modal logic KB and KTB to construct Kripke models and to show the finite model
property of these logics. Our modal language L consists of the following set of ...
Bulletin of the Section of Logic Volume 36:3/4 (2007), pp. 183–193
Yutaka Miyazaki
NORMAL FORMS FOR MODAL LOGICS KB AND KTB Dedicated to Professor Hiroakira Ono in memory of his retirement
Abstract Normal forms for propositional modal logics are used to establish the Kripke completeness, the finite model property, and the decidability for modal logics KB and KTB.
1. Introduction Every modal formula embodies a finite model in which the formula itself is satisfiable. This naive observation helps us to establish the finite model property for some modal logics. Indeed, by using normal forms for modal logics, Kit Fine showed how to construct a finite model for a formula to reject it and established the finite model property for some standard modal systems, and moreover, he proved the finite model property for every modal logic over KD axiomatized by a set of, so called, uniform formulas ([2]). This result implies, in particular, that the modal logic KM is Kripke complete, has the finite model property and, is decidable, where M is McKinsey axiom, i.e. M := 23p → 32p. In this paper, we will show how to deal with normal forms for the modal logic KB and KTB to construct Kripke models and to show the finite model property of these logics. Our modal language L consists of the following set of symbols: (1) a denumerable set of propositional variables {p0 , p1 , . . .}, (2) the classical connectives: ∧, ∨, →, ¬, ⊥, (3) the modal connectives: 2 and 3, and (4) a
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pair of parentheses: (, ). The set Φ = ΦL of all formulas on L is defined as usual. For formulas A and B, A = B means that A is syntactically equal to B. A (normal modal) logic L in L is a set of formulas in Φ, which contains: (1) all classical tautologies in Φ, (2) a formula 2(p0 → p1 ) → (2p0 → 2p1 ), and is closed under (3) modus ponens, (4) uniform substitution, and (5) necessitation. For a logic L and formulas A, B, A is L-equivalent (or equivalent in L) to B (A ≡L B) if (A → B) ∧ (B → A) ∈ L. The smallest normal modal logic in L is denoted by K. For a logic L and Σ ⊆ Φ, L ⊕ Σ denotes the smallest normal modal logic that contains L and Σ. L ⊕ {A} is simply denoted by LA. Particular axioms B and T are defined as: B := p0 → 23p0 and T := 2p0 → p0 . In this paper we mainly consider logics KB and KTB. A Kripke frame, a Kripke model, and the interpretation of formulas in them are defined as usual ([1]). For a formula A, a Kripke model M = hW, R, V i, and a point a ∈ W , we denote (M, a) |= A to mean that A is true at a in M. Furthermore, for a Kripke frame F = hW, Ri, we denote F |= A to mean that A is valid in F. For a logic L and a class C of Kripke frames, L is Kripke complete for C if for any formula A, A ∈ L if and only if A is valid in every Kripke frame in C. L is Kripke complete if there is some class C of Kripke frames such that it is Kripke complete for C. L has the finite model property if there is some class D of finite Kripke frames such that it is Kripke complete for D.
2. Normal forms for K This section is a brief survey of Kit Fine’s paper [2] to introduce the notion of normal forms of modal formulas and some related terminology for future use. Hereafter we fix a finite set of propositional variables V ark := {pi | 0 ≤ i ≤ k}, and we consider formulas and logics only in our restricted language with the variable set V ark . The set of all formulas that are constructed from variables only in V ark is denoted by Φ(V ark ), and we assume that it is enumerated. For A ∈ Φ(V ark ), the degree deg(A) of A is defined as: deg(pi ) = deg(⊥) := 0, deg(A ∨ B) := max{deg(A), deg(B)}, deg(3B) := 1 + deg(B). Then the set Fn of normal forms of degree n is defined inductively as follows.
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(i) n = 0, F0 := {π0 p0 ∧ π1 p1 ∧ · · · ∧ πk pk | πi is blank, or ¬ for each i}. (ii) n ≥ 1, Fn := {B ∧ π0 3A0 ∧ π1 3A1 ∧ · · · ∧ πm 3Am | B ∈ F0 , Aj ∈ Fn−1 , πj is blank, or ¬ for each j}, where Fn−1 = {Aj }m j=0 . (iii) F :=
∞ [
Fn , which is the set of all normal forms in Φ(V ark ).
