that i <

s· (k + 1) and βi= 1} ∨ ✸s·(k+1)⊤.

Case (b′

s, b′′s) = (0k+1, 0k+1). Let φ = !{✸i✷⊥: i ∈ N is such that i <

s· (k + 1) and βi= 1}.

In both cases, the reader may easily verify that for all n ∈ N, if n ≤ s then Vk(φ, n · (k + 1) + k, i) = βn·(k+1)+i for each i ∈ N such that i ≤ k. Hence, for

all n ∈ N, if n ≤ s then Vk(φ, n · (k + 1) + k) = b′n. Thus, for all n ∈ N, if

n ≤ sthen (ak(φ, n), ak(φ, n + 1)) = (b′n, b′′n). Hence, B = gk(φ).

Only if. Suppose B is modally definable. Let φ ∈ AF F be such that B = gk(φ). Let n0 ∈ N be such that either for all n ∈ N, if n0 ≤ n

then ak(φ, n) = 1k+1, or for all n ∈ N, if n0 ≤ n then ak(φ, n) = 0k+1.

Thus, ((ak(φ, 0), ak(φ, 1)), . . . , (ak(φ, n0), ak(φ, n0+ 1))) is a weakly

Hamilto-nian path either ending with (1k+1, 1k+1), or ending with (0k+1, 0k+1). ✷

6 Unification problem: algorithm

As in Section 5, let ψ(x) be an arbitrary formula with at most one atomic formula and k ∈ N be such that deg(ψ(x)) ≤ k. We shall say that an infinite sequence (β0, β1, . . .) of bits respects ψ(x) iff the following conditions hold:

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• for all i ∈ N, if i ≤ k then (β₀, . . . , β_i) |= ψ(x), • for all i ∈ N, (β_i+1, . . . , β_i+k+1) |= ψ(x).

Using the above results, ψ(x) is unifiable iff there exists a modally definable set B of pairs of (k + 1)-tuples of bits from which, by means of its domino relation, an infinite sequence of bits respecting ψ(x) and either ending with 1s, or ending with 0s can be constructed. Hence, in order to determine whether ψ_{(x) is unifiable, it suffices to consider the following procedure:}

procedureU N I(ψ(x)) begin

k:= deg(ψ(x))

guess a tuple(b(0), . . . , b(k)) of bits of size k + 1 bool:= ⊤ i:= 0 whilebool ∧ i ≤ k do begin bool:= M C(b(0), . . . , b(i), ψ(x)) i:= i + 1 end

if¬bool then reject

while(b(0), . . . , b(k)) (= 0k+1∧ (b(0), . . . , b(k)) (= 1k+1do

begin

guess a tuple(b(k + 1), . . . , b(2k + 1)) of bits of size k + 1 bool:= ⊤

i:= 0

whilebool ∧ i ≤ k do begin

bool:= M C(b(i + 1), . . . , b(i + k + 1), ψ(x)) i:= i + 1

end

if¬bool then reject

(b(0), . . . , b(k)) := (b(k + 1), . . . , b(2k + 1)) end

accept end

The function M C(·) takes as input a tuple (b(i), . . . , b(i + j)) of bits and a formula ψ(x) and returns the Boolean value

• M C(b(i), . . . , b(i + j), ψ(x)) = “if (b(i), . . . , b(i + j)) |= ψ(x) then ⊤ else ⊥”.

It can be implemented as a deterministic Turing machine working in polynomial time. The procedure U N I(·) takes as input a formula ψ(x) and accepts it iff, when k = deg(ψ(x)), there exists a modally definable set B of pairs of (k + 1)-tuples of bits from which, by means of its domino relation, an infinite sequence of bits respecting ψ(x) and either ending with 1s, or ending with 0s can be constructed. By Proposition 5.9, the procedure U N I(·) accepts its

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input ψ(x) iff ψ(x) is unifiable. It can be implemented as a nondeterministic Turing machine working in polynomial space. Hence, the unification problem is in N P SP ACE. Since N P SP ACE = P SP ACE, therefore

Proposition 6.1 The unification problem is inP SP ACE.

Still, we do not know whether the unification problem is P SP ACE-hard.

