Download On Decidability of Intuitionistic Modal Logics

On Decidability of Intuitionistic Modal Logics
Natasha Alechina
School of Computer Science and IT
University of Nottingham
[email protected]
Dmitry Shkatov
School of Computer Science and IT
University of Nottingham
[email protected]
Abstract
We prove a general decidability result for a class of intuitionistic modal logics. The proof
is a slight modification of the Ganzinger, Meyer and Veanes [6] result on decidability of the
two variable monadic guarded fragment of first order logic with constraints on the guard
relations expressible in monadic second order logic.
1 Introduction
Intuitionistic modal logics arise naturally in many areas, in particular in computer
science. Intuitionistic logic is a natural underlying logic when dealing with functions and types, and modalities may correspond to various computational phenomena like termination etc., see for example [8,3,2].
There is a large literature on intuitionistic modal logics, for example
[7,12,13,10,4,14,15,16]. Formal properties of various flavours of intuitionistic modal
logics have been studied extensively. In this paper, we are interested in general decidability results for intuitionistic modal logics.
In [15,16] general results on decidability and finite modal property of intuitionistic modal logic have been proved using an embedding of intuitionistic modal
logics with n modalities in classical modal logics with n + 1 modalities. Those
results are typically of the form “Suppose that an intermediate logic Int + Γ is
decidable. Then intuitionistic modal logics IntK ⊕ Γ, IntK ⊕ Γ ⊕ 2p → p and
IntK ⊕ Γ ⊕ 3 (where IntK is intuitionistic logic with axioms and rules for
modal system K added) are decidable” [16], Theorem 13.
In this paper we present a result of the form “If an intuitionistic modal logic L is
defined as a set of formulas valid in a class of Kripke models satisfying conditions
C on the intuitionistic and modal accessibility relations, and conditions in C are
of a certain form, then L is decidable.” The proof is a slight generalisation of the
result in [6] and uses a translation into the two variable monadic guarded fragment
of first order logic. Unfortunately, the decidability proof does not give a good
decision procedure since it proceeds by reduction to satisfiability of formulas of
SkS (monadic second-order theory of trees with constant branching factor k, [11])
where decision procedure has non-elementary complexity.
The reason that we concentrate on applying the result of [6] to intuitionistic
modal logic rather than to classical modal logic is that while applicability to systems such as classical S4 or S5 is immediately obvious (in fact, explicitly stated in
[6]), some additional work needs to be done to apply it to the intuitionistic modal
logic.
2 Intuitionistic modal logics
Intuitionistic modal languages are obtained by adding either or both of the unary
connectives 2 (necessity) and 3 (possibility) to the language of propositional intuitionistic logic, which contains a stock of propositional parameters Par = {p1 , p2 , . . .},
a unary connective ∼, and binary connectives ∧, ∨, and ⇒. Analogously to ∀ and
∃, in intuitionistic logic 2 and 3 are not required to be dual.
Semantics of intuitionistic modal logics extends Kripke semantics for intuitionistic propositional logic. An intuitionistic Kripke model is a structure M =
(W, R, V ) such that (i) W = ∅, (ii) R is a reflexive and transitive binary relation
on W , and (iii) V is a function from Par into the powerset of W such that, for
all w ∈ W and p ∈ Par, if w ∈ V (p) and wRv, then v ∈ V (p) (condition we
will refer to as upward persistency for propositional variables). Elements of W are
called nodes. Truth at a node is defined as follows (→ and ¬ stand for classical
implication and negation, respectively):
M, w |= p
iff w ∈ V (p);
M, w |=∼ ϕ
iff ∀v(R(w, v) → ¬(M, v |= ϕ));
M, w |= ϕ ∧ ψ
iff M, w |= ϕ and M, w |= ψ;
M, w |= ϕ ∨ ψ
iff M, w |= ϕ or M, w |= ψ;
M, w |= ϕ ⇒ ψ iff ∀v(R(w, v) → (¬(M, v |= ϕ) or M, v |= ψ);
To accommodate formulas of the form 2ϕ and 3ϕ, intuitionistic Kripke models
are augmented with binary relations R2 and R3 . There is no single accepted way
of defining the meaning of 2 and 3 in intuitionistic logic. The following clauses
2
are encountered in the literature (see chapter 3 of [13] for a comprehensive survey):
(21 ) M, w |= 2ϕ iff ∀v(wR2 v → M, v |= ϕ)
(22 ) M, w |= 2ϕ iff ∀v(wRv → ∀u(vR2 u → M, u |= ϕ))
(31 ) M, w |= 3ϕ iff ∃v(wR3 v ∧ M, v |= ϕ)
(32 ) M, w |= 3ϕ iff ∀v(wRv → ∃u(vR3 u ∧ M, u |= ϕ))
Observe that definition (32 ) gives rise to a modality which does not distribute
over disjunction (hence to a non-normal modal logic).