n=0
Note that for any C ∈ Fn , all normal forms of degree n − 1 appear in C, in their form, either 3Aj , or ¬3Aj and that the cardinality of Fn depends only on the degree n, so it is finite and constant for every n. For A ∈ Fn , the leading term A` of A is a normal form of degree 0 which is a conjunct of A. For A ∈ Fn and B ∈ Fn−1 , we denote A > B to mean that 3B is a conjunct of A. We can make use of normal forms in the following way, to obtain the finite model property and decidability of the logic K. The reducibility of any formula to normal forms in K is established in the following sense. Theorem 2.1. Every formula A ∈ Φ(V ark ) of degree less or equal to n is equivalent in K to ⊥ or a disjunction of normal forms of degree n. 2 A Kripke model An := hW, R, V i is defined out of normal forms as [ follows: W := Fi , R := {(A, B) ∈ W 2 | A > B}, and A ∈ V (pj ) if and i≤n
only if pj is a conjunct of A` . Then the following holds. Theorem 2.2. For An := hW, R, V i and for any A ∈ W , (An , A) |= A. 2 Based on these two theorems, we have the following theorem for K. Theorem 2.3. The modal logic K is Kripke complete for the class of all Kripke frames. Moreover, K has the finite model property, and so it is decidable. 2 In [2], Fine also showed the finite model property for modal logics KD ⊕ Σ, where D := 2p0 → 3p0 and Σ is an arbitrary set of uniform formulas, KT, and K4, where 4 := 2p0 → 22p0 , by modifying the above argument for K.
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3. Normal forms for KB First, we show one basic fact about normal forms below. Theorem 3.1. For normal forms A, B of the same degree n, if A 6= B, then A → ¬B ∈ K. Proof.
Case analysis on n.
Case n = 0.
Let A :=
k ^
πiA pi , and
i=0
B :=
k ^
πiB pi , where each πiA , πiB is ¬ or blank. Because A and B are
i=0
distinct, there exists at least one index s such that πsA 6= πsB , and so, k _ πsA ps → ¬πsB ps ∈ K. Thus we have A → ¬B ∈ K, since ¬B = ¬πiB pi . Case n ≥ 1. Let A := A` ∧
m ^
πiA 3Ci and B := B` ∧
i=0
m ^
i=0
πiB 3Ci , where
i=0
Fn−1 = {Ci }m i=0 . If A` 6= B` , then our conclusion follows from the same reason as in the case n = 0. So assume A` = B` . Then there exists at least one index s such that πsA 6= πsB , and so, πsA 3Cs → ¬πsB 3Cs ∈ K. Thus we have A → ¬B ∈ K by the similar argument as the previous case. 2 Now we introduce notions of a correlate and a counter-correlate for a normal form in the following. Let A be a normal form of degree n (n ≥ 1). The correlate A0 of A whose degree is n − 1 is defined in Fine’s paper [2] as: For n = 1, A0 := A` . For n ≥ 2, A0 := A` ∧
^
3C ∧
C∈∆
^
¬3C,
C∈Fn−2 \∆
where ∆ = ∆A := {C ∈ Fn−2 | ∃D ∈ Fn−1 , (A > D and D0 = C)}. On the other hand, for a normal form B of degree n (n ≥ 1), the counter-correlate B ◦ of B whose degree is n + 1 is defined as: ^ ^ B ◦ := B` ∧ 3C ∧ ¬3C, C∈Σ
C∈Fn \Σ
where Σ = ΣB := {C ∈ Fn | B > C 0 }.
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Now we will show some properties of correlate and counter-correlate of normal forms to prove the reducibility of formulas in KB to a normal form. Lemma 3.2. For normal forms A, B, where deg(A) = n and deg(B) = n − 1 (n ≥ 2), if A > B, then A0 > B 0 . Proof. If A > B, then B 0 ∈ ∆A , and so, we have A0 > B 0 .