7 Unification type

Following the line of reasoning suggested by Je˘r´abek [16], we consider the for-mula φ(x) = x → ✷x. We also consider the substitution σ⊤defined by σ⊤(x) =

⊤ and for all k ∈ N, the substitution σk defined by σk(x) = ✷<kx ∧ ✷k⊥.

Lemma 7.1 • σ_⊤ is a unifier ofφ(x), • for all k ∈ N, σ_k is a unifier ofφ(x).

Lemma 7.2 Letk, l ∈ N. If k ≤ l then σl, σk.

Lemma 7.3 Letk, l ∈ N. If k < l then σk (, σl.

Proposition 7.4 Letσ be a substitution. The following conditions are

equiv-alent:

(i) σ⊤◦ σ ≃ σ.

(ii) σ⊤, σ.

(iii) |= σ(x).

Proof. (i⇒ii) By definition of ≃ and ,.

(ii⇒iii) Suppose σ⊤, σ. Let τ be a substitution such that σ⊤◦ τ ≃ σ. Thus,

|= τ (σ⊤(x)) ↔ σ(x). Hence, |= ⊤ ↔ σ(x). Consequently, |= σ(x).

(iii⇒i) Suppose |= σ(x). Hence, |= ⊤ ↔ σ(x). Thus, |= σ(σ⊤(x)) ↔ σ(x).

Consequently, σ⊤◦ σ ≃ σ. ✷

Proposition 7.5 Letσ be a unifier of φ(x) and k ∈ N. The following

condi-tions are equivalent:

(i) σk◦ σ ≃ σ.

(ii) σk, σ.

(iii) |= σ(x) → ✷k_⊥.

Proof. (i⇒ii) By definition of ≃ and ,.

(ii⇒iii) Suppose σk , σ. Let τ be a substitution such that σk ◦ τ ≃ σ.

Thus, |= τ (σk(x)) ↔ σ(x). Hence, |= ✷<kτ(x) ∧ ✷k⊥ ↔ σ(x). Consequently,

|= σ(x) → ✷k_⊥.

(iii⇒i) Suppose |= σ(x) → ✷k

⊥. Obviously, |= ✷<k_σ

(x) ∧ ✷k_{⊥ ↔ σ(σ} k(x)).

Hence, |= σ(σk(x)) → σ(x). Since σ is a unifier of φ(x), therefore |= σ(x) →

✷σ(x). Thus, |= σ(x) → ✷<k_σ(x). _{Since |= σ(x) → ✷}k_{⊥, therefore}

|= σ(x) → ✷<k_σ_{(x) ∧ ✷}k_{⊥. Since |= ✷}<k_σ_{(x) ∧ ✷}k_{⊥ ↔ σ(σ}

k(x)), therefore

|= σ(x) → σ(σk(x)). Since |= σ(σk(x)) → σ(x), therefore |= σ(σk(x)) ↔ σ(x).

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Proposition 7.6 Letσ be a unifier of φ(x) and k ∈ N be such that deg(σ(x)) ≤ k. One of the following conditions holds:

(i) |= σ(x).

(ii) |= σ(x) → ✷k_⊥.

Proof. _{Suppose (|= σ(x) and (|= σ(x) → ✷}k_{⊥. Let m, n ∈ N, (U}

0, . . . , Um)

be an m-valuation such that (U0, . . . , Um) (|= σ(x) and (V0, . . . , Vn) be an

n-valuation such that (V0, . . . , Vn) (|= σ(x) → ✷k⊥. Hence, (V0, . . . , Vn) |= σ(x)

and (V0, . . . , Vn) (|= ✷k⊥. Thus, n ≥ k. Since deg(σ(x)) ≤ k, therefore

n ≥deg(σ(x)). Since (V0, . . . , Vn) |= σ(x), therefore (U0, . . . , Um, V0, . . . , Vn) |=

σ(x). Since σ is a unifier of φ(x), therefore |= σ(x) → ✷σ(x). Conse-quently, |= σ(x) → ✷n+1_{σ(x). Since (U}

0, . . . , Um, V0, . . . , Vn) |= σ(x), therefore

(U0, . . . , Um, V0, . . . , Vn) |= ✷n+1σ(x). Hence, (U0, . . . , Um) |= σ(x): a

contra-diction. ✷

Proposition 7.7 φ(x) is nullary.

