On the Finite Model Property of Intuitionistic Modal ... - Semantic Scholar
On the Finite Model Property of Intuitionistic Modal Logics over MIPC? January 30, 1998 Takahito Aoto1 and Hiroyuki Shirasu2 ??
School of Information Science, JAIST Tatsunokuchi, Ishikawa 923-12, Japan E-mail: [email protected] 2 Semantic Group, ETL Mail box 1503, Umezono 1-1-4, Tsukuba, Ibaraki 305, Japan E-mail: [email protected] 1
Abstract. It is shown that every normal intuitionistic modal logic L over MIPC has the nite model property if there exists a universal rstorder sentence 8 such that (1) L is characterized by the class of Kripke frames satisfying 8 and (2) every Kripke frame that validates L satis es 8. Here, MIPC is a well-known intuitionistic modal logic introduced by Prior (1957).
1 Introduction In this paper, we study the nite model property of (propositional) modal logics based on intuitionistic logic|we call them intuitionistic modal logics|with a necessity operator 3 and a possibility operator }. In particular, we focus on those of logics over MIPC in the lattice of 3}-normal intuitionistic modal logics. We show that every 3}-normal intuitionistic modal logic L which includes MIPC has the nite model property if there exists a universal rst-order sentence 8 such that (1) L is characterized by the class of Kripke frames satisfying 8 and (2) every Kripke frame that validates L satis es 8. MIPC is an intuitionistic modal logic introduced by Prior [12], and has been studied by several authors [1][2][3][6][7][8][10][11]. MIPC is one of the S5-like intuitionistic modal logics; that is, if we add p _ :p to MIPC then we get the well-known classical modal logic S5. Another feature of MIPC is that one can obtain an embedding of MIPC into intuitionistic predicate logic by interpreting 3 as 8 and } as 9. Thus, one-variable fragments (i.e. the sets of formulas with ? The
preliminary version of this paper was presented at the workshop
Logic, European
Topics in Polymodal
Summer School in Logic, Language, and Information (ESSLLI'97), August
1997.
?? Supported
by the Center of Excellence (COE) budget of the Science and Technology
Agency (STA) Japan.
one xed variable) of intuitionistic predicate logic can be interpreted in MIPC (see [1], [11]). We also note that MIPC is an extension of FS (Fisher Servi's logic) introduced in [8]. For recent surveys on intuitionistic modal logics we refer the reader to [13], [15]. The nite model property of a logic L ensures the decidability of L when L is nitely axiomatizable, and hence has been one of the central focuses in the study of logics. The nite model property of MIPC was shown in [7], [10] and that of FS in [5]. Several general results on the nite model property of extensions of MIPC are also obtained in [1], [2]. The rest of the paper is organized as follows. In Section 2, we introduce basic notions and x some notation used in this paper. In Section 3, we present a construction of a nite counter MIPC-model for from a given counter MIPC-model for based on a descriptive MIPC-frame. Our main theorem is given at the end of Section 3. We also give sucient conditions of extensions of MIPC to have the nite model property in Section 4. We conclude our result in Section 5.
2 Preliminaries Familiarity with the basic notions of modal logic (as expounded in e.g. [4]) will be helpful in what follows. Let L be the propositional language consisting of in nitely many propositional variables p; q; r; : : : and the usual primitive connectives !; ^; _; ?. (We put > def = ? ! ?.) The set of L-formulas is denoted by For L. A set L For L is called a superintuitionistic logic (abbrev. si-logic) if L includes intuitionistic logic Int and is closed under substitution and modus ponens. For a set 0 For L, the smallest si-logic that includes 0 is denoted by Int + 0; when 0 = f g, Int + 0 will be denoted by Int + . An intuitionistic (general) frame (Int-frame, for short) is a structure hW; R; P i where R is a partial order on W and P is a set of R-cones in W that contains ; and is closed under \; [ and the operation:
def = fx 2 W
z ((xRz and z 2 ) imply z 2 )g:
j 8
Here, W is an R-cone if 8x; y (x 2 and xRy imply y 2 ). It is wellknown (see e.g. [4]) that every Int-frame F = hW;R; P i gives rise to its dual Heyting algebra F+ = hP; \; [; ; ;i; and conversely, every Heyting algebra A = hA; ^; _; !; ?i to its dual Int-frame A+ = hWA ; ; PA i where WA is the set of prime lters in A and PA = fjaj j a 2 Ag with jaj = fX 2 WA j a 2 X g; and that A ' (A+ )+. An Int-frame F is descriptive if F ' (F+)+ . A full Int-frame (or a Kripke Int-frame) is an Int-frame of the form hW; R; Up W i. Here, Up W is the set of all R-cones in W . For an Int-frame F = hW; R; P i, its underlying full Int-frame hW; R; Up W i is denoted by F. An Int-frame F is a 2
frame for a si-logic L when F j= L. A si-logic L is D-persistent if F j= L holds for any descriptive frame F for L.
Let L3} be the propositional bimodal language, i.e. the language L with new unary connectives 3 and }. The sets of (L3} -)variables and (L3} -)formulas are denoted by Var L3} and For L3} , respectively. For a formula , Sub( ) is the set of subformulas of . A set L For L3} is called a 3}-normal intuitionistic modal logic (a 3}-IM-logic, for short) if it includes Int, contains axioms 3(p ^ q ) $ 3p ^ 3q , }(p _ q) $ }p _ }q , 3> and :}?, and is closed under substitution, modus ponens and the monotonicity rules: ! =3 ! 3 and ! =} ! } . We denote by IntK3} the smallest 3}-IM-logic. For a set 0 For L3} and a 3}-IM-logic L, the smallest 3}-IM-logic that includes L and 0 is denoted by L 8 0; when 0 = f g, L 8 0 will be denoted by L 8 . We set MIPC def = IntK3} 8 f3p ! p; p ! }p; }3p ! 3p; }p ! 3}p; }(p ! q) ! (3p ! }q)g. A 3}-IM-logic that includes MIPC is called an MIPC-logic. An MIPC-algebra is a structure A = hA; ^; _; !; ?; 3; }i where hA; ^; _; ! ; ?i is a Heyting algebra and 3; } are unary operations on A such that 3(a^b) = 3a ^ 3b, 3> = >, }(a _ b) = }a _ }b, :}? = >, 3a a, a }a, }3a 3a, }a 3}a and }a ^ }b }(}a ^ b) for any a; b 2 A. It is straightforward to see that for any MIPC-logic L, there exists a class C of MIPC-algebras such that 2 L if and only if 8A 2 C (A j= ); such C is called a class of MIPC-algebras that characterizes L. An (general) MIPC-frame is a structure F = hW; R; R3 ; R} ; P i where (1) hW; R; P i is an Int-frame; (2) R3 ; R} are binary relations on W such that R3 is re exive and transitive, R} = (R3 )01 and R3 = R (R3 \ R} ); and (3) P is closed under the operations:
3 = x W y (xR3 y implies y ) ; f
=
}
2
x2W
f
j 8
2
g
y 2 (xR} y)g
j 9
(see [10], [11]). Here and hereafter, denotes the composition of relations. The following lemma is well-known (see e.g. [4]).
Lemma 2.1 Let A = hA; ^; _; !; ?i be a Heyting algebra. Suppose a 2 A and is a lter in A such that a 2= . Then there exists a prime lter r in A which includes and does not contain a such that r is maximal in the set f j is a lter in A such that and a 2 = g. Using this lemma, it is not hard to show that, for any MIPC-frame F = + def hW; R; R3 ; R} ; P i, F = hP; \; [; ; ;; 3; }i is an MIPC-algebra; that, for any MIPC-algebra A = hA; ^; _; !; ?; 3; }i, A+ def = hWA ; ; R^ 3 ; R^ } ; PA i is an MIPC-frame, where (1) WA is the set of prime lters in A, (2) R^3 if and only if 3a 2 implies a 2 for any a 2 A, (3) R^ } if and only if a 2 implies 3
a 2 for any a 2 A, and (4) PA = fjaj j a 2 Ag with jaj = fX 2 WA j a 2 X g; and that A ' (A+ )+. An MIPC-frame F is descriptive if F ' (F+)+ . Thus, for any MIPC-logic L, there exists a class C of descriptive MIPC-frames such that 2 L i 8F 2 C (F j= ); such C is called a class of descriptive MIPC-frames that characterizes L. A full MIPC-frame (or a Kripke MIPC-frame) is an MIPC-frame of the form hW; R; R3 ; R} ; Up W i. For an MIPC-frame F = hW; R; R3 ; R} ; P i, we denote by F its underlying full MIPC-frame hW;R; R3 ; R} ; Up W i. An MIPC-frame F is a frame for an MIPC-logic L when F j= L. An MIPC-logic L is D-persistent if F j= L holds for any descriptive MIPC-frame F for L. It is straightforward to show that MIPC is D-persistent (see [14]). }
3 Constructing a nite counter MIPC-model First, we review some basic notions on MIPC-models, which will be used in this section. A valuation on an MIPC-frame F = hW; R; R3 ; R} ; P i is a mapping from Var L3} to P . An MIPC-model is a pair M = hF; V i where F is an MIPCframe and V is a valuation on it. M = hF; V i is said to be an MIPC-model based on F. Let M = hF; V i be an MIPC-model based on F = hW; R; R3 ; R} ; P i. For x 2 W and 2 For L3} , we de ne a relation hM; xi j= by induction on the construction of : hM; xi j= p i x 2 V (p); hM; xi j= ^ i hM; xi j= and hM; xi j= ; hM; xi j= _ i hM; xi j= or hM; xi j= ; hM; xi j= ! i 8y 2 W (xRy and hM; yi j= imply hM; xi j= ); hM; xi j= 3 i 8y 2 W (xR3 y implies hM; yi j= ); hM; xi j= } i 9y 2 W (xR} y and hM; yi j= ): When no confusion occurs, hM; xi j= will be abbreviated as x j= . A formula is valid in an MIPC-model M (notation: M j= ) if 8x 2 W (hM; xi j= ); otherwise is refuted in M (or M is a counter model for ). For a set L of formulas, M j= L stands for 8 2 L (M j= ). For an MIPCmodel M and an MIPC-logic L, M is an MIPC-model of L when M j= L. An MIPC-frame F is an MIPC-frame for MIPC-logic L (notation: F j= L) if M j= L for any MIPC-model based on F. If F = hW; R; R3 ; R} ; P i then M = hF; V i is said to be nite if the cardinality of W (notation: jW j) is nite. An MIPC-logic L has the nite model property if, for any 2= L, there exists a nite MIPC-model of L that refutes . We begin our study with showing basic properties of an MIPC-model. We ~ if xRy and not yRx. write xRy 4
De nition 3.1 Let M = hF; V i be an MIPC-model based on an MIPC-frame W; R; R3 ; R} ; P i and 2 For L3} . Then y 2 W is said to be R-maximal ~ implies z j= ). The set of relative to if and only if y j= = and 8z 2 W (y Rz all R-successors of x that are R-maximal relative to is written as R-max(x; ).