2
Lemma 3.3. For any normal form A of degree n (n ≥ 1), (A◦ )0 = A. Proof. Induction on n. Case n = ^1. For an arbitrary C ∈ F0 such that A > C, define B := C ∧ 3C ∧ ¬3D. Then, because deg(B) = 1 D∈F0 \{C}
and A > C = B 0 , we have B ∈ ΣA . Thus B 0 = C ∈ ∆A◦ , and so, (A◦ )0 > C. Conversely, suppose (A◦ )0 > C for an arbitrary C ∈ F0 . Then, by the fact that C ∈ ∆A◦ , there exists D ∈ F1 such that A◦ > D and D0 = C. The former means that D ∈ ΣA and so, we have A > D0 = C. Case n ≥ 2. For C ∈ Fn−1 , suppose A > C. By induction hypothesis, A > C = (C ◦ )0 , and so we have C ◦ ∈ ΣA . Therefore A◦ > C ◦ , but then we have C ∈ ∆A◦ . Thus (A◦ )0 > C. For the converse, suppose (A◦ )0 > C for C ∈ Fn−1 . Then, C ∈ ∆A◦ , therefore, there exists D ∈ Fn such that A◦ > D and D0 = C. Thus we have A > D0 = C since D ∈ ΣA . 2 Lemma 3.4. For normal forms A, B, where deg(A) = n and deg(B) = n − 1 (n ≥ 2), if A > B, then A◦ > B ◦ . Proof. By Lemma 3.3, A > B = (B ◦ )0 , and so we have B ◦ ∈ ΣA . Thus A◦ > B ◦ . 2 By the definition of (·)◦ , A◦ cannot be specified, in general, when deg(A) = 0. But for B ∈ F1 , we define (B` )◦ to be B, then we can establish that the correlate and the counter-correlate of a normal form are the dual operations of each other in the sense of Lemma 3.3 and the following lemma. Lemma 3.5. For any normal form A of degree n (n ≥ 1), (A0 )◦ = A. Proof. Induction on n. Case n = 1. By the above convention, it is obvious that (A0 )◦ = (A` )◦ = A.
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Case n ≥ 2. For an arbitrary C ∈ Fn−1 such that A > C, by Lemma 3.2, we have A0 > C 0 , and so, C ∈ ΣA0 . Thus (A0 )◦ > C. Conversely, suppose (A0 )◦ > C for an arbitrary C ∈ Fn−1 . Then C ∈ ΣA0 , so A0 > C 0 . This means that C 0 ∈ ΣA , therefore there exists D ∈ Fn−1 such that A > D and D0 = C 0 . By the induction hypothesis, D = (D0 )◦ = (C 0 )◦ = C. Thus A > C. 2 Next, we will see a few facts on reducibility of normal forms. Lemma 3.6. (See [2]) For a normal form A of degree n (n ≥ 1), A → A0 ∈ K. Proof. Induction on n. Case n = 1. This case is trivial since A0 = A` . Case n ≥ 2. We will show that A → C ∈ K for every conjunct C of A0 . (a) Case C = A` . This is also trivial. (b) Case C = 3B. By the fact that A0 > B, there is D ∈ Fn−1 such that A > D and D0 = B. The former implies that A → 3D ∈ K. By the induction hypothesis, we have D → D0 = D → B ∈ K, and so, 3D → 3B ∈ K. Thus we can say that A → 3B = A → C ∈ K. (c) Case C = ¬3B. Let Π := {D ∈ Fn−1 | D0 = B}. For each D ∈ Π, ¬3D is a conjunct of A, because, by A0 6> B, it is certain ^ that for any E, E 0 = B implies A 6> E. Therefore we have A → ¬3D ∈ K. D∈Π
On the other hand, by Theorem 3.1, for any E ∈ Fn−1 \ Π, B → ¬E 0 ∈ K, since E 0 6= B and deg(E 0 ) = deg(B) . Then by the induction hypothesis, we have E → E 0 ∈ K, and so, B → ¬E _ ∈ K. Since deg(B) = n − 2 and by Theorem 2.1, B ≡K ⊥, or B ≡K D D∈Γ
for some Γ ⊆ Fn−1 , in both of which we have B →
_
D ∈ K.
D∈Fn−1
Therefore B →
_
D ∈ K, in particular. The last fact implies that
D∈Π
3B →
_
3D ∈ K, and so,
D∈Π
^
¬3D → ¬3B ∈ K. Thus, we have
D∈Π
A → ¬3B = A → C ∈ K.
2
Corollary 3.7. For a normal form A of degree n (n ≥ 1), A◦ → A ∈ K. Proof. By Lemma 3.3 and 3.6, A◦ → A = A◦ → (A◦ )0 ∈ K.
2
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Here the set of KB-suitable normal forms is defined inductively on the degree n of normal forms. Case n = 0: Every normal form of degree 0 is KB-suitable. Case n ≥ 1: A normal form A ∈ Fn is KB-suitable if for any B ∈ Fn such that A > B 0 , B 0 is KB-suitable and B > A0 . The following simple observation on the logic KB is a key fact for us. Fact 3.8. For formulas C and D, C → ¬3D ∈ KB if and only if D → ¬3C ∈ KB. Proof. Easy calculation by using B = p0 → 23p0 .