Proof. Let Σ = {σ⊤} ∪ {σk : k ∈ N}. By Lemma 7.1 and Propositions 7.4–

7.6, Σ is a complete set of unifiers of φ(x). Suppose there exists a minimal complete set of unifiers of φ(x). Let Γ be a minimal complete set of unifiers of φ_{(x). Let γ ∈ Γ be such that γ , σ}0. Since Σ is a complete set of unifiers of

φ_{(x), therefore let σ ∈ Σ be such that σ , γ. Now, we consider the following} 2 cases.

Case σ= σ⊤. Since γ , σ0, therefore σ , σ0. Let υ be a substitution such

that σ ◦ υ ≃ σ0. Hence, |= υ(σ(x)) ↔ σ0(x). Thus, |= ⊤ ↔ ⊥: a contradiction.

Caseσ= σk for somek ∈ N. By Lemma 7.3, σ (, σk+1. Let γ′∈ Γ be such

that γ′ _{, σ}

k+1. By Lemma 7.2, since σ , γ, therefore γ′ , γ. Since Γ is a

minimal complete set of unifiers of φ(x), therefore γ′_{= γ. Since γ}′_{, σ} k+1and

σ, γ, therefore σ , σk+1: a contradiction. ✷

8 Sub-Boolean variants

In this section, we study sub-Boolean variants of the unification problem. 8.1 {✷, ⊤, ∧}-fragment

In the {✷, ⊤, ∧}-fragment, formulas are defined as follows:

• φ::= x | ⊤ | (φ ∧ ψ) | ✷φ.

Lemma 8.1 Let σ, τ, υ be substitutions such that for all variables x, var(σ(x)) ∩ var(τ (x)) = ∅ and |= υ(x) ↔ σ(x) ∧ τ (x). Then υ , σ and υ, τ .

Lemma 8.2 Letφ be a formula and σ, τ, υ be substitutions. If for all variables x,|= υ(x) ↔ σ(x) ∧ τ (x) then |= υ(φ) ↔ σ(φ) ∧ τ (φ).

The ✷-integer-set of a variable x with respect to a formula φ, in symbols is✷(x, φ), is inductively defined as follows:

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• is_✷(x, y) = ∅ if x (= y, • is_✷(x, ⊤) = ∅,

• is_✷(x, φ ∧ ψ) = is_✷(x, φ) ∪ is_✷(x, ψ), • is_✷(x, ✷φ) = {i + 1 : i ∈ is_✷(x, φ)}.

For instance, is_✷(x, y ∧ ✷✷z) = ∅, is✷(y, y ∧ ✷✷z) = {0} and is✷(z, y ∧ ✷✷z) =

{2}. Let φ be an arbitrary formula.

Lemma 8.3 For allx ∈ AF , is_✷(x, φ) is a finite set such that is✷(x, φ) (= ∅

iffx ∈ var(φ). Moreover, |= φ ↔"{✷i_x_{: x ∈ AF & i ∈ is}

✷(x, φ)}.

Lemma 8.4 Letσ be a substitution andx be a variable. If|= σ(φ) ↔ x then

there exists a variabley such that0 ∈ is✷(y, φ) and |= σ(y) ↔ x.

Lemma 8.5 Letσ be a substitution,i ≥0 and x, y be variables. If i ∈ is_✷(x, φ)

and|= σ(x) ↔ y then i ∈ is✷(y, σ(φ)).

In the {✷, ⊤, ∧}-fragment, unification problems are finite sets of pairs of formulas. We shall say that a finite set S = {(φ1, ψ1), . . . , (φn, ψn)} of pairs of

formulas is unifiable iff there exists a substitution σ such that |= σ(φ1) ↔ σ(ψ1),

. . ., |= σ(φn) ↔ σ(ψn). In that case, σ is called a unifier of S. Of course,

now, substitutions are functions associating to each variable a formula in the {✷, ⊤, ∧}-fragment. Obviously, if a finite set of pairs of formulas possesses a unifier then it possesses a closed unifier. Moreover, by Lemma 8.3, every atom-free formula is equivalent to ⊤. As a result,

Proposition 8.6 Every finite set of pairs of formulas possesses a unifier.