h
Lemma 3.2 Let M = hF; V i be an MIPC-model based on a descriptive MIPCframe hW; R; R3 ; R} ; P i. For any 2 For L3} and x 2 W such that x j= = , R-max(x; ) is non-empty. Proof. Since F is descriptive, without loss of generality, we can assume F = A+ for some MIPC-algebra A. Then there exists a 2 A such that V ( ) = jaj. Note that every y 2 W is a prime lter in A and it holds that y j= i a 2 y. Suppose that x j= = , i.e. a 2= x. Put := x. Then by Lemma 2.1, we obtain a prime lter r satisfying a 2= r and x r such that, for any lter y, r ( y implies a 2 y . Since R = in A+ , xRr and r is R-maximal relative to .
It is easy to see that R3 \ R} is an equivalence relation. In the sequel, we abbreviate R3 \ R} as . One easily checks that hM; xi j= } if and only if 9y 2 W (x y and hM; y i j= ). For x 2 W , the -equivalence class that contains x is written as [x]. For W , is said to be a precluster if [x] for some x 2 W ; and when this is the case, we write [ ] for [x].
Lemma 3.3 Let M = hF; V i be an MIPC-model based on an MIPC-frame hW; R; R3 ; R} ; P i. Let be a formula of the form 3 or } for some 2 For L3} . Then for any x; y 2 W with x y , we have x j= i y j= . Proof. Easy.
Thus for any precluster and x; y 2 , we have that x j= 3 i y j= 3, and that x j= } i y j= } for any 2 For L3} . We write j= 3 and j= } when x j= 3 and x j= } for some (and hence any) x 2 , respectively.
Lemma 3.4 Let M = hF; V i be an MIPC-model based on a descriptive MIPCframe hW; R; R3 ; R} ; P i. Suppose that x 2 W is R-maximal relative to 3 . Then for any y 2 [x], we have R-max(y; 3 ) [x]. Proof. Suppose that y x = z and z 2 R-max(y; 3 ). Since x yRz, there ~ . exists w 2 W such that xRw z . Since w z = x, w 2= [x]. Hence xRw Then by R-maximality of x, we have w j= 3 . On the other hand, z j= = 3 by R-maximality of z . But this contradicts Lemma 3.3.
Thus for any -equivalence class that contains an element that is Rmaximal relative to 3 , we have R-max(x; 3 ) for any x 2 ; when this is the case, is also said to be R-maximal relative to 3 . 5
Lemma 3.5 Let M = hF; V i be an MIPC-model based on a descriptive MIPCframe hW; R; R3 ; R} ; P i. For any -equivalence class that is R-maximal relative to 3 , there exists y 2 such that y j= = .
Proof. By assumption, there exists x 2 that is R-maximal relative to 3. Since x j= = 3 , there exists y; z 2 W such that xRz y and y j= = . Suppose ~ . By R-maximality of x, we have z j= 3 . y 2= . Then x = z , and so xRz Together with z y, this implies y j= , which is not the case.
De ne a relation on the set of -equivalence classes by 9y 2 (xRy ). Then we obtain the following.
i 9x
2
Lemma 3.6 The relation on the set of -equivalence classes de ned as above is a partial order, i.e. it is a re exive, transitive and antisymmetric relation. Proof. Easy.
Note that if and only if 8x ~ ). < then 8x 2 9y 2 (xRy
2
8y
2
(x(R )y). Moreover, if
We are now going to construct a nite counter MIPC-model M for from a given MIPC-model M that refutes 2 For L3} . To this end, we x 2 For L3} and MIPC-model M = hF; V i based on a descriptive MIPCframe F = hW; R; R3 ; R} ; P i such that M j= = . We will nd a suitable subset W of W , based on which the counter model M will be constructed. For this, we de ne some procedures which act on the subsets X; Y and Z of W satisfying the constraints: all X; Y and Z are nite; X is a precluster; [z ] 2 [y ] for any y 2 Y and z 2 Z ; [X ] 2 [y] for any y 2 Y ; [z ] 2 [X ] for any z 2 Z .
(1) (2) (3) (4) (5)
Roughly speaking, our construction is carried out by \completing" preclusters one by one in an ascending order. During the construction, X is the precluster that we are focusing now, Y is the set of already selected points consisting of completed preclusters, and Z is a set of points that we should think of later (see Figure 1). Note that it follows from the above constraints that [X ]; f[y ] j y 2 Y g and f[z ] j z 2 Z g are pairwise disjoint.
De nition 3.7 COP Y (X; Y ) is a procedure that updates X as follows: Let Y 0 = fy 2 Y j y is maximal w.r.t. R in Y , 9x 2 X (y ( R)x); and not 9x 2 X (yRx)g: 6
1100 01 11001100 11001100 10 00 11 0011 00 0011 11 11001100X 1100 11 00 11 00 01 11001100 101100 101100110011000011 10101100110011001100001101 1100 0011 11001100110011001100 1100101000111010 00 11 11 1100110000111100 Y 1100 00 11001100 001111001100 00110011 Z
Figure 1: X , Y , Z during the construction Then, by the de nition of an MIPC-frame, for each y 2 Y 0, we can take xy 2 [X ] such that yRxy . We put X := X [ fxy j y 2 Y 0g. De nition 3.8 SAT 3 (X; Z ) is a procedure that updates X and Z as follows: Let 0 = f3 2 Sub j [X ] is R-maximal relative to 3 g: Then by Lemma 3.5, for each 3 2 0, we can take x 2 [X ] such that x j= = . We put X := X [ fx j 3 2 0g. Let
6 = f3 2 Sub j [X ] j= = 3 and [X ] is not R-maximal relative to 3g: Fix an arbitrary x 2 X . By Lemma 3.2, for each 3 2 6, we can take x 2 Rmax(x; 3 ). Note that x 2= [X ] by Lemma 3.5. We put Z := Z [ fx j 3 2 6g.
De nition 3.9 SAT } (X; Y ) is a procedure that updates X as follows: Let 0 = f} 2 Sub j [X ] j= } and 8x 2 X (x j= = )g: Then by de nition, for each } 2 0, we can take x 2 [X ] such that x j= . We put X := X [ fx j } 2 0g. De nition 3.10 SAT ! (X; Z ) is a procedure that updates X and Z as follows: First, we de ne sequences X0 ; X1; : : : and Z0; Z1 ; : : : by induction. Let X1 = X
7
and X0 = Z0 = Z1 = ;. Suppose Xk and Zk (k n + 1) are already de ned. We de ne Xn+2 and Zn+2. For each x 2 Xn+1nXn, let
0x = f ! 2 Sub j x j= = ! and x is not R-maximal relative to ! g: Then by Lemma 3.2, for each ! 2 0x , we can take x ! 2 R-max(x; ! ). We let
Xn+2 = (fx ! j x 2 Xn+1nXn and ! 2 0x g \ [X ]) [ Xn+1 Zn+2 = fx ! j x 2 Xn+1nXn and ! 2 0x gn[X ]: Finally, put X :=
S
S
2 Xn and Z := Z [ n2! Zn. Lemma 3.11 Updated X , Y and Z by the above procedures again satisfy the constraints (1) (5). n !
Proof. Let us denote the updated X , Y and Z by X^ , Y^ and Z^ , respectively. 1. COP Y (X; Y ). Since fxy j y 2 Y 0g [X ], X^ is again a precluster and [X ] = [X^ ]. Accordingly (2) (5) hold. By the assumption that Y is nite, Y 0 is nite. Together with niteness of X , it follows that X^ is nite. Thus (1) is also satis ed. 2. SAT 3 (X; Z ). First, we show that the rst part of the procedure preserves the constraints. Since fx j 3 2 0g [X ], X^ is again a precluster and [X ] = [X^ ]. Accordingly (2) (5) hold. Since Sub is nite, 0 is nite. Together with niteness of X , it follows that X^ is nite. Thus (1) is also satis ed. Next, we show the latter part of the procedure preserves the constraints. Since Sub is nite, 6 is nite. Together with niteness of Z , it follows that Z^ is nite. Thus (1) is satis ed. Clearly, it remains to show (3) and (5). Since xRx and x 2= [X^ ], we have [X^ ] < [x ], and hence (3) and (5) hold. 3. SAT } (X; Y ). Similar to the rst part of SAT 3 (X; Z ).