2
Now we are in a position to show the reduction theorem in the logic KB. Theorem 3.9. Every formula A ∈ Φ(V ark ) of degree less or equal to n is equivalent in KB to ⊥ or a disjunction of KB-suitable normal forms of degree n. Proof. If A ≡KB ⊥, we are done. So suppose A 6≡KB ⊥. By_ Theorem 2.1, there exist some normal forms {Ai }i∈I ⊆ Fn such that A ≡K Ai . Among i∈I
{Ai }i∈I , we may assume that all normal forms not KB-equivalent to ⊥ are recollected and renumbered in the set {A0 , A1 , . . . , At }. So we have t _ A ≡KB Aj . Then we have only to show that each Aj is KB-suitable by j=0
induction on n = deg(Aj ). Case n = 0. Of course Aj is KB-suitable. Case n ≥ 1. Suppose Aj > B 0 for an arbitrary B ∈ Fn . Then deg(B 0 ) = n − 1, by the induction hypothesis, B 0 is KB-suitable. Now assume that B 6> A0j . Then we have B → ¬3A0j ∈ K ⊆ KB. Therefore by Fact 3.8, A0j → ¬3B ∈ KB, and so, by Lemma 3.6 and Corollary 3.7 we have A◦j → ¬3B ∈ KB. Thus we have A◦j 6> B, which is equivalent to Aj 6> B 0 since Aj > B 0 implies A◦j > (B 0 )◦ = B by Lemma 3.5, and A◦j > B implies Aj = (A◦j )0 > B 0 by Lemma 3.3. 2 We need one more proposition to construct a model for KB out of normal forms, and to prove Kripke completeness for it.
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Lemma 3.10. For a normal form A of degree n, there is a KB-suitable normal form B of degree n + 1 such that B 0 = A. ^ Proof. Induction on n. If n = 0, define B ∈ F1 as B := A ∧ 3C∧ C∈Γ
^
0
0
¬3C, where Γ := {D ∈ F0 |D > A}. Then, B = B` = A. To prove
C∈F0 \Γ
B to be KB-suitable, suppose B > D0 for an arbitrary D ∈ F1 . Then, of course, D0 is KB-suitable, and by the definition of Γ, D > A = B 0 . If n ≥ 1, define B := A◦ . Then obviously B 0 = (A◦ )0 = A. Suppose B > C 0 for an arbitrary C ∈ Fn . Then, by our induction hypothesis, we have C 0 is KB-suitable. And we can also show that C > B 0 by the same argument as in the proof of the previous theorem. 2 Now we define a Kripke model Bn := hW, R, V i for n ≥ 0 as follows: W := {A ∈ Fn | A is KB-suitable}, R := {(A, B) ∈ W 2 | A > B 0 }, and A ∈ V (pj ) if and only if pj is a conjunct of A. For a normal form A, and for m (0 ≤ m ≤ deg(A)), a formula A(m) is defined inductively as: A(0) := A, and A(m) := (A0 )(m−1) ([2]). Then the following holds. Theorem 3.11. For 0 ≤ m ≤ n, the model Bn := hW, R, V i, and for any A ∈ W , (Bn , A) |= A(m) . Proof. Induction on k := deg(A(m) ) = n − m. Case k = 0. A(m) = A` , so obviously (Bn , A) |= A(m) by the definition of V . Case k ≥ 1. For each conjunct C of A(m) , we will show that (Bn , A) |= C. (a) Case C = (A(m) )` . Then, C = A` , and so this is also trivial. (b) Case C = 3B. By the fact that A(m) > B, there exists D0 ∈ Fk such that A(m−1) > D0 and D00 = B. The former implies that there exists D1 ∈ Fk+1 such that A(m−2) > D1 and D10 = D0 . Again, the former implies that there exists D2 ∈ Fk+2 such that A(m−3) > D2 and D20 = D1 . These inferences can be iterated to reach that there exists 0 Dm−1 ∈ Fn−1 such that A(0) > Dm−1 and Dm−1 = Dm−2 . Put D := Dm−1 , then we have that A > D and that D(m) = B. Then, by Lemma 3.10, there is a KB-suitable normal form E of degree n such that E 0 = D. For this E, we have A > E 0 , E (m+1) = B, and deg(E (m+1) ) = deg(E) − (m + 1) = k − 1. Therefore, by the
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Normal forms for ...