The simplicity of unification problems in the {✷, ⊤, ∧}-fragment does not entail that every finite set of pairs of formulas possesses a minimal complete set of unifiers. Following the line of reasoning suggested by Baader [2], we consider the formulas φ(x, y, z) = ✷x ∧ ✷y and ψ(x, y, z) = y ∧ ✷✷z. We also consider for all k ∈ N, the substitution σk defined by σk(x) = tk, σk(y) = ✷✷<k+1tk

and σk(z) = ✷ktk. We will assume that for all k, l ∈ N, k (= l, the variables tk

and tlare distinct.

Lemma 8.7 For allk ∈ N, σk is a unifier of{(φ(x, y, z), ψ(x, y, z))}.

For all k ∈ N, we consider

• the substitution γ_k inductively defined as follows:

· γ0= σ0,

· γk+1is the substitution defined by γk+1(x) = γk(x) ∧ σk+1(x), γk+1(y) =

γk(y) ∧ σk+1(y) and γk+1(z) = γk(z) ∧ σk+1(z).

Lemma 8.8 For allk ∈ N, γk is a unifier of{(φ(x, y, z), ψ(x, y, z))}.

Lemma 8.9 Letk, l ∈ N. If k ≤ l then γl, σk.

Lemma 8.10 Letk, l ∈ N. If k ≤ l then γl, γk.

Proposition 8.11 Letσ be a unifier of{(φ(x, y, z), ψ(x, y, z))} and k ∈ N. If σ, σk then there exists a variableu such that k ∈ is✷(u, σ(z)).

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Proof. Suppose σ , σk. Let τ be a substitution such that σ ◦ τ ≃ σk. Thus,

|= τ (σ(x)) ↔ σk(x). Hence, |= τ (σ(x)) ↔ tk. By Lemma 8.4, let u be a

variable such that 0 ∈ is✷(u, σ(x)) and |= τ (u) ↔ tk. Since σ is a unifier of

{(φ(x, y, z), ψ(x, y, z))}, therefore |= ✷σ(x) ∧ ✷σ(y) ↔ σ(y) ∧ ✷✷σ(z). Since 0 ∈ is✷(u, σ(x)), therefore 1 ∈ is✷(u, σ(y)).

Claim: Let i ≥ 1. If i ∈ is✷(u, σ(y)) and i − 1 (∈ is✷(u, σ(z)) then i + 1 ∈

is✷(u, σ(y)).

Proof of the Claim:Suppose i ∈ is✷(u, σ(y)) and i − 1 (∈ is✷(u, σ(z)). Since

|= ✷σ(x) ∧ ✷σ(y) ↔ σ(y) ∧ ✷✷σ(z), therefore i + 1 ∈ is✷(u, σ(y)).

By the above Claim, let i ≥ 1 be such that i − 1 ∈ is✷(u, σ(z)). By Lemma 8.5,

since |= τ (u) ↔ tk, therefore i−1 ∈ is✷(tk, τ(σ(z))). Since σ ◦τ ≃ σk, therefore

|= τ (σ(z)) ↔ σk(z). Hence, |= τ (σ(z)) ↔ ✷ktk. Since i − 1 ∈ is✷(tk, τ(σ(z))),

therefore i − 1 = k. Since i − 1 ∈ is✷(u, σ(z)), therefore k ∈ is✷(u, σ(z)). ✷

Lemma 8.12 Letσ be substitution. If σ is a unifier of{(φ(x, y, z), ψ(x, y, z))}

thenσ, γk for at most finitely manyk ∈ N.

Proposition 8.13 There exists no minimal complete set of unifiers of

{(φ(x, y, z), ψ(x, y, z))}.

Proof. Let ∆ be a minimal complete set of unifiers of {(φ(x, y, z), ψ(x, y, z))}. Let δ ∈ ∆ be such that δ , σ0. Hence, δ , γ0. By Lemmas 8.10 and 8.12,

let k ∈ N be such that δ , γk and δ (, γk+1. Without loss of generality, we

can assume that var(δ(x)) ∩ var(γk+1(x)) = ∅, var(δ(y)) ∩ var(γk+1(y)) = ∅

and var(δ(z)) ∩ var(γk+1(z)) = ∅. Let ǫ be the substitution defined by ǫ(x) =

δ(x)∧γk+1(x), ǫ(y) = δ(y)∧γk+1(y) and ǫ(z) = δ(z)∧γk+1(z). By Lemmas 8.1,

8.2 and 8.8, ǫ is a unifier of {(φ(x, y, z), ψ(x, y, z))}, ǫ , δ and ǫ , γk+1. Since