4. SAT ! (X; Z ). First, we show that there exists some l > 0 such that Xl+1nXl = ;. Suppose contrarily that Xn+1 nXn = ; for all n > 0. Then for any k 2 !, there exists R-chain x1Rx2R 1 1 1 xk of length k such that, for any i (k > i > 0), there exists ! 2 Sub satisfying (a) xi j= = ! , (b) xi is not R-maximal relative to ! , and (c) xi+1 2 R-max(xi ; ! ). 8
Let
x_ = f ! 2 Sub j x j= ! or x is R-maximal relative to ! g: Then it follows that x_ i ) x_ i+1 for any i (k > i > 0). But since each j x_ i j is bounded by j Sub j, there is no R-chain longer than j Sub j, which is not the case. Thus there exists l > 0 such that Xl+1nXl = ;. Then by construction, it follows that Xl = Xl+1 = 1 1 1 and ; = Zl+1 = Zl+2 = 1 1 1. Clearly, each Xn and Zn is nite, hence so are X^ and Z^. Thus (1) is satis ed. (2) is also satis ed, since Xi [X ] for each i 2 !. Clearly, it remains to show (3) and (5). We show for any z 2 Z^ nZ , [X^ ] < [z]. Without loss of generality, we suppose z 2 Zn+2. Then there exists x 2 Xn+1 nXn such that xRz . Together with z 2= [X ], we have [X^ ] = [x] < [z ]. The constraints (3) and (5) immediately follow from this. We are now going to present the construction of model M .
De nition 3.12 Let 2 For L3} , and suppose M = hF; V i is a counter MIPC-model for based on a descriptive MIPC-frame F = hW; R; R3 ; R} ; P i. We construct a model M as follows: First, take any point x0 2 W such that x0 j= = (such x0 always exists since M is a counter MIPC-model for ). Next, we de ne sequences U0 ; U1 ; : : : and V0 ; V1 ; : : : by induction. Put U0 = ; and V0 = fx0g. Suppose Uk and Vk are already de ned. Step 1. Select an element X in Vk = such that [X ] is minimal w.r.t. in f[v ] j v 2 Vk g, and put Y := Uk and Z := Vk nX . Step 2. Update X by COP Y (X; Y ). Step 3. Update X and Z SAT 3 (X; Z ).
Step 4. Update X by SAT } (X; Y ). Step 5. Update X and Z by SAT ! (X; Z ).
And then de ne Uk+1 = USk [ X and Vk+1 = Z . Finally, de ne W = n2! Un and M = hF ; V i, where
F = W ; RW ; R3 W ; R} W ; Up W ; h
i
V (p) = V (p) \ W for any p 2 Var L3} : 9
Lemma 3.13 Above construction is well-de ned. Moreover, for any X W =, there exists some l such that X = Ul+1 nUl (hence X is nite).
2
Proof. We show by induction on n that
i. Un and Vn are nite, ii. [u] 2 [v] for any u 2 Un and v 2 Vn . Base step is obvious. For the induction step, suppose i and ii are satis ed for n = k. Then the precluster X selected in step 1 is nite, since Vk is nite. Also, it is clear that X , Y and Z set in step 1 satisfy the constraints (1) (5) by induction hypothesis. In step 2 { 5, by Lemma 3.11, updated X , Y and Z also satisfy the constraints (1) (5). Finally, by the constraints (1) (5), it is easy to see i and ii are again satis ed by Uk+1 and Vk+1 . Now, by the constraints (3) (5) and ii, every element of W = is Ul+1 nUl for some l; it is nite by (1).
Lemma 3.14 M is a nite MIPC-model. Proof. We will show that W is nite. Let Msub = f 2 Sub j } for some 2 For L3} g. For each 0 Msub , let W (0) = fx 2 W j for any 2 Msub , x j= i 2 0g. Note, by Lemma 3.3, for any precluster in W there exists a unique 0 Msub such that W (0). Thus, it suces to show W (0) is nite for each 0 Msub , since P (Msub ) is nite. We show this by induction on j0j. Suppose for any 6 Msub , j6j < j0j implies that W (6) is nite. This means, in particular, for any 6 ( 0, W (6) is nite. Since every precluster in W (0) is nite by Lemma 3.13, it suces to show that W (0)= is nite. Moreover, since W (0)= is a poset and there is no in nite descending chain in it by the construction, it suces to show that there are neither in nite ascending chain nor in nite antichain in W (0)=. First, we suppose that there is an in nite antichain in W (0)= and show that there is also an in nite ascending chain in W (0)=, so that it will be enough to show that no in nite ascending chain exists in W (0)=. For this, we introduce a relation < on W (0)=: Let 2 W (0)=. Then, by Lemma 3.13, = Ul+1 nUl for some l. We let where k = minfi 2 ! j \ Vi 6= ;g, and = Uk nUk01 2 W =. Clearly, < , and 1 = 2 whenever 1 S and 2 . Moreover, it is easy to observe that S 2 ( 60 W (6))=. Thus, we get an in nite -tree rooted at U1 nU0 in ( 60 W (6))=. This tree is nitely branching by the de nition of the procedures, and hence there must be an in nite ascending -chain (and hence in nite ascending chain) in itSby Konig's Lemma. But by induction hypothesis, there is no such chain in ( 6(0 W (6))=, and so this in nite ascending chain is contained in W (0)=.
3 or
10
Suppose that 0 < 1 < 1 1 1 is an in nite ascending chain in W (0)=. Let T = M = B = H =
[ [ [ [ f
2
W (0)= j is maximal in W (0)=g;
f
2
W (0)= j is neither maximal nor minimal in W (0)=g;
f
2
W (0)= j is minimal in W (0)=g and
6(0
W (6):
First note that i is contained in M [ B for all i, because otherwise i+1 2= W (0)= and this contradicts our assumption. Let R0 R~ be a relation on M [ B [ H de ned as follows: vR0 w if either (i) v = y and z = xy in a stage by COP Y (X; Y ), (ii) v = x and z = x in a stage by SAT ! (X; Z ) or (iii) v = x and z = x ! in a stage by the latter half of SAT 3 (X; Z ). By de nition, R0 is nitely branching. We are now going to claim that M [ B [ H has nitely many R0-minimal elements. Let x 2 M and k = minfi 2 ! j x 2 Vi [ Ui g. Then x is added to W at the k-stage either by COP Y (X; Y ), SAT ! (X; Z ), or the latter half of SAT 3 (X; Z ). Indeed, if x is added to W by SAT } (X; Z ), then [x] \ W is contained in B; and if x is added to W by the former half of SAT 3 (X; Z ), then [x] \ W is contained in T. Consequently, for any element x 2 M, there exists y 2 M [ B [ H such that yR0 x. By induction hypothesis H is nite, and it can be shown B= is nite exactly as above; hence B [ H is nite by Lemma 3.13. Thus, M [ B [ H has nitely many R0 -minimal elements. Next note that there exists in nitely many pairs x; y 2 M [ B [ H such that xR0 y and x 2 R-max(y; ) for some 2 Sub . For, (M [ B [ H)= is in nite, and, for each 2 (M [ B [ H)=, there exists an element in which is add to W as an element of Z in SAT ! (X; Z ) or the latter half of SAT 3 (X; Z ). Now, let us regard M [ B [ H as an R0 -DAG. Then, by Konig's Lemma, there exists an R0 -chain x0R0 x1 R0 1 1 1 (and hence R-chain x0 Rx1 R 1 1 1) such that xi+1 2 R-max(xi ; ) for some 2 Sub for in nitely many i 2 ! . But this is impossible, and thus there exists no in nite ascending chain 0 < 1 < 1 1 1 in W (0)=. It remains to show M is an MIPC-model, but this easily follows from the construction.
Lemma 3.15 For any 2 Sub and x 2 W , hM ; xi j= if and only if hM; xi j= . Proof. Our proof proceeds by induction on the construction of . When 2 Var L3} or ?, it follows from the de nition of V . When ^ or _ , it immediately follows from the induction hypothesis. Next we verify the remaining cases. In each case, (() direction is straightforward, and hence we concentrate on the other. 11
1.
. Suppose x is R-maximal relative to ! . By de nition, ~ , we have hM; z i j= ! . But for any z 2 W such that xRz ! . Thus, hM; xi j= and hM; xi j= = . Then, by induction hypothesis, hM ; xi j= and hM ; xi j= = . Hence, hM ; xi j= = ! . Otherwise suppose x is not R-maximal relative to ! . Then by Lemma 3.13, x 2 Uk+1 nUk for some k and hence there exists y 2 W such that y 2 Rmax(x; ! ) by SAT ! (X; Z ). Then, analogously, we have hM ; y i j= and hM ; yi j= = . Hence hM ; xi j= = ! .
h
2.
3.
!
M; x == ij
3 . Suppose x is R -maximal relative to 3 . Then since [x] is Rmaximal relative to 3 , there exists x0 2 [x] \ W such that hM; x0 i j= = by SAT 3 (X; Z ). Then by induction hypothesis, we have hM ; x0 i j= = and hence hM ; xi j= = 3 . Otherwise suppose [x] is not R-maximal relative to 3 . Then there exists x0 2 [x] \ W and y 2 W such that y 2 Rmax(x0 ; 3 ) by SAT 3 (X; Z ). Then, analogously, there exists y0 2 [y ] \ W such that hM; y0 i j= = . Then by induction hypothesis, we have hM ; y0 i j= = . But since x x0 Ry y0, we have x(R )y 0 . Thus hM ; xi j= = 3 . } . Suppose hM; xi j= } . Then there exists y 2 W such that x y and hM; yi j= by SAT } (X; Y ). Then by induction hypothesis, we have hM ; yi j= . Thus hM ; xi j= } .
