Tableau Calculi for the Logics of Finite k-Ary

Trees

Mauro Ferrari

1

, Camillo Fiorentini

1

, and Guido Fiorino

2

1

Dipartimento di Scienze dell’Informazione, Universit`a degli Studi di Milano

Via Comelico, 39, 20135 Milano, Italy

2

CRII, Universit`a dell’Insubria,Via Ravasi 2, 21100 Varese, Italy

Abstract. We present tableau calculi for the logics D

k

(k ≥ 2) seman-

tically characterized by the classes of Kripke models built on ﬁnite k-ary

trees. Our tableau calculi use the signs T and F, some tableau rules for

Intuitionistic Logic and two rules formulated in a hypertableau fashion.

We prove the Soundness and Completeness Theorems for our calculi.

Finally, we use them to prove the main properties of the logics D

k

,in

particular their constructivity and their decidability.

1 Introduction

In recent years there has been a growing interest (see [1,3,4,6,7,8,10,11]) in proof-

theoretical characterization of propositional intermediate logics, that is logics

laying between Intuitionistic and Classical Logic. This interest is motivated by

the applications of some of these logics. As an example we recall Dummett-

G¨odel Logic, studied for its relationship with multi-valued and fuzzy logics [14];

Jankov Logic and here-and-there Logic, studied for their application to Logic

Programming [15,16].

Apart from the cases of Intuitionistic and Classical Logic, the proof-

theoretical characterization of Intermediate Logics given in the literature relies

on variations of the standard sequent calculi or tableau calculi. As an example,

the tableau calculi for the interpolable Intermediate Logics described in [1,10,

11] use new signs besides the usual signs T and F (we remark that the calculi of

[10,11] give rise to space-eﬃcient decision procedures). However, this approach

seems hard to apply to several families of interesting Intermediate Logics. An-

other approach relies on hypersequent calculi, a natural generalization of sequent

calculi; e.g., in [3] a hypersequent characterization of Dummett-G¨odel Logic is

presented, while in [7] the authors extend this approach to some families of

Intermediate Logics with bounded Kripke models. However, also the approach

based on hypersequent calculi or hypertableau calculi (the dualized version of

hypersequents presented in [6]) seems to be inadequate to treat some Intermedi-

ate Logics and further variations are needed. An example is given in [6], where

the notion of path-hypertableau calculus is introduced to treat the intermediate

logic of ﬁnite-depth Kripke models.

Despite the wide research in this ﬁeld, we remark that all the intermedi-

ate logics studied in the above mentioned papers fail to be constructive, where

U. Egly and C.G. Ferm¨uller (Eds.): TABLEAUX 2002, LNAI 2381, pp. 115–129, 2002.

c

Springer-Verlag Berlin Heidelberg 2002

116 M. Ferrari, C. Fiorentini, and G. Fiorino

we call constructive any intermediate logic L satisfying the disjunction prop-

erty: A ∨ B ∈ L implies A ∈ L or B ∈ L. As it is well-known, there exists a

continuum of constructive intermediate logics [5,9], but, as far as we know, no

proof-theoretical characterizations of constructive logics are known, apart from

those given in [2]. In that paper generalized tableau calculi for the constructive

logics D

k

(k ≥ 2) and for the constructive Kreisel-Putnam Logic are presented;

however, such calculi are far from being genuine tableau calculi and are highly

ineﬃcient. Indeed, they are obtained by adding to the intuitionistic tableau cal-

culus a special rule allowing us to introduce, at any point of the derivation, a

suitable T-signed instance of the schema characterizing the logic.

In this paper we provide tableau calculi for the intermediate constructive

logics D

k

(k ≥ 2) of ﬁnite k-ary trees. D

k

is the set of all the formulas valid

in every Kripke model built on a ﬁnite k-ary tree. These logics have been in-

troduced in [13], where a ﬁnite axiomatization of every D

k

is given, and their

decidability is proved. Our proof-theoretical characterization is based on a hy-

brid tableau calculus that uses the two usual signs T and F, some tableau rules

for Intuitionistic Logic and two rules formulated in a hypertableau fashion (a

structural rule and a purely logical rule). Then we use such calculi to provide a

proof of the main properties of the logics D

k

, in particular their constructivity

and their decidability.

The paper is organized as follows: in Section 2 we introduce the logics D

k

providing both the axiomatization and the semantical characterization in terms

of families of Kripke models. In Section 3 we introduce the calculi TD

k

and we

prove that they characterize the logics D

k

. Finally, in Section 4 we use these

calculi to prove the main properties of the logics D

k

.