∆ is a minimal complete set of unifiers of {(φ(x, y, z), ψ(x, y, z))}, therefore let δ′ _{∈ ∆ be such that δ}′ _{, ǫ. Since ǫ , δ, therefore δ}′ _{, δ. Since ∆ is a}

minimal complete set of unifiers of {(φ(x, y, z), ψ(x, y, z))}, therefore δ′ _{= δ.}

Since ǫ , γk+1and δ′, ǫ, therefore δ , γk+1: a contradiction. ✷

8.2 _{{✸, ⊤, ∧}-fragment}

In the {✸, ⊤, ∧}-fragment, formulas are defined as follows:

• φ::= x | ⊤ | (φ ∧ ψ) | ✸φ.

The ✸-integer-set of a variable x with respect to a formula φ, in symbols is✸(x, φ), is inductively defined as has been defined the ✷-integer-set of x with

respect to φ. Let φ be an arbitrary formula.

Lemma 8.14 For all x ∈ AF , is✸(x, φ) is a finite set such that is✸(x, φ) (= ∅

iffx ∈ var(φ). Moreover, |= φ ↔"{✸i_x_{: x ∈ AF & i ∈ is}

✸(x, φ)}∧✸deg(φ)⊤.

As before, if a finite set of pairs of formulas possesses a unifier then it possesses a closed unifier. Unlike the {✷, ⊤, ∧}-fragment, there exists non-unifiable finite sets of pairs of formulas. The truth is that many atom-free formulas are not equivalent to ⊤. Nevertheless, by Lemma 8.14, for all atom-free formulas φ, φ is equivalent to ✸deg(φ)_⊤.

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Lemma 8.15 Let φ be a formula. For all closed substitutions σ, |= σ(φ) ↔ "

{✸i_{σ(x) : x ∈ var(φ) & i = max is}

✸(x, φ)} ∧ ✸deg(φ)⊤.

Lemma 8.16 Let S be a finite set of pairs of formulas. Let φ, ψ, φ′_{, ψ}′ _be

formulas such that deg(φ) = deg(φ′_{), var(φ) = var(φ}′_{), deg(ψ) = deg(ψ}′₎

and var(ψ) = var(ψ′_{). If for all x ∈ AF , max is}

✸(x, φ) = max is✸(x, φ′)

and max is✸(x, ψ) = max is✸(x, ψ′) then S ∪ {(φ, ψ)} possesses a unifier iff

S ∪{(φ′_{, ψ}′_{)} possesses a unifier.}

Let the normal formulas be defined as follows:

• φ::= (x₁∧ . . . ∧ x_α) | ⊤ | ((x₁∧ . . . ∧ x_α) ∧ ✸φ) | ✸φ.

For example, the formula ✸x ∧ ✸y is not normal and the formula y ∧ ✸✸z is normal. In the above definition of normal formulas, we use the conjunction (x1∧ . . . ∧ xα) of the variables x1, . . . , xα. In such a situation, we will always

consider that α ≥ 1. We shall say that a formula φ is minimalist if for all x ∈ AF, x occurs at most once in φ. For instance, the formula"{✸i_x_{: x ∈}

var(φ) & i = max is✸(x, φ)} is minimalist for each formula φ.

Lemma 8.17 Letφ be a formula. There exists a normal formula φ′_{such that}

|= φ ↔ φ′_{. Moreover, if}_{φ is minimalist then φ}′ _{is minimalist too. Finally,}_φ′

can be easily computed fromφ in polynomial time.

For example, the non-normal formula ✸x ∧ ✸y is equivalent to the normal formula ✸(x ∧ y) and the non-normal formula y ∧ ✸⊤ ∧ ✸✸z is equivalent to the normal formula y ∧ ✸✸z.

Lemma 8.18 LetS be a finite set of pairs of formulas. There exists a finite set S′ _{of pairs of minimalist normal formulas such that}_{S possesses a unifier iff S}′

possesses a unifier. Moreover,S′ _{can be easily computed from}_{S in polynomial}

time.