Thus, we arrive at
Theorem 3.16 Let L be an MIPC-logic such that for any descriptive MIPCframe F = hW; R; R3 ; R} ; P i,
F = L implies V j
8
W (FV j= MIPC implies FV j= L);
where FV = hV; RV; R3 V; R} V; Up V i. Then L has the nite model property. Proof. Suppose 2= L. Then there exists a descriptive MIPC-frame F = W; R; R3 ; R} ; P i for L and a model M based on it such that M j= = . Then by Lemma 3.14 and Lemma 3.15, there exists a nite subset V of W , such that FV is an MIPC-frame and a model M based on it such that M j= = . By assumption, FV j= L. Thus, M is a nite counter MIPC-model of L for . h
4 MIPC-logics with the nite model property In this section, we give some corollaries of Theorem 3.16 in accessible forms and present some examples of MIPC-logics with the nite model property. For this, we use the following result given in [14]. 12
Proposition 4.1 If a 3}-IM-logic L is characterized by a class of full 3}frames which is closed under elementary equivalence (in the rst-order language with the predicates =, R, R3 and R} ), then L is D-persistent. From this and Theorem 3.16, we obtain
Corollary 4.2 Let L be an MIPC-logic. If there exists a universal rst-order sentence 8 with the predicates =, R, R3 and R} such that 1. L is characterized by the class of full MIPC-frames satisfying 8 and 2. every full MIPC-frame that validates L satis es 8, then L has the nite model property. Proof. We show that the condition of Theorem 3.16 is satis ed: Let F = hW; R; R3 ; R} ; P i be a descriptive MIPC-frame for L. By condition 1 and Proposition 4.1, L is D-persistent. Consequently, F j= L, and so F satis es 8 by condition 2. Since 8 is a universal sentence, the class of full MIPC-frames satisfying 8 is closed under substructure (see e.g. [9]). Hence, for any V W , FV satis es 8, and thus FV j= L by condition 1.
Example 4.3 The following MIPC-logics have the nite model property; hence are decidable. 1. MIPC 8 sc, where sc = 3(3p ! q ) _ 3(3q ! p).
2. MIPC 8 (3p ! q) _ (3q ! p).
3. MIPC 8 3(p ! q) _ 3(q ! p).
Proof. 1. It is straightforward to show that every full MIPC-frame satis es
x; y; z (xR3 y ^ xR3 z ! zR3 y _ yR3 z )
8
(6)
if and only if it validates sc. Thus, it remains to show that MIPC 8 sc is characterized by the class of full MIPC-frames satisfying (6). We are now going to show that every descriptive frame F = hW; R; R3 ; R} ; P i for MIPC 8 sc satis es condition (6). Let M be an MIPC-model based on F. Without loss of generality, we can assume F = A+ for some MIPC-algebra A. Suppose that F does not satisfy condition (6). Then xR3 y; xR3 z; :(yR3 z ) and :(zR3 y ). From :(yR3 z ), we know that there exist a 2 A such that y 2 3a and z 2= a. Similarly, from :(zR3 y ), b 2 A such that z 2 3b and y 2= b. Thus, y 2= 3a ! b and z 2= 3b ! a. Then since xR3 y and xR3 z , x 2= 3(3a ! b) and x 2= 3(3b ! a). Thus, x 2= 3(3a ! b) _ 3(3b ! a), i.e. F j== 3(3a ! b) _ 3(3b ! a). 13
Let C be the class of full MIPC-frames that satisfy condition (6). If MIPC 8 sc j= = , then there exists a descriptive MIPC-frame F with F j= = and F j= sc. Then, by the argument above, F satis es condition (6), and hence so does F. Thus, F 2 C , F j= MIPC 8 sc and F j= = . On the other hand, if 2 MIPC 8 sc, then F j= for all F 2 C . Thus, MIPC 8 sc is characterized by the class C . Therefore, by Corollary 4.2, MIPC 8 sc has the nite model property. 2 and 3 can be shown similarly by replacing condition (6) with
x; y; z (xRy ^ xRz ! zR3 y _ yR3 z ) 8x; y; z (xR3 y ^ xR3 z ! zRy _ yRz ) 8
(7) (8)
respectively. In [1], it has been proved that if a si-logic Int + 0 is tabular then MIPC 8 [3p=p]p2Var( ) j 2 0g has the nite model property. Here, [3p=p]p2Var( ) denotes the formula obtained from by replacing every propositional variable p appearing in by 3p. Also, the complete list of pre-tabular MIPC-logics has been obtained in [1]. However, these results are not directly applicable to the MIPC-logics in Example 4.3. f
Next, we turn our attention to non-modal extensions of MIPC.
Corollary 4.4 For 0 For L, If there exists a universal rst-order sentence 8 with the predicates =, R such that 1. Int + 0 is characterized by the class of full Int-frames satisfying 8 and 2. every full Int-frame that validates 0 satis es 8, then MIPC 8 0 has the nite model property. Proof. Suppose 0 For L and Int + 0 is characterized by the class of full Int-frames satisfying a universal rst-order sentence 8. Then it is well-known that Int + 0 is D-persistent (see Theorem 10.22 in [4]). We are now going to show MIPC 8 0 is characterized by the class of full MIPC-frames satisfying 8. Let 2= MIPC 8 0. Then F j= = for some descriptive MIPC-frame F such that F j= MIPC 8 0. It is clear that F j= = ; thus, it remains to show that F satis es 8. By the D-persistency of Int + 0, we have F j= 0, since F is also a descriptive Int-frame for Int + 0. Hence, by condition 2, F satis es 8. On the other hand, if a full MIPC-frames satis es 8, then F j= 0 by condition 1, and so F j= MIPC 8 0. We next show that for any full MIPC-frames F, F satis es 8 whenever F j= MIPC 8 0. Indeed, if F j= MIPC 8 0 then F j= 0, and so F satis es 8 by assumption 2. Therefore, by Corollary 4.2, MIPC 8 0 has the nite model property.
14
Example 4.5 The following MIPC-logics have the nite model property; hence are decidable. 1. MIPC 8 bdn (n 1), where bdn is de ned inductively as: (1) bd1 = p1 _ :p1 ; (2) bdn+1 = pn+1 _ (pn+1 ! bdn). 2. MIPC 8 bwn (n 1), where bwn =
Wn
(pi !
W
6 p j ). 3. MIPC 8 bcn (n 1), where bcn = p0 _ (p0 ! p1 ) _ 1 1 1 _ (p0 ^ 1 1 1 ^ pn01 ! p n ). i=0
i=j
Proof. It immediately follows from Corollary 4.4 and the well-known characterizations of Int + bdn , Int + bwn and Int + bcn (see e.g. [4]).
It should be noted that in [2] the nite model property of MIPC 8 bwn and that of (every extensions of) MIPC 8 bdn have been already obtained. Also, it is shown in [1] that, for every 0 For L, MIPC 8 0 has the nite model property if either (1) 0 consists of f^; !; ?g-formulas; or (2) Int + 0 is locally tabular.
5 Conclusion In this paper we gave a sucient condition for intuitionistic modal logics over MIPC (in the lattice of 3}-normal intuitionistic modal logics) to have the nite model property. We showed that every MIPC-logic has the nite model property if there exists a universal rst-order sentence 8 such that (1) L is characterized by the class of full MIPC-frames satisfying 8 and (2) every full MIPC-frame that validates L satis es 8. We also applied our result to show the nite model property of some intuitionistic modal logics over MIPC whose nite model property had been previously unknown.
Acknowledgments Deep appreciation goes to Frank Wolter, who leads the authors into this topic. Thanks are also due to Guram Bezhanishvili for helpful comments.
References [1] G. Bezhanishvili. An Algebraic Approach to Intuitionistic Modal Logics over MIPC. PhD Thesis, Tokyo Institute of Technology, 1997. [2] G. Bezhanishvili and M. Zakharyaschev. Logics over MIPC. In Proceedings of Sequent Calculi and Kripke Semantics for Non-Classical Logics, RIMS, Kyoto University. To appear. 15
[3] R. A. Bull. MIPC as the formalization of an intuitionist concept of modality. Journal of Symbolic Logic, 31(4):609{616, 1966. [4] A. Chagrov and M. Zakharyaschev. Modal Logic, volume 35 of Oxford Logic Guide. Oxford University Press, 1997. [5] C. Grefe. Fisher Servi's intuitionistic modal logic has the nite model property. In Advances in Modal Logic '96, M. de Rijke et al. (eds.). CSLI, 1997. [6] G. Fischer Servi. On modal logic with an intuitionistic base. Studia Logica, 36(3):141{149, 1977. [7] G. Fischer Servi. The nite model property for MIPQ and some consequences. Notre Dame Journal of Formal Logic, XIX:687{692, 1978. [8] G. Fischer Servi. Axiomatizations for some intuitionistic modal calculi. Rendiconti del Seminaro Matematico Universita e Politecnico di Torino, 42:179{ 194, 1984. [9] H. J. Keisler. Fundamentals of model theory. In Handbook of Mathematical Logic, J. Barwise (ed.). North-Holland, 1977. [10] H. Ono. On some intuitionistic modal logics. Publications of the Research Institute for Mathematical Science, 13(3):687{722, 1977. [11] H. Ono and N.-Y. Suzuki. Relations between intuitionistic modal logics and intermediate predicate logics. Reports on Mathematical Logic, 22:65{ 87, 1988. [12] A. Prior. Time and Modality. Clarendon Press, 1957. [13] A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD Thesis, University of Ediburgh, 1994. [14] F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 36:73{92, 1997. [15] M. Zakharyaschev, F. Wolter and A. Chagrov. Advanced Modal logic. In Handbook of Philosophical Logic, Gabbay et al. (eds.). To appear.