On top of the requirement that R is reflexive and transitive, some additional
conditions are usually imposed on R, R2 , and R3 . As a rule, these conditions
specify the way R, R2 , and R3 interact. For example, the following conditions
usually accompany truth clauses (21 ) and (31 ) (see [14]):
(1)
R ◦ R2 ◦ R = R2
(2)
−1
R ◦ R−1
3 ◦ R = R3
Another condition occurring in the literature (see for example [5]) stipulates
that
(3)
R3 ⊆ R
In section 4 we will define a large class of conditions on R, R2 , R3 including reflexivity, transitivity, and conditions (1) - (3) above, which following [6] we
call mso closure conditions or closure constraints. We will show that intuitionistic
modal logics defined using those conditions are decidable.
3 Embedding into two-variable monadic fragment
Our decidability proof hinges on the result of [6] that a monadic two-variable
2
of classical first-order logic, where in addition, some
guarded fragment GFmon
relations satisfy conditions that can be expressed as monadic second-order definable closure constraints, is decidable. For our purposes, we need a slightly more
general result, since the decidability proof of [6] does not accommodate conditions
involving several relations (other than equality)—that is such conditions as (1), (2),
and (3)—which are crucial for intuitionistic modal logics.
Our proof consists of two parts. First, we show that every intuitionistic modal
logic L defined semantically with any of the truth clauses (21 ) − (32 ) can be
2
translated into GFmon
. Secondly, we show that if L is in addition the logic of
a class M of Kripke models defined by an acyclic set of mso definable closure
conditions on relations R, R2 , and R3 (some terminology will be defined later)
2
then L is decidable, since GFmon
over such models is (the last result being an easy
generalisation of that of [6]).
2
We start by defining GFmon
. In the following definitions, F V (ϕ) stands for the
set of free variables of ϕ, x stands for a sequence of variables. We assume a first
order language which contains predicate letters of arbitrary arity, including equality
3
=, and no constants or functional symbols.
Definition 3.1 The guarded fragment GF of first-order logic is the smallest set
that contains all first-order atoms and is closed under boolean connectives and the
following rule: if ρ is an atom, ϕ ∈ GF, and x ⊆ F V (ϕ) ⊆ F V (ρ), then ∃x(ρ∧ϕ)
and ∀x(ρ → ϕ) ∈ GF (in such a case ρ is called a guard).
2
Definition 3.2 The monadic two-variable guarded fragment GFmon
is a subset of
GF containing formulas ϕ such that (i) ϕ has no more than two variables (free or
bound), and (ii) all non-unary predicate letters of ϕ occur in guards.