Let the thin formulas be defined as follows:

• φ::= x | ⊤ | (x ∧ ✸φ) | ✸φ.

For example, the formula ✸x ∧ ✸y is not thin and the formula y ∧ ✸✸z is thin. Remark that for all formulas φ, if φ is thin then φ is normal.

Proposition 8.19 LetS be a finite set of pairs of minimalist normal formulas

with variablesx1, . . . , xα. Let, be a total order on 1, . . . , α. Let S′ be a finite

set of pairs of thin minimalist formulas obtained fromS and , by replacing each

conjunct of the form(xβ1∧ . . . ∧ xβn) in S by xβ whereβ= max%{β1, . . . , βn}.

SupposeS′ _{possesses a closed unifier}_{σ such that}

• for all β= 1 . . . α, there exists k_β ∈ N such that σ(x_β) = ✸kβ_⊤,

• for all β, γ= 1 . . . α, if β , γ then k_β ≤ k_γ.

Thenσ is also a unifier ofS.

Proof. Let (xβ1∧. . .∧xβn) be a conjunct in S and β = max%{β1, . . . , βn}. Let

kβ1, . . . , kβn∈ N be such that σ(xβ1) = ✸

k_β1_{⊤, . . ., σ(x} βn) = ✸

kβn_{⊤. Since for}

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kβ = max%{kβ1, . . . , kβn}. Hence, |= ✸

k_β1_{⊤ ∧ . . . ∧ ✸}kβn_{⊤ ↔ ✸}kβ_{⊤. Thus,}

|= σ(xβ1∧ . . . ∧ xβn) ↔ σ(xβ). Since σ is a unifier of S

′_{, therefore σ is a unifier}

of S. ✷

Proposition 8.20 LetS be a finite set of pairs of minimalist normal formulas

with variablesx1, . . . , xα. Suppose S possesses a closed unifier σ such that • for all β= 1 . . . α, there exists k_β ∈ N such that σ(x_β) = ✸kβ_⊤.

Let, be a total order on 1, . . . , α such that

• for all β, γ= 1 . . . α, if β , γ then kβ ≤ kγ.

LetS′ _{be a finite set of pairs of thin minimalist formulas obtained from}_{S and}

, by replacing each conjunct of the form (xβ1∧ . . . ∧ xβn) in S by xβ where

β= max%{β1, . . . , βn}. Then σ is also a unifier of S′.

Proof. Similar to the proof of Proposition 8.19. ✷

In Propositions 8.19 and 8.20, the finite set S′ _{of pairs of thin minimalist}

formulas obtained from S and , is called a thin ,-subset of S. Using the above results, a given finite set S of pairs of minimalist normal formulas with variables x1, . . . , xα is unifiable iff there exists a total order , on 1, . . . , α and

a thin ,-subset of S possessing a unifier. Now, in order to determine whether a given finite set S of pairs of thin minimalist normal formulas is unifiable, it suffices to consider the following procedure:

procedureU N ISET(S) begin

recursively replace each pair of the form_{(✸φ, ✸ψ) in S by (φ, ψ)} bool:= BC(S)

ifbool ∧ var(S) (= ∅ then begin

guess a subsetAF(S) of var(S) for allx ∈ AF(S) do

replace inS each occurrence of x by ⊤ for allx ∈ var(S) \ AF (S) do

replace inS each occurrence of x in S by ✸x

transformS into an equivalent finite set of pairs of thin minimalist normal formulas

U N ISET(S) end

if¬bool then reject accept

end

The function BC(·) takes as input a finite set S of pairs of thin minimalist normal formulas and returns the Boolean value

• BC(S) =“if neither S contains pairs of the form (✸φ, ⊤), nor S contains

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It can be implemented as a deterministic Turing machine working in polynomial time. The procedure U N ISET (·) takes as input a finite set of pairs of thin min-imalist normal formulas and accepts it iff it is unifiable. It can be implemented as a nondeterministic Turing machine working in polynomial space. Hence, the unification problem is in N P SP ACE. Since N P SP ACE = P SP ACE, therefore

Proposition 8.21 The unification problem is inP SP ACE.

Still, we do not know whether the unification problem is P SP ACE-hard.