16
School of Information Science, JAIST Tatsunokuchi, Ishikawa 923-12, Japan E-mail: [email protected] 2 Semantic Group, ETL Mail box 1503, Umezono 1-1-4, Tsukuba, Ibaraki 305, Japan E-mail: [email protected] 1
Abstract. It is shown that every normal intuitionistic modal logic L over MIPC has the nite model property if there exists a universal rstorder sentence 8 such that (1) L is characterized by the class of Kripke frames satisfying 8 and (2) every Kripke frame that validates L satis es 8. Here, MIPC is a well-known intuitionistic modal logic introduced by Prior (1957).
1 Introduction In this paper, we study the nite model property of (propositional) modal logics based on intuitionistic logic|we call them intuitionistic modal logics|with a necessity operator 3 and a possibility operator }. In particular, we focus on those of logics over MIPC in the lattice of 3}-normal intuitionistic modal logics. We show that every 3}-normal intuitionistic modal logic L which includes MIPC has the nite model property if there exists a universal rst-order sentence 8 such that (1) L is characterized by the class of Kripke frames satisfying 8 and (2) every Kripke frame that validates L satis es 8. MIPC is an intuitionistic modal logic introduced by Prior [12], and has been studied by several authors [1][2][3][6][7][8][10][11]. MIPC is one of the S5-like intuitionistic modal logics; that is, if we add p _ :p to MIPC then we get the well-known classical modal logic S5. Another feature of MIPC is that one can obtain an embedding of MIPC into intuitionistic predicate logic by interpreting 3 as 8 and } as 9. Thus, one-variable fragments (i.e. the sets of formulas with ? The
preliminary version of this paper was presented at the workshop
Logic, European
Topics in Polymodal
Summer School in Logic, Language, and Information (ESSLLI'97), August
1997.
?? Supported
by the Center of Excellence (COE) budget of the Science and Technology
Agency (STA) Japan.
one xed variable) of intuitionistic predicate logic can be interpreted in MIPC (see [1], [11]). We also note that MIPC is an extension of FS (Fisher Servi's logic) introduced in [8]. For recent surveys on intuitionistic modal logics we refer the reader to [13], [15]. The nite model property of a logic L ensures the decidability of L when L is nitely axiomatizable, and hence has been one of the central focuses in the study of logics. The nite model property of MIPC was shown in [7], [10] and that of FS in [5]. Several general results on the nite model property of extensions of MIPC are also obtained in [1], [2]. The rest of the paper is organized as follows. In Section 2, we introduce basic notions and x some notation used in this paper. In Section 3, we present a construction of a nite counter MIPC-model for from a given counter MIPC-model for based on a descriptive MIPC-frame. Our main theorem is given at the end of Section 3. We also give sucient conditions of extensions of MIPC to have the nite model property in Section 4. We conclude our result in Section 5.
2 Preliminaries Familiarity with the basic notions of modal logic (as expounded in e.g. [4]) will be helpful in what follows. Let L be the propositional language consisting of in nitely many propositional variables p; q; r; : : : and the usual primitive connectives !; ^; _; ?. (We put > def = ? ! ?.) The set of L-formulas is denoted by For L. A set L For L is called a superintuitionistic logic (abbrev. si-logic) if L includes intuitionistic logic Int and is closed under substitution and modus ponens. For a set 0 For L, the smallest si-logic that includes 0 is denoted by Int + 0; when 0 = f g, Int + 0 will be denoted by Int + . An intuitionistic (general) frame (Int-frame, for short) is a structure hW; R; P i where R is a partial order on W and P is a set of R-cones in W that contains ; and is closed under \; [ and the operation:
def = fx 2 W
z ((xRz and z 2 ) imply z 2 )g:
j 8
Here, W is an R-cone if 8x; y (x 2 and xRy imply y 2 ). It is wellknown (see e.g. [4]) that every Int-frame F = hW;R; P i gives rise to its dual Heyting algebra F+ = hP; \; [; ; ;i; and conversely, every Heyting algebra A = hA; ^; _; !; ?i to its dual Int-frame A+ = hWA ; ; PA i where WA is the set of prime lters in A and PA = fjaj j a 2 Ag with jaj = fX 2 WA j a 2 X g; and that A ' (A+ )+. An Int-frame F is descriptive if F ' (F+)+ . A full Int-frame (or a Kripke Int-frame) is an Int-frame of the form hW; R; Up W i. Here, Up W is the set of all R-cones in W . For an Int-frame F = hW; R; P i, its underlying full Int-frame hW; R; Up W i is denoted by F. An Int-frame F is a 2
frame for a si-logic L when F j= L. A si-logic L is D-persistent if F j= L holds for any descriptive frame F for L.
Let L3} be the propositional bimodal language, i.e. the language L with new unary connectives 3 and }. The sets of (L3} -)variables and (L3} -)formulas are denoted by Var L3} and For L3} , respectively. For a formula , Sub( ) is the set of subformulas of . A set L For L3} is called a 3}-normal intuitionistic modal logic (a 3}-IM-logic, for short) if it includes Int, contains axioms 3(p ^ q ) $ 3p ^ 3q , }(p _ q) $ }p _ }q , 3> and :}?, and is closed under substitution, modus ponens and the monotonicity rules: ! =3 ! 3 and ! =} ! } . We denote by IntK3} the smallest 3}-IM-logic. For a set 0 For L3} and a 3}-IM-logic L, the smallest 3}-IM-logic that includes L and 0 is denoted by L 8 0; when 0 = f g, L 8 0 will be denoted by L 8 . We set MIPC def = IntK3} 8 f3p ! p; p ! }p; }3p ! 3p; }p ! 3}p; }(p ! q) ! (3p ! }q)g. A 3}-IM-logic that includes MIPC is called an MIPC-logic. An MIPC-algebra is a structure A = hA; ^; _; !; ?; 3; }i where hA; ^; _; ! ; ?i is a Heyting algebra and 3; } are unary operations on A such that 3(a^b) = 3a ^ 3b, 3> = >, }(a _ b) = }a _ }b, :}? = >, 3a a, a }a, }3a 3a, }a 3}a and }a ^ }b }(}a ^ b) for any a; b 2 A. It is straightforward to see that for any MIPC-logic L, there exists a class C of MIPC-algebras such that 2 L if and only if 8A 2 C (A j= ); such C is called a class of MIPC-algebras that characterizes L. An (general) MIPC-frame is a structure F = hW; R; R3 ; R} ; P i where (1) hW; R; P i is an Int-frame; (2) R3 ; R} are binary relations on W such that R3 is re exive and transitive, R} = (R3 )01 and R3 = R (R3 \ R} ); and (3) P is closed under the operations:
3 = x W y (xR3 y implies y ) ; f
=
}
2
x2W
f
j 8
2
g
y 2 (xR} y)g
j 9
(see [10], [11]). Here and hereafter, denotes the composition of relations. The following lemma is well-known (see e.g. [4]).
Lemma 2.1 Let A = hA; ^; _; !; ?i be a Heyting algebra. Suppose a 2 A and is a lter in A such that a 2= . Then there exists a prime lter r in A which includes and does not contain a such that r is maximal in the set f j is a lter in A such that and a 2 = g. Using this lemma, it is not hard to show that, for any MIPC-frame F = + def hW; R; R3 ; R} ; P i, F = hP; \; [; ; ;; 3; }i is an MIPC-algebra; that, for any MIPC-algebra A = hA; ^; _; !; ?; 3; }i, A+ def = hWA ; ; R^ 3 ; R^ } ; PA i is an MIPC-frame, where (1) WA is the set of prime lters in A, (2) R^3 if and only if 3a 2 implies a 2 for any a 2 A, (3) R^ } if and only if a 2 implies 3
a 2 for any a 2 A, and (4) PA = fjaj j a 2 Ag with jaj = fX 2 WA j a 2 X g; and that A ' (A+ )+. An MIPC-frame F is descriptive if F ' (F+)+ . Thus, for any MIPC-logic L, there exists a class C of descriptive MIPC-frames such that 2 L i 8F 2 C (F j= ); such C is called a class of descriptive MIPC-frames that characterizes L. A full MIPC-frame (or a Kripke MIPC-frame) is an MIPC-frame of the form hW; R; R3 ; R} ; Up W i. For an MIPC-frame F = hW; R; R3 ; R} ; P i, we denote by F its underlying full MIPC-frame hW;R; R3 ; R} ; Up W i. An MIPC-frame F is a frame for an MIPC-logic L when F j= L. An MIPC-logic L is D-persistent if F j= L holds for any descriptive MIPC-frame F for L. It is straightforward to show that MIPC is D-persistent (see [14]). }
3 Constructing a nite counter MIPC-model First, we review some basic notions on MIPC-models, which will be used in this section. A valuation on an MIPC-frame F = hW; R; R3 ; R} ; P i is a mapping from Var L3} to P . An MIPC-model is a pair M = hF; V i where F is an MIPCframe and V is a valuation on it. M = hF; V i is said to be an MIPC-model based on F. Let M = hF; V i be an MIPC-model based on F = hW; R; R3 ; R} ; P i. For x 2 W and 2 For L3} , we de ne a relation hM; xi j= by induction on the construction of : hM; xi j= p i x 2 V (p); hM; xi j= ^ i hM; xi j= and hM; xi j= ; hM; xi j= _ i hM; xi j= or hM; xi j= ; hM; xi j= ! i 8y 2 W (xRy and hM; yi j= imply hM; xi j= ); hM; xi j= 3 i 8y 2 W (xR3 y implies hM; yi j= ); hM; xi j= } i 9y 2 W (xR} y and hM; yi j= ): When no confusion occurs, hM; xi j= will be abbreviated as x j= . A formula is valid in an MIPC-model M (notation: M j= ) if 8x 2 W (hM; xi j= ); otherwise is refuted in M (or M is a counter model for ). For a set L of formulas, M j= L stands for 8 2 L (M j= ). For an MIPCmodel M and an MIPC-logic L, M is an MIPC-model of L when M j= L. An MIPC-frame F is an MIPC-frame for MIPC-logic L (notation: F j= L) if M j= L for any MIPC-model based on F. If F = hW; R; R3 ; R} ; P i then M = hF; V i is said to be nite if the cardinality of W (notation: jW j) is nite. An MIPC-logic L has the nite model property if, for any 2= L, there exists a nite MIPC-model of L that refutes . We begin our study with showing basic properties of an MIPC-model. We ~ if xRy and not yRx. write xRy 4
De nition 3.1 Let M = hF; V i be an MIPC-model based on an MIPC-frame W; R; R3 ; R} ; P i and 2 For L3} . Then y 2 W is said to be R-maximal ~ implies z j= ). The set of relative to if and only if y j= = and 8z 2 W (y Rz all R-successors of x that are R-maximal relative to is written as R-max(x; ).
h
Lemma 3.2 Let M = hF; V i be an MIPC-model based on a descriptive MIPCframe hW; R; R3 ; R} ; P i. For any 2 For L3} and x 2 W such that x j= = , R-max(x; ) is non-empty. Proof. Since F is descriptive, without loss of generality, we can assume F = A+ for some MIPC-algebra A. Then there exists a 2 A such that V ( ) = jaj. Note that every y 2 W is a prime lter in A and it holds that y j= i a 2 y. Suppose that x j= = , i.e. a 2= x. Put := x. Then by Lemma 2.1, we obtain a prime lter r satisfying a 2= r and x r such that, for any lter y, r ( y implies a 2 y . Since R = in A+ , xRr and r is R-maximal relative to .
It is easy to see that R3 \ R} is an equivalence relation. In the sequel, we abbreviate R3 \ R} as . One easily checks that hM; xi j= } if and only if 9y 2 W (x y and hM; y i j= ). For x 2 W , the -equivalence class that contains x is written as [x]. For W , is said to be a precluster if [x] for some x 2 W ; and when this is the case, we write [ ] for [x].
Lemma 3.3 Let M = hF; V i be an MIPC-model based on an MIPC-frame hW; R; R3 ; R} ; P i. Let be a formula of the form 3 or } for some 2 For L3} . Then for any x; y 2 W with x y , we have x j= i y j= . Proof. Easy.
Thus for any precluster and x; y 2 , we have that x j= 3 i y j= 3, and that x j= } i y j= } for any 2 For L3} . We write j= 3 and j= } when x j= 3 and x j= } for some (and hence any) x 2 , respectively.
Lemma 3.4 Let M = hF; V i be an MIPC-model based on a descriptive MIPCframe hW; R; R3 ; R} ; P i. Suppose that x 2 W is R-maximal relative to 3 . Then for any y 2 [x], we have R-max(y; 3 ) [x]. Proof. Suppose that y x = z and z 2 R-max(y; 3 ). Since x yRz, there ~ . exists w 2 W such that xRw z . Since w z = x, w 2= [x]. Hence xRw Then by R-maximality of x, we have w j= 3 . On the other hand, z j= = 3 by R-maximality of z . But this contradicts Lemma 3.3.
Thus for any -equivalence class that contains an element that is Rmaximal relative to 3 , we have R-max(x; 3 ) for any x 2 ; when this is the case, is also said to be R-maximal relative to 3 . 5
Lemma 3.5 Let M = hF; V i be an MIPC-model based on a descriptive MIPCframe hW; R; R3 ; R} ; P i. For any -equivalence class that is R-maximal relative to 3 , there exists y 2 such that y j= = .
Proof. By assumption, there exists x 2 that is R-maximal relative to 3. Since x j= = 3 , there exists y; z 2 W such that xRz y and y j= = . Suppose ~ . By R-maximality of x, we have z j= 3 . y 2= . Then x = z , and so xRz Together with z y, this implies y j= , which is not the case.
De ne a relation on the set of -equivalence classes by 9y 2 (xRy ). Then we obtain the following.
i 9x
2
Lemma 3.6 The relation on the set of -equivalence classes de ned as above is a partial order, i.e. it is a re exive, transitive and antisymmetric relation. Proof. Easy.
Note that if and only if 8x ~ ). < then 8x 2 9y 2 (xRy
2
8y
2
(x(R )y). Moreover, if
We are now going to construct a nite counter MIPC-model M for from a given MIPC-model M that refutes 2 For L3} . To this end, we x 2 For L3} and MIPC-model M = hF; V i based on a descriptive MIPCframe F = hW; R; R3 ; R} ; P i such that M j= = . We will nd a suitable subset W of W , based on which the counter model M will be constructed. For this, we de ne some procedures which act on the subsets X; Y and Z of W satisfying the constraints: all X; Y and Z are nite; X is a precluster; [z ] 2 [y ] for any y 2 Y and z 2 Z ; [X ] 2 [y] for any y 2 Y ; [z ] 2 [X ] for any z 2 Z .
(1) (2) (3) (4) (5)
Roughly speaking, our construction is carried out by \completing" preclusters one by one in an ascending order. During the construction, X is the precluster that we are focusing now, Y is the set of already selected points consisting of completed preclusters, and Z is a set of points that we should think of later (see Figure 1). Note that it follows from the above constraints that [X ]; f[y ] j y 2 Y g and f[z ] j z 2 Z g are pairwise disjoint.
De nition 3.7 COP Y (X; Y ) is a procedure that updates X as follows: Let Y 0 = fy 2 Y j y is maximal w.r.t. R in Y , 9x 2 X (y ( R)x); and not 9x 2 X (yRx)g: 6
1100 01 11001100 11001100 10 00 11 0011 00 0011 11 11001100X 1100 11 00 11 00 01 11001100 101100 101100110011000011 10101100110011001100001101 1100 0011 11001100110011001100 1100101000111010 00 11 11 1100110000111100 Y 1100 00 11001100 001111001100 00110011 Z
Figure 1: X , Y , Z during the construction Then, by the de nition of an MIPC-frame, for each y 2 Y 0, we can take xy 2 [X ] such that yRxy . We put X := X [ fxy j y 2 Y 0g. De nition 3.8 SAT 3 (X; Z ) is a procedure that updates X and Z as follows: Let 0 = f3 2 Sub j [X ] is R-maximal relative to 3 g: Then by Lemma 3.5, for each 3 2 0, we can take x 2 [X ] such that x j= = . We put X := X [ fx j 3 2 0g. Let
6 = f3 2 Sub j [X ] j= = 3 and [X ] is not R-maximal relative to 3g: Fix an arbitrary x 2 X . By Lemma 3.2, for each 3 2 6, we can take x 2 Rmax(x; 3 ). Note that x 2= [X ] by Lemma 3.5. We put Z := Z [ fx j 3 2 6g.
De nition 3.9 SAT } (X; Y ) is a procedure that updates X as follows: Let 0 = f} 2 Sub j [X ] j= } and 8x 2 X (x j= = )g: Then by de nition, for each } 2 0, we can take x 2 [X ] such that x j= . We put X := X [ fx j } 2 0g. De nition 3.10 SAT ! (X; Z ) is a procedure that updates X and Z as follows: First, we de ne sequences X0 ; X1; : : : and Z0; Z1 ; : : : by induction. Let X1 = X
7
and X0 = Z0 = Z1 = ;. Suppose Xk and Zk (k n + 1) are already de ned. We de ne Xn+2 and Zn+2. For each x 2 Xn+1nXn, let
0x = f ! 2 Sub j x j= = ! and x is not R-maximal relative to ! g: Then by Lemma 3.2, for each ! 2 0x , we can take x ! 2 R-max(x; ! ). We let
Xn+2 = (fx ! j x 2 Xn+1nXn and ! 2 0x g \ [X ]) [ Xn+1 Zn+2 = fx ! j x 2 Xn+1nXn and ! 2 0x gn[X ]: Finally, put X :=
S
S
2 Xn and Z := Z [ n2! Zn. Lemma 3.11 Updated X , Y and Z by the above procedures again satisfy the constraints (1) (5). n !
Proof. Let us denote the updated X , Y and Z by X^ , Y^ and Z^ , respectively. 1. COP Y (X; Y ). Since fxy j y 2 Y 0g [X ], X^ is again a precluster and [X ] = [X^ ]. Accordingly (2) (5) hold. By the assumption that Y is nite, Y 0 is nite. Together with niteness of X , it follows that X^ is nite. Thus (1) is also satis ed. 2. SAT 3 (X; Z ). First, we show that the rst part of the procedure preserves the constraints. Since fx j 3 2 0g [X ], X^ is again a precluster and [X ] = [X^ ]. Accordingly (2) (5) hold. Since Sub is nite, 0 is nite. Together with niteness of X , it follows that X^ is nite. Thus (1) is also satis ed. Next, we show the latter part of the procedure preserves the constraints. Since Sub is nite, 6 is nite. Together with niteness of Z , it follows that Z^ is nite. Thus (1) is satis ed. Clearly, it remains to show (3) and (5). Since xRx and x 2= [X^ ], we have [X^ ] < [x ], and hence (3) and (5) hold. 3. SAT } (X; Y ). Similar to the rst part of SAT 3 (X; Z ).
4. SAT ! (X; Z ). First, we show that there exists some l > 0 such that Xl+1nXl = ;. Suppose contrarily that Xn+1 nXn = ; for all n > 0. Then for any k 2 !, there exists R-chain x1Rx2R 1 1 1 xk of length k such that, for any i (k > i > 0), there exists ! 2 Sub satisfying (a) xi j= = ! , (b) xi is not R-maximal relative to ! , and (c) xi+1 2 R-max(xi ; ! ). 8
Let
x_ = f ! 2 Sub j x j= ! or x is R-maximal relative to ! g: Then it follows that x_ i ) x_ i+1 for any i (k > i > 0). But since each j x_ i j is bounded by j Sub j, there is no R-chain longer than j Sub j, which is not the case. Thus there exists l > 0 such that Xl+1nXl = ;. Then by construction, it follows that Xl = Xl+1 = 1 1 1 and ; = Zl+1 = Zl+2 = 1 1 1. Clearly, each Xn and Zn is nite, hence so are X^ and Z^. Thus (1) is satis ed. (2) is also satis ed, since Xi [X ] for each i 2 !. Clearly, it remains to show (3) and (5). We show for any z 2 Z^ nZ , [X^ ] < [z]. Without loss of generality, we suppose z 2 Zn+2. Then there exists x 2 Xn+1 nXn such that xRz . Together with z 2= [X ], we have [X^ ] = [x] < [z ]. The constraints (3) and (5) immediately follow from this. We are now going to present the construction of model M .
De nition 3.12 Let 2 For L3} , and suppose M = hF; V i is a counter MIPC-model for based on a descriptive MIPC-frame F = hW; R; R3 ; R} ; P i. We construct a model M as follows: First, take any point x0 2 W such that x0 j= = (such x0 always exists since M is a counter MIPC-model for ). Next, we de ne sequences U0 ; U1 ; : : : and V0 ; V1 ; : : : by induction. Put U0 = ; and V0 = fx0g. Suppose Uk and Vk are already de ned. Step 1. Select an element X in Vk = such that [X ] is minimal w.r.t. in f[v ] j v 2 Vk g, and put Y := Uk and Z := Vk nX . Step 2. Update X by COP Y (X; Y ). Step 3. Update X and Z SAT 3 (X; Z ).
Step 4. Update X by SAT } (X; Y ). Step 5. Update X and Z by SAT ! (X; Z ).
And then de ne Uk+1 = USk [ X and Vk+1 = Z . Finally, de ne W = n2! Un and M = hF ; V i, where
F = W ; RW ; R3 W ; R} W ; Up W ; h
i
V (p) = V (p) \ W for any p 2 Var L3} : 9
Lemma 3.13 Above construction is well-de ned. Moreover, for any X W =, there exists some l such that X = Ul+1 nUl (hence X is nite).
2
Proof. We show by induction on n that
i. Un and Vn are nite, ii. [u] 2 [v] for any u 2 Un and v 2 Vn . Base step is obvious. For the induction step, suppose i and ii are satis ed for n = k. Then the precluster X selected in step 1 is nite, since Vk is nite. Also, it is clear that X , Y and Z set in step 1 satisfy the constraints (1) (5) by induction hypothesis. In step 2 { 5, by Lemma 3.11, updated X , Y and Z also satisfy the constraints (1) (5). Finally, by the constraints (1) (5), it is easy to see i and ii are again satis ed by Uk+1 and Vk+1 . Now, by the constraints (3) (5) and ii, every element of W = is Ul+1 nUl for some l; it is nite by (1).
Lemma 3.14 M is a nite MIPC-model. Proof. We will show that W is nite. Let Msub = f 2 Sub j } for some 2 For L3} g. For each 0 Msub , let W (0) = fx 2 W j for any 2 Msub , x j= i 2 0g. Note, by Lemma 3.3, for any precluster in W there exists a unique 0 Msub such that W (0). Thus, it suces to show W (0) is nite for each 0 Msub , since P (Msub ) is nite. We show this by induction on j0j. Suppose for any 6 Msub , j6j < j0j implies that W (6) is nite. This means, in particular, for any 6 ( 0, W (6) is nite. Since every precluster in W (0) is nite by Lemma 3.13, it suces to show that W (0)= is nite. Moreover, since W (0)= is a poset and there is no in nite descending chain in it by the construction, it suces to show that there are neither in nite ascending chain nor in nite antichain in W (0)=. First, we suppose that there is an in nite antichain in W (0)= and show that there is also an in nite ascending chain in W (0)=, so that it will be enough to show that no in nite ascending chain exists in W (0)=. For this, we introduce a relation < on W (0)=: Let 2 W (0)=. Then, by Lemma 3.13, = Ul+1 nUl for some l. We let where k = minfi 2 ! j \ Vi 6= ;g, and = Uk nUk01 2 W =. Clearly, < , and 1 = 2 whenever 1 S and 2 . Moreover, it is easy to observe that S 2 ( 60 W (6))=. Thus, we get an in nite -tree rooted at U1 nU0 in ( 60 W (6))=. This tree is nitely branching by the de nition of the procedures, and hence there must be an in nite ascending -chain (and hence in nite ascending chain) in itSby Konig's Lemma. But by induction hypothesis, there is no such chain in ( 6(0 W (6))=, and so this in nite ascending chain is contained in W (0)=.
3 or
10
Suppose that 0 < 1 < 1 1 1 is an in nite ascending chain in W (0)=. Let T = M = B = H =
[ [ [ [ f
2
W (0)= j is maximal in W (0)=g;
f
2
W (0)= j is neither maximal nor minimal in W (0)=g;
f
2
W (0)= j is minimal in W (0)=g and
6(0
W (6):
First note that i is contained in M [ B for all i, because otherwise i+1 2= W (0)= and this contradicts our assumption. Let R0 R~ be a relation on M [ B [ H de ned as follows: vR0 w if either (i) v = y and z = xy in a stage by COP Y (X; Y ), (ii) v = x and z = x in a stage by SAT ! (X; Z ) or (iii) v = x and z = x ! in a stage by the latter half of SAT 3 (X; Z ). By de nition, R0 is nitely branching. We are now going to claim that M [ B [ H has nitely many R0-minimal elements. Let x 2 M and k = minfi 2 ! j x 2 Vi [ Ui g. Then x is added to W at the k-stage either by COP Y (X; Y ), SAT ! (X; Z ), or the latter half of SAT 3 (X; Z ). Indeed, if x is added to W by SAT } (X; Z ), then [x] \ W is contained in B; and if x is added to W by the former half of SAT 3 (X; Z ), then [x] \ W is contained in T. Consequently, for any element x 2 M, there exists y 2 M [ B [ H such that yR0 x. By induction hypothesis H is nite, and it can be shown B= is nite exactly as above; hence B [ H is nite by Lemma 3.13. Thus, M [ B [ H has nitely many R0 -minimal elements. Next note that there exists in nitely many pairs x; y 2 M [ B [ H such that xR0 y and x 2 R-max(y; ) for some 2 Sub . For, (M [ B [ H)= is in nite, and, for each 2 (M [ B [ H)=, there exists an element in which is add to W as an element of Z in SAT ! (X; Z ) or the latter half of SAT 3 (X; Z ). Now, let us regard M [ B [ H as an R0 -DAG. Then, by Konig's Lemma, there exists an R0 -chain x0R0 x1 R0 1 1 1 (and hence R-chain x0 Rx1 R 1 1 1) such that xi+1 2 R-max(xi ; ) for some 2 Sub for in nitely many i 2 ! . But this is impossible, and thus there exists no in nite ascending chain 0 < 1 < 1 1 1 in W (0)=. It remains to show M is an MIPC-model, but this easily follows from the construction.
Lemma 3.15 For any 2 Sub and x 2 W , hM ; xi j= if and only if hM; xi j= . Proof. Our proof proceeds by induction on the construction of . When 2 Var L3} or ?, it follows from the de nition of V . When ^ or _ , it immediately follows from the induction hypothesis. Next we verify the remaining cases. In each case, (() direction is straightforward, and hence we concentrate on the other. 11
1.
. Suppose x is R-maximal relative to ! . By de nition, ~ , we have hM; z i j= ! . But for any z 2 W such that xRz ! . Thus, hM; xi j= and hM; xi j= = . Then, by induction hypothesis, hM ; xi j= and hM ; xi j= = . Hence, hM ; xi j= = ! . Otherwise suppose x is not R-maximal relative to ! . Then by Lemma 3.13, x 2 Uk+1 nUk for some k and hence there exists y 2 W such that y 2 Rmax(x; ! ) by SAT ! (X; Z ). Then, analogously, we have hM ; y i j= and hM ; yi j= = . Hence hM ; xi j= = ! .
h
2.
3.
!
M; x == ij
3 . Suppose x is R -maximal relative to 3 . Then since [x] is Rmaximal relative to 3 , there exists x0 2 [x] \ W such that hM; x0 i j= = by SAT 3 (X; Z ). Then by induction hypothesis, we have hM ; x0 i j= = and hence hM ; xi j= = 3 . Otherwise suppose [x] is not R-maximal relative to 3 . Then there exists x0 2 [x] \ W and y 2 W such that y 2 Rmax(x0 ; 3 ) by SAT 3 (X; Z ). Then, analogously, there exists y0 2 [y ] \ W such that hM; y0 i j= = . Then by induction hypothesis, we have hM ; y0 i j= = . But since x x0 Ry y0, we have x(R )y 0 . Thus hM ; xi j= = 3 . } . Suppose hM; xi j= } . Then there exists y 2 W such that x y and hM; yi j= by SAT } (X; Y ). Then by induction hypothesis, we have hM ; yi j= . Thus hM ; xi j= } .
Thus, we arrive at
Theorem 3.16 Let L be an MIPC-logic such that for any descriptive MIPCframe F = hW; R; R3 ; R} ; P i,
F = L implies V j
8
W (FV j= MIPC implies FV j= L);
where FV = hV; RV; R3 V; R} V; Up V i. Then L has the nite model property. Proof. Suppose 2= L. Then there exists a descriptive MIPC-frame F = W; R; R3 ; R} ; P i for L and a model M based on it such that M j= = . Then by Lemma 3.14 and Lemma 3.15, there exists a nite subset V of W , such that FV is an MIPC-frame and a model M based on it such that M j= = . By assumption, FV j= L. Thus, M is a nite counter MIPC-model of L for . h
4 MIPC-logics with the nite model property In this section, we give some corollaries of Theorem 3.16 in accessible forms and present some examples of MIPC-logics with the nite model property. For this, we use the following result given in [14]. 12
Proposition 4.1 If a 3}-IM-logic L is characterized by a class of full 3}frames which is closed under elementary equivalence (in the rst-order language with the predicates =, R, R3 and R} ), then L is D-persistent. From this and Theorem 3.16, we obtain
Corollary 4.2 Let L be an MIPC-logic. If there exists a universal rst-order sentence 8 with the predicates =, R, R3 and R} such that 1. L is characterized by the class of full MIPC-frames satisfying 8 and 2. every full MIPC-frame that validates L satis es 8, then L has the nite model property. Proof. We show that the condition of Theorem 3.16 is satis ed: Let F = hW; R; R3 ; R} ; P i be a descriptive MIPC-frame for L. By condition 1 and Proposition 4.1, L is D-persistent. Consequently, F j= L, and so F satis es 8 by condition 2. Since 8 is a universal sentence, the class of full MIPC-frames satisfying 8 is closed under substructure (see e.g. [9]). Hence, for any V W , FV satis es 8, and thus FV j= L by condition 1.
Example 4.3 The following MIPC-logics have the nite model property; hence are decidable. 1. MIPC 8 sc, where sc = 3(3p ! q ) _ 3(3q ! p).
2. MIPC 8 (3p ! q) _ (3q ! p).
3. MIPC 8 3(p ! q) _ 3(q ! p).
Proof. 1. It is straightforward to show that every full MIPC-frame satis es
x; y; z (xR3 y ^ xR3 z ! zR3 y _ yR3 z )
8
(6)
if and only if it validates sc. Thus, it remains to show that MIPC 8 sc is characterized by the class of full MIPC-frames satisfying (6). We are now going to show that every descriptive frame F = hW; R; R3 ; R} ; P i for MIPC 8 sc satis es condition (6). Let M be an MIPC-model based on F. Without loss of generality, we can assume F = A+ for some MIPC-algebra A. Suppose that F does not satisfy condition (6). Then xR3 y; xR3 z; :(yR3 z ) and :(zR3 y ). From :(yR3 z ), we know that there exist a 2 A such that y 2 3a and z 2= a. Similarly, from :(zR3 y ), b 2 A such that z 2 3b and y 2= b. Thus, y 2= 3a ! b and z 2= 3b ! a. Then since xR3 y and xR3 z , x 2= 3(3a ! b) and x 2= 3(3b ! a). Thus, x 2= 3(3a ! b) _ 3(3b ! a), i.e. F j== 3(3a ! b) _ 3(3b ! a). 13
Let C be the class of full MIPC-frames that satisfy condition (6). If MIPC 8 sc j= = , then there exists a descriptive MIPC-frame F with F j= = and F j= sc. Then, by the argument above, F satis es condition (6), and hence so does F. Thus, F 2 C , F j= MIPC 8 sc and F j= = . On the other hand, if 2 MIPC 8 sc, then F j= for all F 2 C . Thus, MIPC 8 sc is characterized by the class C . Therefore, by Corollary 4.2, MIPC 8 sc has the nite model property. 2 and 3 can be shown similarly by replacing condition (6) with
x; y; z (xRy ^ xRz ! zR3 y _ yR3 z ) 8x; y; z (xR3 y ^ xR3 z ! zRy _ yRz ) 8
(7) (8)
respectively. In [1], it has been proved that if a si-logic Int + 0 is tabular then MIPC 8 [3p=p]p2Var( ) j 2 0g has the nite model property. Here, [3p=p]p2Var( ) denotes the formula obtained from by replacing every propositional variable p appearing in by 3p. Also, the complete list of pre-tabular MIPC-logics has been obtained in [1]. However, these results are not directly applicable to the MIPC-logics in Example 4.3. f
Next, we turn our attention to non-modal extensions of MIPC.
Corollary 4.4 For 0 For L, If there exists a universal rst-order sentence 8 with the predicates =, R such that 1. Int + 0 is characterized by the class of full Int-frames satisfying 8 and 2. every full Int-frame that validates 0 satis es 8, then MIPC 8 0 has the nite model property. Proof. Suppose 0 For L and Int + 0 is characterized by the class of full Int-frames satisfying a universal rst-order sentence 8. Then it is well-known that Int + 0 is D-persistent (see Theorem 10.22 in [4]). We are now going to show MIPC 8 0 is characterized by the class of full MIPC-frames satisfying 8. Let 2= MIPC 8 0. Then F j= = for some descriptive MIPC-frame F such that F j= MIPC 8 0. It is clear that F j= = ; thus, it remains to show that F satis es 8. By the D-persistency of Int + 0, we have F j= 0, since F is also a descriptive Int-frame for Int + 0. Hence, by condition 2, F satis es 8. On the other hand, if a full MIPC-frames satis es 8, then F j= 0 by condition 1, and so F j= MIPC 8 0. We next show that for any full MIPC-frames F, F satis es 8 whenever F j= MIPC 8 0. Indeed, if F j= MIPC 8 0 then F j= 0, and so F satis es 8 by assumption 2. Therefore, by Corollary 4.2, MIPC 8 0 has the nite model property.
14
Example 4.5 The following MIPC-logics have the nite model property; hence are decidable. 1. MIPC 8 bdn (n 1), where bdn is de ned inductively as: (1) bd1 = p1 _ :p1 ; (2) bdn+1 = pn+1 _ (pn+1 ! bdn). 2. MIPC 8 bwn (n 1), where bwn =
Wn
(pi !
W
6 p j ). 3. MIPC 8 bcn (n 1), where bcn = p0 _ (p0 ! p1 ) _ 1 1 1 _ (p0 ^ 1 1 1 ^ pn01 ! p n ). i=0
i=j
Proof. It immediately follows from Corollary 4.4 and the well-known characterizations of Int + bdn , Int + bwn and Int + bcn (see e.g. [4]).
It should be noted that in [2] the nite model property of MIPC 8 bwn and that of (every extensions of) MIPC 8 bdn have been already obtained. Also, it is shown in [1] that, for every 0 For L, MIPC 8 0 has the nite model property if either (1) 0 consists of f^; !; ?g-formulas; or (2) Int + 0 is locally tabular.
5 Conclusion In this paper we gave a sucient condition for intuitionistic modal logics over MIPC (in the lattice of 3}-normal intuitionistic modal logics) to have the nite model property. We showed that every MIPC-logic has the nite model property if there exists a universal rst-order sentence 8 such that (1) L is characterized by the class of full MIPC-frames satisfying 8 and (2) every full MIPC-frame that validates L satis es 8. We also applied our result to show the nite model property of some intuitionistic modal logics over MIPC whose nite model property had been previously unknown.
Acknowledgments Deep appreciation goes to Frank Wolter, who leads the authors into this topic. Thanks are also due to Guram Bezhanishvili for helpful comments.
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[3] R. A. Bull. MIPC as the formalization of an intuitionist concept of modality. Journal of Symbolic Logic, 31(4):609{616, 1966. [4] A. Chagrov and M. Zakharyaschev. Modal Logic, volume 35 of Oxford Logic Guide. Oxford University Press, 1997. [5] C. Grefe. Fisher Servi's intuitionistic modal logic has the nite model property. In Advances in Modal Logic '96, M. de Rijke et al. (eds.). CSLI, 1997. [6] G. Fischer Servi. On modal logic with an intuitionistic base. Studia Logica, 36(3):141{149, 1977. [7] G. Fischer Servi. The nite model property for MIPQ and some consequences. Notre Dame Journal of Formal Logic, XIX:687{692, 1978. [8] G. Fischer Servi. Axiomatizations for some intuitionistic modal calculi. Rendiconti del Seminaro Matematico Universita e Politecnico di Torino, 42:179{ 194, 1984. [9] H. J. Keisler. Fundamentals of model theory. In Handbook of Mathematical Logic, J. Barwise (ed.). North-Holland, 1977. [10] H. Ono. On some intuitionistic modal logics. Publications of the Research Institute for Mathematical Science, 13(3):687{722, 1977. [11] H. Ono and N.-Y. Suzuki. Relations between intuitionistic modal logics and intermediate predicate logics. Reports on Mathematical Logic, 22:65{ 87, 1988. [12] A. Prior. Time and Modality. Clarendon Press, 1957. [13] A. K. Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD Thesis, University of Ediburgh, 1994. [14] F. Wolter and M. Zakharyaschev. On the relation between intuitionistic and classical modal logics. Algebra and Logic, 36:73{92, 1997. [15] M. Zakharyaschev, F. Wolter and A. Chagrov. Advanced Modal logic. In Handbook of Philosophical Logic, Gabbay et al. (eds.). To appear.
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