2
Next, we show that intuitionistic modal logics can be translated into GFmon
without equality. To that end, we define, by mutual recursion, two translations, τx
and τy , so that a first-order formula τv (ϕ) (v ∈ {x, y}) contains a sole free variable
v, which intuitively stands for the world at which ϕ is being evaluated in the Kripke
model. τx is defined by
τx (p) := P (x)
τx (∼ ϕ) := ∀y(R(x, y) → ¬τy (ϕ))
τx (ϕ ∧ ψ) := τx (ϕ) ∧ τx (ψ)
τx (ϕ ∨ ψ) := τx (ϕ) ∨ τx (ψ)
τx (ϕ ⇒ ψ) := ∀y(R(x, y) → (¬τy (ϕ) ∨ τy (ψ)))
τx (2ϕ) := ∀y(R(x, y) → ∀x(R2 (y, x) → τx (ϕ)))
τx (3ϕ) := ∀y(R(x, y) → ∃x(R3 (y, x) ∧ τx (ϕ)))
τy is defined analogously, switching the roles of x and y. This translation assumes
modal truth clauses (22 ) and (32 ). Clauses for (21 ) and (31 ) are even simpler
(and familiar from classical modal logic):
τx (2ϕ) := ∀y(R2 (x, y) → τy (ϕ))
τx (3ϕ) := ∃y(R3 (x, y) ∧ τy (ϕ))
Theorem 3.3 Let φ be an intuitionistic modal formula and M be a class of models
of intuitionistic modal logic. Let M ∈ M. Then, M, w |= φ iff M |= τx (φ)[w]
(where M is taken as a model of first order logic with R, R2 , R3 interpreting
R, R2 , R3 ).
2
over M is
From the theorem it follows that if satisfiability problem of GFmon
decidable, then satisfiability problem of intuitionistic modal logic over M is decidable.
It is well known that the guarded fragment is decidable over the class of all first
2
order models ([1]). Decidability of GFmon
over models with reflexive transitive
guards is proved in [6]. From this and from the fact that upward persistency for
2
propositional variables occurring in φ is expressible in GFmon
it follows immediately that basic intuitionistic modal logic (with no conditions connecting R, R2 ,
4
and R3 ) is decidable. The purpose of this paper is to generalise the result of [6] to
include classes of models defined using conditions involving interaction between
R, R2 and R3 .
4 Closure conditions
In this section we define the form of conditions which yield decidable intuitionistic
modal logics.
Definition 4.1 Let W be a non-empty set. A unary function C on the powerset of
W n is a simple closure operator if, for all P, P ⊆ W n ,
(i) P ⊆ C(P) (C is increasing),
(ii) P ⊆ P implies C(P) ⊆ C(P ) (C is monotone)
(iii) C(P) = C(C(P)) (C is idempotent).
An m+1-ary function C on the powerset of W n is a parametrised closure operator
if, given any choice of m relations P1 , . . . , Pm ⊆ W n , it gives rise to a unary
function C P1 ,...,Pm (parametrised by P1 , . . . , Pm ) that is a simple closure operator
on the powerset of W n . (C P1 ,...,Pm may be viewed as the curried version of C.)
Example 4.2 A reflexive, transitive closure operator for binary relations T C(P) is
a simple closure operator.
Example 4.3 A function InclP (P) = P ∪ P is a closure operator parametrised
by P . With P fixed, it gives rise to a simple closure operator.
Definition 4.4 A condition on relation P is a simple closure condition if it can be
expressed in the form C(P) = P, where C is a simple closure operator.
A condition on relation P is a parametrised closure condition if it can be expressed in the form C P1 ,...,Pm (P) = P, where C is a parametrised closure operator.
Example 4.5 Reflexivity-and-transitivity is a simple closure condition, since it can
be expressed in the form T C(P) = P.
Example 4.6 Condition P ⊆ P is a closure condition on P parametrised by P ,
since it can be stated as InclP (P) = P.
Given a set of closure conditions on a collection of relations S, we want to
preclude circularity while closing off relations in S.
Definition 4.7 Let S be a set of relations and C a set of closure conditions on
relations in S. Let us say for P, P ∈ S, that P depends on P if C contains
a parametrised condition of the form C P1 ,...,P ,...Pm (P) = P. A set of closure
conditions C is acyclic, if its associated “depends on” relation is acyclic.
Furthermore, we are not interested in arbitrary closure operators, but only in
those definable in monadic second-order logic. Let ϕ(x1 , . . . , xn )M stand for
the set of n-tuples satisfying ϕ in model M.
