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Book ChapterDOI

Tableau Calculi for the Logics of Finite k-Ary Trees

30 Jul 2002-pp 115-129

TL;DR: Tableau calculi for the logics Dk (k ? 2) semantically characterized by the classes of Kripke models built on finite k-ary trees are presented and the Soundness and Completeness Theorems for these calculi are proved.
Abstract: We present tableau calculi for the logics Dk (k ? 2) semantically 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 Dk, in particular their constructivity and their decidability.
Topics: Soundness (57%), Intuitionistic logic (57%), Completeness (logic) (55%), Decidability (55%)

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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-
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
}
k2
has the following properties:
k2
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
TAtom 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(AB) 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:

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