9 Conclusion

Much remains to be done. For example, there is the related admissibility problem: given an inference rule ψ1(x1,...,xn),...,ψk(x1,...,xn)

χ(x1,...,xn) , determine whether

for all formulas φ1, . . . , φn, if |= ψ1(φ1, . . . , φn), . . ., |= ψk(φ1, . . . , φn) then

|= χ(φ1, . . . , φn). One may also consider the unification problem when the

ordinary modal language is extended by a set AP of parameters (denoted p, q, etc). In this case, the unification problem is to determine, given a formula ψ(p1, . . . , pα, x1, . . . , xβ), whether there exists formulas φ1, . . . , φβ such that |=

ψ(p1, . . . , pα, φ1, . . . , φβ). For each k ≥ 2, one may also consider the unification

problem in Altk, the least normal logic containing the formula ✸(x1∧¬x2∧. . .∧

¬xk−1∧¬xk)∧. . .∧✸(¬x1∧¬x2∧. . .∧¬xk−1∧xk) → ✷(x1∨x2∨. . .∨xk−1∨xk).

Its decidability is open. Finally, what becomes of these problems when the ordinary modal language is extended by the master modality, the universal modality or the difference modality?

Acknowledgements

Special acknowledgement is heartly granted to Joseph Boudou, Wojciech Dzik, Ç i˘gdem Gencer, Silvio Ghilardi, Rosalie Iemhoff, Emil Je˘rábek and Michael Rusinowitch. We also would like to thank the referees for the feed-back we have obtained from them. Philippe Balbiani was partially sup-ported by the Bulgarian National Science Fund (contract DID02/32/209) and Tinko Tinchev was partially supported by the Centre international de mathématiques et d’informatique (contract ANR-11-LABX-0040-CIMI within the programme ANR-11-IDEX-0002-02). Philippe Balbiani and Tinko Tinchev were partially supported by the programme RILA (contracts 34269VB and DRILA01/2/2015).

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Annex

Proof of Proposition 3.1: _{(i)⇒(ii): It suffices to remark that for all n ∈ N,} every n-valuation can be considered as a model based on a deterministic frame. (ii)⇒(i): It suffices to remark that for all n ∈ N, every generated submodel of a model based on a deterministic frame is n-bisimilar to a k-valuation for some k ∈ N such that k ≤ n.

Proof of Lemma 5.2: By induction on ψ(x).

Proof of Lemma 5.3: Let φ ∈ AF F . The following conditions are equivalent: (1) |= ✸k_{⊤ → ψ(φ); (2) for all n ∈ N, |=}

n ✸k⊤ → ψ(φ); (3) for

all n ∈ N, if |=n ✸k⊤ then |=n ψ(φ); (4) for all n ∈ N, if k ≤ n then

(Vk(φ, n, 0), . . . , Vk(φ, n, k)) |= ψ(x). The reasons for these equivalences to

hold are the following: the equivalence between (1) and (2) follows from the definition of |=, the equivalence between (2) and (3) follows from the fact that φ ∈ AF F and the equivalence between (3) and (4) follows from Lemma 5.2.

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Proof of Lemma 5.5: By definitions of ≡kand fk and Lemma 5.3.

Proof of Lemma 5.8: By Lemma 5.5 and Proposition 5.7.

Proof of Lemma 7.2: Suppose k ≤ l. Let υ be the substitution defined by υ(x) = x ∧ ✷k_{⊥. The reader may easily verify that |= υ(σ}

l(x)) ↔ σk(x).

Hence, σl, σk.

Proof of Lemma 7.3: Suppose k < l and σk , σl. Let υ be a

sub-stitution such that σk ◦ υ ≃ σl. Hence, |= υ(σk(x)) ↔ σl(x). Thus,

|= ✷<l_{x ∧ ✷}l_{⊥ → ✷}k_{⊥. Consequently, |= ✷}l_{⊥ → ✷}k_{⊥. Hence, l ≤ k: a}

contradiction.

Proof of Lemma 8.1: Let θ and µ be the substitutions defined by

• θ(x) = x if x ∈ var(σ(x)) and θ(x) = ⊤ otherwise, • µ(x) = x if x ∈ var(τ (x)) and µ(x) = ⊤ otherwise.

The reader may easily verify that for all variables x, |= θ(υ(x)) ↔ σ(x) and |= µ(υ(x)) ↔ τ (x). Hence, υ , σ and υ , τ .

Proof of Lemma 8.2: By induction on φ. Proof of Lemma 8.3: By induction on φ. Proof of Lemma 8.4: By induction on φ. Proof of Lemma 8.5: By induction on φ. Proof of Lemma 8.8: By Lemmas 8.2 and 8.7.

Proof of Lemma 8.9: Suppose k ≤ l. Let υ be the substitution de-fined by υ(ti) = tk if i = k and υ(ti) = ⊤ otherwise. The reader may easily

verify that |= υ(γl(x)) ↔ σk(x), |= υ(γl(y)) ↔ σk(y) and |= υ(γl(z)) ↔ σk(z).

Hence, γl, σk.

Proof of Lemma 8.10: Suppose k ≤ l. Let υ be the substitution de-fined by υ(ti) = ti if i ≤ k and υ(ti) = ⊤ otherwise. The reader may easily

verify that |= υ(γl(x)) ↔ γk(x), |= υ(γl(y)) ↔ γk(y) and |= υ(γl(z)) ↔ γk(z).

Hence, γl, γk.

Proof of Lemma 8.12: By Lemma 8.9 and Proposition 8.11. Proof of Lemma 8.14: By induction on φ.

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x ∈ AF & i ∈ is✸(x, φ)} ∧ ✸deg(φ)⊤ and "{✸iσ(x) : x ∈ var(φ) & i ∈

is✸(x, φ)} ∧ ✸deg(φ)⊤ is a consequence of Lemma 8.14. The

equiva-lence between "{✸i_{σ(x) :} _{x ∈ var(φ) & i ∈ is}

✸(x, φ)} ∧ ✸deg(φ)⊤ and

"

{✸i_{σ(x) : x ∈ var(φ) & i = max is}

✸(x, φ)} ∧ ✸deg(φ)⊤ is a consequence of

the fact that for all i, j ∈ N, if i ≤ j then |= ✸i_{⊤ ∧ ✸}j_{⊤ ↔ ✸}j_⊤.

Proof of Lemma 8.16: Suppose for all x ∈ AF , max is✸(x, φ) =

max is✸(x, φ′) and max is✸(x, ψ) = max is✸(x, ψ′). Let σ be a

closed substitution. By Lemma 8.15, |= σ(φ) ↔ "{✸i_{σ(x) :} _{x ∈}

var(φ) & i = max is✸(x, φ)} ∧ ✸deg(φ)⊤, |= σ(φ′) ↔

"

{✸i_{σ(x) :} _{x ∈}

var(φ′_{) & i = max is}

✸(x, φ′)} ∧ ✸deg(φ

′₎

⊤, |= σ(ψ) ↔ "{✸i_{σ(x) :} _{x ∈}

var(ψ) & i = max is✸(x, ψ)} ∧ ✸deg(ψ)⊤ and |= σ(ψ′) ↔ "{✸iσ(x) :

x ∈ var(ψ′_{) & i = max is}

✸(x, ψ′)} ∧ ✸deg(ψ

′₎

⊤. Since deg(φ) = deg(φ′_),

var(φ) = var(φ′_{), deg(ψ) = deg(ψ}′_{), var(ψ) = var(ψ}′_{) and for all x ∈ AF ,}

max is✸(x, φ) = max is✸(x, φ′) and max is✸(x, ψ) = max is✸(x, ψ′), therefore

|= σ(φ) ↔ σ(φ′_{) and |= σ(ψ) ↔ σ(ψ}′_{). Hence, S ∪ {(φ, ψ)} possesses a unifier}

iff S ∪ {(φ′_{, ψ}′_{)} possesses a unifier.}

Proof of Lemma 8.18: Let S′ _{be the finite set of pairs of minimalist}

formulas obtained by replacing each pair (φ, ψ) in S of formulas by the pair ("{✸i_x _: _{x ∈ var(φ) & i = max is}

✸(x, φ)} ∧ ✸deg(φ)⊤,"{✸ix :

x ∈ var(ψ) & i = max is✸(x, ψ)} ∧ ✸deg(ψ)⊤) of minimalist formulas. By

Lemma 8.16, S possesses a unifier iff S′ _{possesses a unifier. By Lemma 8.17,}

S′ _{can be easily transformed into an equivalent finite set of pairs of minimalist}