5
Definition 4.8 A closure operator C P1 ,...,Pm on n-ary relations is mso (-definable),
if there exists a monadic second-order formula CPP1 ,...,Pm with predicate parameters
P1 , . . . , Pm and P , such that, for any model M and any n-ary formula φ,
C P1 ,...,Pm (φM) = CPP1 ,...,Pm (φ/P ))M.
Example 4.9 Closure operator T C is definable by the mso formula
T CP (z1 , z2 ) = ∀X(X(z1 ) ∧ ∀x, y(X(x) ∧ P (x, y) → X(y)) → X(z2 ))
Example 4.10 Closure operator InclP is definable by the mso (in fact, first-order)
formula
InclPP (z1 , z2 ) = P (z1 , z2 ) ∨ P (z1 , z2 )
2
Theorem 4.11 Let φ ∈ GFmon
and C be an acyclic set of mso closure conditions
on relations in φ so that at most one closure condition is associated with each
relation. It is decidable whether φ is satisfiable in a model satisfying C.
2
Proof. See appendix A.
Theorems 3.3 and 4.11 give us our main theorem:
Theorem 4.12 Let M be a class of intuitionistic modal models defined by an acyclic
set of mso closure conditions on R, R2 , and R3 so that at most one closure condition is associated with each relation, and let φ be an intuitionistic modal formula.
Then, it is decidable whether φ is satisfiable in M.
5 Examples
In this section, we state several decidability results just to illustrate our approach.
The first example is by no means a surprise, although we doubt if anyone has
proved this for all possible combinations of truth definitions for modalities. Essentially this is decidability of several flavours of basic intuitionistic modal logic (no
conditions on the modal accessibility relation).
Example 5.1 An intuitionistic modal logic L with two modalities 2 and 3, defined
by a class of models where
−1
R ◦ R−1
3 ◦ R = R3
R ◦ R2 ◦ R = R2
and employing any of the truth definitions for modalities (21 ), (22 ), (31 ), (32) (in
any combination, e.g. (21 ) with (32 ); possibly with more modalities provided all
2
), is decidable.
truth definitions are translatable in GFmon
Proof. The class of models of L is defined by the following closure conditions on
R2 , R3 and R:
(i) R is reflexive and transitive;
−1
(ii) R ◦ R−1
3 ◦ R = R3
6
(iii) R ◦ R2 ◦ R = R2
There is clearly at most one condition for each of the relations R, R3 and R2 , and
the set of conditions is acyclic. We have shown in Examples 4.2 and 4.5 that the
condition on R is a closure condition and in Example 4.9 that it is mso-definable. It
remains to be shown that conditions on R2 and R3 correspond to an mso-definable
closure conditions.
Consider a function CompP (P) = P ◦ P ◦ P . Assume that P is reflexive
and transitive. Then P ⊆ P ◦ P ◦ P by the reflexivity of P ; P ◦ P ◦ P is
obviously monotone in P; and CompP is idempotent because of the transitivity
of P . This proves that CompP is a closure operator provided that P is reflexive
and transitive. Conditions of the form P ◦ P ◦ P = P can be expressed as
closure conditions: CompP (P) = P. In particular, our condition on R2 can be
expressed as CompR (R2 ) = R2 and the condition on R3 can be expressed as
−1
CompR (R−1
3 ) = R3 . The condition is also mso-definable; in fact, it is definable
by a first order formula:
CompPP (z1 , z2 ) = ∃x∃y(P (z1 , x) ∧ P (x, y) ∧ P (y, z2))
We have shown that the class of models of L conforms to the conditions of
Theorem 4.12 which proves that L is decidable.
2
The next example looks like a known result (decidability of P LL [5]), but for
a slightly different logic (without fallible worlds):
Example 5.2 An intuitionistic modal logic L with one modality 3, defined by a
class of models where
R3 is reflexive and transitive;
R3 ⊆ R
and employing the truth definition (32 ) for the modality, is decidable.
Proof. The class of models of L is defined by the following closure conditions:
(i) T C(R3 ) = R3 ;
(ii) T C(R) = R;
(iii) InclR3 (R) = R (see Examples 4.3 and 4.6).
This set of conditions is acyclic and each condition is mso definable. However
there are two constraints associated with R: it is required to be closed both with
respect to T C and to InclR3 . To satisfy the conditions of Theorem 4.12 we need
to combine them into one mso definable closure condition. Luckily, T C ◦ InclP
is a closure operator with the property that for any relation P,
T C(InclP (P)) = P ⇔ T C(P) = P and InclP (P) = P 1
1
In general, for any two closure operators C1 and C2 there is a closure operator C3 such that, for
7
so the conditions can be reformulated as
(i) T C(R3 ) = R3 ;
(ii) T C(InclR3 (R)) = R;
and it is straightforward to show that the second condition is mso definable.
2
Finally, a non-example: we failed to reformulate the condition R2 ◦R ⊆ R◦R2
defining an intuitionistic modal logic in [9] as a closure condition.
6 Conclusions
We have described a general method for proving decidability of an intuitionistic
modal logic by translating it into monadic GF 2 and showing that conditions on
the intuitionistic and modal accessibility relations can be expressed using mso definable closure operators. We illustrate this method by showing that it works for
various truth definitions for modalities and various conditions on the intuitionistic
and modal accessibility occurring in the literature. Most of the decidability results
for particular logics obtained as illustrations of our proof are already known, but
we believe that our method can easily yield new results, especially for logics with
non-normal modalities defined using the truth definition (32 ) which are less well
studied. Obviously, the same method works for intuitionistic logic with more than
two modalities, provided all truth definition are translatable in GFmon 2 .
Acknowledgement
This work was supported by the EPSRC grant GR/M98050/01.
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any relation P,
C3 (P) = P iff C1 (P) = P and C2 (P) = P.
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also be defined by giving a single condition on P.
8
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transitive relations, in: Proc. 14th IEEE Symposium on Logic in Computer Science
(1999), pp. 24–34.
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(1976), pp. 31 – 42, 21 – 52.
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(1991), pp. 55–92.
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semantics for constructive modal logics, in: Proceedings CSL’01 (2001), pp. 292–307.
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editor, Theoretical Aspects of Reasoning about Knowledge, 1986.
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Transactions of the American Mathematical Society 141 (1969), pp. 1–35.
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pp. 533–546.
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A Proof of Theorem 4.11
This appendix contains the proof of theorem 4.11, which is a slight generalisation
of the proof of Theorem 4 from [6]. In that proof all relations are assumed to be
closed under equivalence (to show decidability of the fragment with equality). Note
that closure under equivalence is a special case of a parametrised closure constraint
so we do not need to treat it separately here.
2
Let φ ∈ GFmon
and let C be an acyclic set of mso closure conditions on relations in φ. φ is satisfiable in a model satisfying C iff N, the Skolemised form of φ,
is satisfiable in a Herbrand model in which all conditions from C hold. The idea of
the decidability proof is to reduce the latter problem to satisfiability of formulas of
SkS (mso theory of trees with constant branching factor k), where k is the number
of Skolem function symbols in N. We construct an mso formula MSON , in the
vocabulary of SkS, such that MSON is satisfiable iff N has a Herbrand model
satisfying closure conditions from C. The construction proceeds in three stages.
9
Stage 1. For each predicate P in N, construct formula ϕP , as follows. Let
P (t1 ), . . . , P (tm ) be all positive literals of N containing P (each ti is either ti , or
ti1 , ti2 , depending on the arity of P ). Let t[s] be the result of substituting a term s for
the free variable of t (note that positive literals contain at most one free variable).
First, for each of the above P (ti ), we introduce a new unary second-order variable
XP (ti ) . Second, for a binary predicate P , we put
ϕP (z1 , z2 ) =
m
∃z(XP (ti ) (z) ∧ z1 = ti1 [z] ∧ z2 = ti2 [z])
i=1
(for unary predicates, just ignore z2 ). Intuitively, the relation defined by ϕP is
the minimal extension of P . Next, for each predicate that has a closure condition
imposed on it, we define the closure ψP of ϕP with respect to the closure condition
on P .
For each such P we have a single closure condition CP , which may be parametrised
by other predicates. For simplicity, assume that CP is parametrised by a single
predicate P that, in its own turn, has a simple closure condition CP . We know,
then, that CP is definable by an MSO formula CP (z1 , z2 ) containing P and CP
is definable by an MSO formula CPP (z1 , z2 ), containing P and P . First, we define
the closure of P with respect to its simple closure condition:
ψP (z1 , z2 ) = CP (z1 , z2 )[ϕP /P ]
that is, we replace every occurrence of P in CP (z1 , z2 ) with ϕP .
Next, we define the closure of P with respect to its parametrised condition:
ψP (z1 , z2 ) = CPP (z1 , z2 )[ψP /P , ϕP /P ]
In general, for any acyclic set C of conditions on the collection of relations S,
we first define the simple closures, then the closures parametrised by relations with
simple closure conditions, etc. The acyclicity of C ensures that this procedure can
be carried out.
Stage 2. For each clause χ = {ρ1 , . . . , ρl } in N (which is thought of as a
conjunction of clauses), we construct formula MSOχ , as follows. First, for every
 literal ρ in χ, we define formula MSOρ , according to the rule MSOρ =


X (x), if ρ is a non-ground atom containing x

 ρ
∃zXρ (z), if ρ is a ground atom



 ¬ψ (t), if ρ is ¬P (t)
P
where ψP is the formula constructed at stage 1. Secondly, we put MSOχ =
ρ∈χ MSOρ .
Stage 3. Lastly, let MSON = ∃X∀x χ∈N MSOχ .
Having defined MSON , we now prove that N has a Herbrand model satisfying
the closure conditions in C iff MSON is satisfiable in a tree. We can assume,
10
without loss of generality, that the nodes of a tree are terms of a Herbrand universe
of N and the ith successor of node s is node fi (s). Let’s denote this tree by T .
(⇐) Assume that N has a Herbrand model A satisfying the closure conditions
in C. We want to show that T satisfies MSON . Fix witnesses for second-order
variables Xρ of MSON as follows:
(i) If ti is non-ground, then XP (ti ) = {a : A |= P (ti [a])}.
(ii) If ti is ground, then XP (ti ) is a non-empty set.
We know that for each clause χ of N, and each tuple a, A |= χ(a). This
means that for each a, there is a literal ρ in χ such that A |= ρ(a). We show that
for any a and ρ, if A |= ρ(a), then T |= MSOρ (a). Hence A |= χ(a) implies
T |= MSOχ (a).
We have three cases to consider, depending on the form of ρ. The first two
(non-ground atom P (ti ) and ground atom) are exactly the same as in [6]. If ρ
is a negative literal ¬P (ti ), we need to show that T |= ¬ψP (t)(a). It suffices
to show that ψP A ⊆ P A . Indeed, this, together with our assumption that A |=
¬P (t)[a], implies T |= ¬ψP (a). First, the definition of T guarantees that ϕP A ⊆
PA
PA
P A . Hence, by monotonicity of closure operators, CP 1 (ϕP A ) ⊆ CP 1 (P A ). By
PA
definition of ψP , CP 1 (ϕP A ) = ψP A ; furthermore, since A satisfies conditions
PA
in C, CP 1 (P A ) = P A ; hence, ψP A ⊆ P A .
(⇒) Assume that MSON is true in T . We have to find a Herbrand model A
satisfying conditions in C where N is true. Define A as follows. The universe
of A is the set of nodes of T , and P A = ψP . First, we prove that A satisfies
closure conditions C. To this end, we have to show that CPP1 (P A) = P A . Indeed,
ψP ψP ψP PA
PA
CP 1 (P A ) = CP 1 (ψP ) = CP 1 (CP 1 ( ϕP )) = CP 1 (ϕP ) = ψP =
P A.
Second, we need to show that A satisfies all clauses in N. This part of the proof
is exactly the same as in [6].
11