2 Preliminaries

Here we consider the propositional language based on a denumerable set of

atomic symbols and the logical constants ⊥, ∧, ∨, →. We denote with p,q,...,

possibly with indexes, the atomic symbols and with A,B,..., possibly with

indexes, arbitrary formulas. Moreover, as usual in the setting of intermediate

logics, ¬A is deﬁned as A →⊥. Int and Cl denote respectively the set of

intuitionistically and classically valid formulas.

An intermediate propositional logic (see, e.g., [5]) is any set L of formulas

satisfying the following conditions: (i) L is consistent; (ii) Int ⊆ L; (iii) L is

closed under modus ponens; (iv) L is closed under propositional substitution

(where a propositional substitution is any function mapping every propositional

variable to a formula). It is well-known that, for any intermediate logic L, L ⊆ Cl.

Many intermediate logics can be semantically characterized by families of

Kripke models. A (propositional) Kripke model (see, e.g., [5]) is a structure

K

= P, ≤, , where P, ≤ is a poset (partially ordered set), and (the forcing

relation) is a binary relation between elements of P and atomic symbols such

that, for every atomic symbol p, α p implies β p for every β ∈ P such that

α ≤ β. The forcing relation is extended to arbitrary formulas as follows:

Tableau Calculi for the Logics of Finite k-Ary Trees 117

1. α ⊥;

2. α B ∧ C iﬀ α B and α C;

3. α B ∨ C iﬀ α B or α C;

4. α B → C iﬀ, for every β ∈ P such that α ≤ β, β B implies β C.

We write α A to mean that α A does not hold. We remark that, according

to the above interpretation, α ¬A iﬀ, for every β ∈ P such that α ≤ β,we

have β A.

It is easy to check that the forcing relation meets the monotonicity condition:

Proposition 1. For every formula A, Kripke model K

= P, ≤, and element

α in K

,ifα A then β A for every β ∈ P such that α ≤ β.

Given a Kripke model K

= P, ≤, , we write α<βto mean that α ≤ β

and α = β. Given α ∈ P , we call immediate successor of α (in K

)anyβ ∈ P

such that α<βand, for every γ ∈ P ,ifα ≤ γ ≤ β then either γ = α or γ = β.

We call ﬁnal element of K

any φ ∈ P such that, for every α ∈ P ,ifφ ≤ α then

φ = α. It is easy to check that a ﬁnal element φ of K

behaves like a classical

interpretation, that is, for every formula A, either φ A or φ ¬A.

A formula A is valid in a Kripke model K

if α A for all α ∈ P.IfK is a

non empty class of Kripke models, A is valid in K if it is valid in every model

of K.

For every k ≥ 2, let (D

k

) be the axiom schema

k

i=0

(p

i

→

j=i

p

j

) →

j=i

p

j

→

k

i=0

p

i

and let D

k

denote the closure under modus ponens and propositional substitu-

tion of the set containing Int and all the instances of the axiom schema (D

k

).

As shown in [13] every D

k

(k ≥ 2) is an intermediate logic and the sequence

{D

k

}

k≥2

has the following properties:

– ∩

k≥2

D

k

= Int;

– For every k ≥ 2, D

k

⊃ D

k+1

;

– For every k ≥ 2, D

k

has the disjunction property: that is, for every formula

of the kind A ∨ B,ifA ∨ B ∈ D

k

, then A ∈ D

k

or B ∈ D

k

;

– Every D

k

is decidable.

Now, let T

k

be the the family of all the Kripke models K = P, ≤, where:

– P, ≤ is a ﬁnite tree;

– Given α ∈ P , α has at most k immediate successors in P, ≤.

In [13] the following result is proved:

Theorem 1. For every k ≥ 2, A ∈ D

k

iﬀ A is valid in T

k

.

Hence the above result shows that every logic D

k

is characterized by the class

of ﬁnite k-ary trees.

118 M. Ferrari, C. Fiorentini, and G. Fiorino

The above quoted properties of the logics D

k

are proved in [13] by means of

semantical tools. In particular the decidability of D

k

relies on the decidability

of the second order theory describing the validity of formulas in T

k

and the

disjunction property follows from a property of the class of models T

k

.Inthe

following sections we introduce a tableau calculus for every TD

k

and then we use

the properties of such a calculus to deduce the decidability and the disjunction

property for D

k

.

3 The Sequence of Tableau Calculi TD

k

(k ≥ 2)

A signed formula (swﬀ for short) is an expression of the form TX or FX where

X is any formula. The meaning of the signs T and F is as follows: given a

Kripke model K

= P, ≤, and a swﬀ H, α ∈ P realizes H (in symbols α ✄ H)

if H ≡ TX and α X,orH ≡ FX and α X. α H means that α ✄ H does

not hold. α realizes a set of swﬀ’s S (α ✄ S)ifα realizes every swﬀ in S.By

Proposition 1, if α ✄ TX then β ✄ TX for every β ∈ P such that α ≤ β.Onthe

other hand, if α ✄ FX, then can exist β ∈ P such that α ≤ β and β FX.