induction hypothesis, (Bn , E) |= E (m+1) . Since the first fact means A RE , finally we have (Bn , A) |= 3B. (c) Case C = ¬3B. We will show that (Bn , A) 6|= 3B. Consider an arbitrary D ∈ W such that A RD , which means that A > D0 . Now deg(D(m+1) ) = deg(D) − (m + 1) = k − 1, so by our induction hypothesis we have (Bn , D) |= D(m+1) . By the fact that A > D0 , A(m) > D(m+1) by Lemma 3.2. On the other hand, we also know that A(m) 6> B. Both facts imply that D(m+1) 6= B. Since deg(B) = deg(D(m+1) ), we can conclude, with the aid of Theorem 3.1, that (Bn , D) 6|= B, and so, we have (B, A) 6|= 3B. 2 Finally, we show our main theorem. Theorem 3.12. The modal logic KB is Kripke complete for the class of symmetric Kripke frames. Moreover it has the finite model property, and it is decidable. Proof. Let the frame Fn := hW, Ri of Bn = hW, R, V i. To check this frame is symmetric, suppose A RB for A, B ∈ W . This means that A > B 0 , and since A is KB-suitable, we have B > A0 , which says B RA . Therefore as is well known, Fn |= KB. Conversely, consider an arbitrary formula A of degree n such that A 6∈ KB. Then ¬A 6≡KB ⊥. By Theorem 3.9, there are p _ KB-suitable normal forms B0 , B1 , . . . , Bp ∈ Fn such that ¬A ≡KB Bj . j=0
Here, we have (Bn , Bj ) |= Bj for each j, and so, (Bn , B0 ) |= ¬A, in particular. Hence, (Bn , B0 ) 6|= A. 2
4. Normal forms for KTB The treatment for the logic KB and that for the logic KT, which is in [2], are combined to show that the modal logic KTB can also be treated in a similar fashion. Here, we see the outline of this treatment for KTB. A KTB-suitable normal form is defined to be a normal form of both KT-suitable and KB-suitable, that is, it is defined inductively on the degree n of a normal form in the following:
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Case n = 0: Every normal form of degree 0 is KTB-suitable. Case n ≥ 1: A normal form A ∈ Fn is KTB-suitable if A > A0 , and for any B ∈ Fn such that A > B 0 , B 0 is KTB-suitable and B > A0 . Then we have the reducibility of formulas to KTB-suitable normal forms as before, in the following way. Theorem 4.1. Every formula A ∈ Φ(V ark ) of degree less or equal to n is equivalent in KTB to ⊥ or a disjunction of KTB-suitable normal forms of degree n. 2 Furthermore, we are able to show a similar lemma as Lemma 3.10, also for KTB-suitable normal forms, that is used for proving Kripke completeness for KTB. Lemma 4.2. For a normal form A of degree n, there is a KTB-suitable normal form B of degree n + 1 such that B 0 = A. 2 Now we define a Kripke model Cn := hW, R, V i for n ≥ 0 as follows: W := {A ∈ Fn | A is KTB-suitable}, R := {(A, B) ∈ W 2 | A > B 0 }, and A ∈ V (pj ) if and only if Pj is a conjunct of A. Then the following satisfiability theorem holds. Theorem 4.3. For 0 ≤ m ≤ n, the model Cn := hW, R, V i, and for any A ∈ W , (Cn , A) |= A(m) . 2 Finally, we can also prove the Kripke completeness, the finite model property, and decidability for the modal logic KTB through its suitable normal forms. Theorem 4.4. The modal logic KTB is Kripke complete for the class of reflexive and symmetric Kripke frames. Moreover it has the finite model property, and it is decidable. 2
Remark Methods with normal forms are both elegant and constructive ([2]). However, we do not have a systematic way to characterize some class of modal logics in general through normal forms, except for a blessed class of modal logics.
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Let us consider a finitely axiomatizable modal logic L := K⊕A for some A ∈ Φ(V ark ). It seems that each L-suitable normal form B of degree n should fulfill the following requirement: Let An be a Kripke model defined in Section 2 for K, and Fn the frame part of An . Then the requirement seems to be that the generated subframe FB of Fn from the formula B ∈ W should validate A (see [2]). What is a general way to define L-suitable normal form to meet this requirement for any given A? Furthermore, for what kind of axioms does there exist a systematic way of defining the Lsuitability of normal forms to meet this requirement?
References [1] A. Chagrov, M. Zakharyaschev, Modal logic, Oxford University Press, 1997. [2] K. Fine, Normal forms in modal logic, Nortre Dame Journal of Formal Logic, 16, No.2 (1975), pp. 229–237.
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