A hyperset is an expression of the form

S

1

| ... | S

n

where, for all i =1,...,n, S

i

is a set of swﬀ’s. S

i

is called a component of the

hyperset. We call simple hyperset a hyperset containing exactly one component.

A conﬁguration is an expression of the form

Ψ

1

... Ψ

m

where, for all i =1,...,m, Ψ

i

is a hyperset. Ψ

i

is called a component of the

conﬁguration. A simple conﬁguration is a conﬁguration where every component

is a simple hyperset.

The intended meaning of the symbol | is conjunctive while the one of the

symbol is disjunctive. Formally, given a Kripke model K

= P, ≤, , K

realizes a hyperset S

1

| ... | S

n

if, for every i =1,...,n, there exists α

i

∈ P

such that α

i

✄ S

i

. On the other hand, K realizes a conﬁguration Ψ

1

... Ψ

m

if there exists a hyperset Ψ

j

, with j ∈{1,...,m}, such that K realizes Ψ

j

.

The rules of Table 1 are common to all the calculi TD

k

for k ≥ 2 and are

independent of the parameter k. The rule properly characterizing the tableau

calculus TD

k

is D

k

and it will be introduced in a while. Before that we introduce

some notations. First of all, in the rules of the calculus TD

k

, we simply denote

with S, H

1

,...,H

h

the set S ∪{H

1

,...,H

h

}. The rules apply to conﬁgurations

but, to simplify the notation, we omit the components of the conﬁguration not

involved in the rule. E.g., the schema

S

1

... S, T(A ∧ B) ... S

n

S

1

... S, TA, TB ... S

n

T∧

illustrates an application of the T∧-rule. In every rule we distinguish two parts:

the premise, that is the conﬁguration above the line, and the conclusion, that

is the conﬁguration below the line. We remark that all the rules of Table 1 but

Tableau Calculi for the Logics of Finite k-Ary Trees 119

Table 1. Rules common to all the TD

k

S, T(A ∧ B)

S, TA, TB

T∧

S, F(A ∧ B)

S, FA S, FB

F∧

S, T(A ∨ B)

S, TA S, TB

T∨

S, F(A ∨ B)

S, FA

F∨

1

S, F(A ∨ B)

S, FB

F∨

2

S, TA, T(A → B)

S, TA, TB

T→Atom with A an atom

S, T((A ∧ B) → C)

S, T(A → (B → C))

T→∧

S, T((A ∨ B) → C)

S, T(A → C), T(B → C)

T→∨

S, T((A → B) → C)

S, F(A → B), T((A → B) → C) S, TC

T→→

S

1

| ... | S

i

| ... | S

n

S

i

Weak

Weak only involve simple conﬁgurations (both the premise and the consequence

of such rules are simple conﬁgurations). On the other hand, Weak has a non

simple conﬁguration as premise and a simple conﬁguration as consequence; on

the contrary, as we will see, the rule D

k

properly characterizing TD

k

has a

simple conﬁguration as premise and (in general) a non simple conﬁguration as

consequence.

We call main set of swﬀ’s of a rule the set of swﬀ’s that are in evidence in the

premise of the rule; when the main set of swﬀ’s of a rule contains just a swﬀ we

call it the main swﬀ of the rule. As an example, T(A∧B) is the main swﬀ of the

rule T∧ while {TA, T(A → B)} is the main set of swﬀ’s of the rule T → Atom.

The rule Weak is a structural rule; it acts on components of a hyperset and it

does not have a main set of swﬀ’s.

The rule properly characterizing the calculus TD

k

is D

k

that applies to the

premise

S, F(A

1

→ B

1

),...,F(A

n

→ B

n

), T((C

1

→ D

1

) → E

1

),...,T((C

m

→ D

m

) → E

m

)

Let

U = {F(A

1

→ B

1

),...,F(A

n

→ B

n

)}

V = {T((C

1

→ D

1

) → E

1

),...,T((C

m

→ D

m

) → E

m

)}

and let U ∪ V be the main set of swﬀ’s of the rule. Now, let Σ

U

be the set

containing all the subsets of U diﬀerent from U itself and the empty set. We

remark that the cardinality of Σ

U

is 2

n

− 2. We denote with Σ

k

U

the set of all

the complete k-sequences of Σ

U

, that is the set of all the sequences Φ

1

,...,Φ

h

of elements of Σ

U

such that: