SOME REMARKS ON MAEHARA S METHOD. Abstract

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1 Bulletin of the Section of Logic Volume 30/3 (2001), pp Takahiro Seki SOME REMARKS ON MAEHARA S METHOD Abstract In proving the interpolation theorem in terms of sequent calculus, Maehara s method is usually used. This paper shows that the interpolation theorem for substructural logics obtained by deleting constants from FL, FLe, FLec, CFLe and CFLec can be proved by putting some restriction on partitions in Maehara s method. 1. Introduction The following claim, called the interpolation theorem, is first proved by W.Craig in 1957 (see [1] and [2]): if A B is a theorem and if A and B have at least one predicate symbol in common, then there is an intermediate formula D such that both A D and D B are theorems and that all predicate symbols occurring in D also occur both in A and in B. Also, if A B and if A and B have no predicate symbol in common, then either A or B is a theorem. After that, S.Maehara showed in 1960 that the interpolation theorem follows from the cut elimination theorem for LK. The method which he used in the proof is called now Maehara s method. According to [3], we explain Maehara s method concisely. First, let and be constants and add and

2 148 Takahiro Seki to initial sequent. Then we generalize the interpolation theorem as follows: if a sequent Γ is provable, and if Γ and divide any two parts Γ 1, Γ 2 and 1, 2, respectively, then there exists a formula D such that both Γ 1 D, 1 and Γ 2, D 2 are provable and all predicate symbols occurring in D also occur both in Γ 1, 1 and in Γ 2, 2. Maehara s method gives many positive results on the interpolation theorem for the logics which admit the cut elimination theorem. If we want to apply this method to the logic without constants, then we once introduce constants and eliminate them after proving the generalized interpolation theorem. This procedure is considered to simplify the proof. It is known that Maehara s method cannot be applied directly to substrucutural logics FL, FL e, FL ec, CFL e and CFL ec, obtained from FL, FLe, FLec, CFLe and CFLec, respectively. As shown in this paper, the interpolation theorem for above logics can be proved by putting some restriction on partitions in Maehara s method. Then the proof is similar way to Maehara s method without the procedure introducing and eliminating constants. Further, we notice that this method gives the proof of the interpolation theorem for LK etc. without using constants. 2. Preliminaries The language of our logics consists of predicate symbols and logical connectives,,, and (and +, if possible) and quantifiers and. The formulas are defined in the usual way and denoted by A, B, C,. Further, Greek capital letters Γ,, denote (finite, possibly empty) sequence of formulas. A sequent is of the form A 1,, A m B 1,, B n, where A 1,, A m, B 1,, B n (m, n 0) are formulas. The sequent calculus CFL ec is as follows, where t and a is a term and a variable, respectively.

3 Some Remarks on Maehara s Method 149 Initial sequent Contraction rules A A Γ, A, A, Σ Γ, A, Σ (c ) Γ Λ, A, A, Θ ( c). Γ Λ, A, Θ Exchange rules Γ, A, B, Σ Γ, B, A, Σ (e ) Γ Λ, A, B, Θ ( e) Γ Λ, B, A, Θ Cut rule Γ A, Θ Σ, A, Π Γ, Σ, Π, Θ Rules for logical connectives Γ A, Θ Γ, ( ) A, Γ Θ A Θ ( ) Γ A, Θ Γ A, Θ Π, B, Σ Π, A B, Γ, Σ, Θ ( ) Γ, A B, Θ Γ A B, Θ ( ) Γ, A, Σ Γ, A B, Σ ( 1) Γ, B, Σ ( 2) Γ, A B, Σ Γ Λ, A, Θ Γ Λ, B, Θ ( ) Γ Λ, A B, Θ Γ, A, Σ Γ, B, Σ ( ) Γ, A B, Σ Γ Λ, A, Θ Γ ( 1) Γ Λ, A B, Θ Λ, B, Θ ( 2) Γ Λ, A B, Θ Γ, A, B, Σ Γ, A B, Σ ( ) Γ A, Λ Σ B, Θ ( ) Γ, Σ A B, Λ, Θ Γ, A Λ Σ, B Θ Γ, Σ, A + B Λ, Θ (+ ) Γ Π, A, B, Σ ( +) Γ Π, A + B, Σ

4 150 Takahiro Seki Rules for quantifiers Γ, A(t), Σ Γ, xa(x), Σ ( ) Γ Λ, A(a), Θ ( ) Γ Λ, xa(x), Θ Γ, A(a), Σ Γ, xa(x), Σ ( ) Γ Λ, A(a), Θ ( ) Γ Λ, xa(x), Θ For ( ) and ( ), there are restrictions on variables: the variable a must occur in neither Γ nor. CFL e is obtained from CFL ec by deleting contraction rules. FL ec is obtained from CFL ecby restricting as follows: (1) at most one formula occurs in, (2) Λ and Θ must be empty, and (3) deleting ( c), ( e), (+ ) and ( +). FL e is obtained from FL ec by deleting (c ). FL is obtained from FL e by deleting (e ). The definition of proofs and provability are as usual ways. In particular, a formula A is provable if a sequent A is provable. The following cut elimination theorem is well-known. For more information, see [4]. Theorem 1. If a sequent Γ is provable in FL, FL e, FL ec, CFL e or CFL ec, then it is also provable in FL, FL e, FL ec, CFL e or CFL ecwithout using the cut rule. 3. Proof of interpolation theorem We will introduce some notions through this section. A sequent Γ is standard if at least two formulas appears in it. For any formula A, V (A) denotes the set of all predicate symbols appearing in A. When Γ is a sequence of formulas A 1,, A n, V (Γ) denotes V (A 1 ) V (A n ). First, we prove the interpolation theorem for CFL e and CFL ec. To do this, we will introduce some notions. For a given (not necessary standard) sequent Γ, a pair (Γ 1 : 1 ); (Γ 2 : 2 ) of the pair of (possibly empty) sequences of formulas (Γ 1 : 1 ) and (Γ 2 : 2 ) is called a partition of Γ, if each component of Γ appears just once either in Γ 1 or in Γ 2, and if each component of appears just once either in 1 or 2. Furthermore, a partition (Γ 1 : 1 ); (Γ 2 : 2 ) of Γ is conditional, if both

5 Some Remarks on Maehara s Method 151 Γ 1, 1 and Γ 2, 2 are not empty. Therefore conditional partitions (below, c-partitions) are defined only for standard sequents. Then in order to prove the interpolation theorem for CFL e and CFL ec, it is sufficient to show the following. Lemma 2. Let L be CFL e or CFL ec. If a standard sequent Γ is provable in L and (Γ 1 : 1 ); (Γ 2 : 2 ) is a c-partition of Γ, then there exists a formula D (called interpolant) such that (a) both Γ 1 1, D and D, Γ 2 2 are provable in L (b) V (D) V (Γ 1, 1 ) V (Γ 2, 2 ). Proof. By Theorem 1, there is a cut-free proof P of Γ C. This lemma is proved by induction on the length n of P. Here we will prove only the case that L is CFL e. Suppose that n = 1. In this case, Γ must be an initial sequent A A. It is clear that an initial sequent is standard and that V (A) V (A). Then a c-partition is either (A : ); ( : A) or ( : A); (A : ). For the former, we may take A as an interpolant. For the latter, we may take A as an interpolant. Next, suppose that n > 1. Let I be the last rule of inference in P. Here, we prove the problematic case that I is ( ). Since L has exchange rules, it may assume that I has the following form: Γ A, Θ B, Σ A B, Γ, Σ, Θ. 1) Consider a c-partition (A B, Γ 1, Σ 1 : 1, Θ 1 ); (Γ 2, Σ 2 : 2, Θ 2 ) of A B, Γ, Σ, Θ. We should consider the following three cases. 1.1) Suppose that both Γ 2, Θ 2 and Σ 2, 2 are not empty. Take a c- partition (Γ 1 : A, Θ 1 ); (Γ 2 : Θ 2 ) of Γ A, Θ and a c-partition (B, Σ 1 : 1 ); (Σ 2 : 2 ) of B, Σ. By the hypotheses of induction, there exist formulas D and E such that (a) both Γ 1 A, Θ 1, D and D, Γ 2 Θ 2 are provable (b) V (D) V (Γ 1, A, Θ 1 ) V (Γ 2, Θ 2 ) (c) both B, Σ 1 1, E and E, Σ 2 2 are provable (d) V (E) V (B, Σ 1, 1 ) V (Σ 2, 2 ). Using ( e), ( ), (+ ) and ( +), we see that

6 152 Takahiro Seki (e) both A B, Γ 1, Σ 1 1, Θ 1, D + E and D + E, Γ 2, Σ 2 2, Θ 2 are provable (f) V (D + E) V (A B, Γ 1, Σ 1, 1, Λ 1 ) V (Γ 2, Σ 2, 2, Λ 2 ). 1.2) Suppose that Γ 2, Θ 2 is not empty and Σ 2, 2 is empty. Then B, Σ 1 1 is provable. Take a c-partition (Γ 1 : A, Θ 1 ); (Γ 2 : Θ 2 ) of Γ A, Θ. By the hypothesis of induction, there exists a formula D such that (a) both Γ 1 A, Θ 1, D and D, Γ 2 Θ 2 are provable (b) V (D) V (Γ 1, A, Θ 1 ) V (Γ 2, Θ 2 ). Using ( e) and ( ), we see that D is an interpolant. 1.3) Suppose that Γ 2, Θ 2 is empty and Σ 2, 2 is not empty. Similar to (1.2). 2) Consider a c-partition (Γ 1, Σ 1 : 1, Θ 1 ); (A B, Γ 2, Σ 2 : 2, Θ 2 ) of A B, Γ, Σ, Θ. Similar to (1). Next, we prove the interpolation theorem for FL e and FL ec. To do this, we will introduce some notions. For a given (not necessary standard) sequent Γ C, a pair Γ 1 ; Γ 2 of (possibly empty) sequences of formulas Γ 1 and Γ 2 is called a partition of Γ C, if each component of Γ appears just once either in Γ 1 or in Γ 2. Furthermore, a partition Γ 1 ; Γ 2 of Γ C is conditional, if both Γ 1 and Γ 2, C are not empty. Then in order to prove the interpolation theorem for FL e and FL ec, it is sufficient to show the following. The proof is as in Lemma 2. Lemma 3. Let L be FL e or FL ec. If a standard sequent Γ C is provable in L and Γ 1 ; Γ 2 is a c-partition of Γ C, then there exists a formula D such that (a) both Γ 1 D and D, Γ 2 C are provable in L and (b) V (D) V (Γ 1 ) V (Γ 2, C). Finally, we prove the interpolation theorem for FL. To do this, we will introduce some notions. For a given (not necessary standard) sequent Γ C, we call a triple Γ 1 ; Γ 2 ; Γ 3 of (possibly empty) sequences of formulas Γ 1, Γ 2 and Γ 3, partition of Γ C, if each component of Γ appears just once either in Γ 1, Γ 2 or in Γ 3 preserving the order of occurrence in Γ, i.e., Γ 1, Γ 2, Γ 3 is just Γ. Furthermore, a partition Γ 1 ; Γ 2 ; Γ 3 of Γ C is conditional, if both Γ 2 and Γ 1, Γ 3, C are not empty.

7 Some Remarks on Maehara s Method 153 The following lemma plays an essential role in proving the interpolation theorem for FL. The proof is analogue to Lemma 2. Lemma 4. Let a standard sequent Γ C be provable in FL and Γ 1 ; Γ 2 ; Γ 3 be a c-partition of Γ C. Then there exists a formula D such that (a) both Γ 2 D and Γ 1, D, Γ 3 C are provable in FL, and (b) V (D) V (Γ 2 ) V (Γ 1, Γ 3, C). From Lemmas 2 through 4, it is easy to see the following. Theorem 5. Let L be FL, FL e, FL ec, CFL e or CFL ec. If A B is provable in L, then there exists a formula D such that (a) both A D and D B are provable in L (b) V (D) V (A) V (B). 4. Concluding remarks By using above method, we can also prove the interpolation theorem for LK, LJ and so on. For LK, it is sufficient to show the following, where terminologies are as in Section 3. Suppose that a standard sequent Γ is provable in LK and (Γ 1 : 1 ); (Γ 2 : 2 ) is a c-partition of Γ. Then either (i) or (ii) holds: (i) if V (Γ 1, 1 ) V (Γ 2, 2 ), there exists a formula D such that (a) both Γ 1 1, D and D, Γ 2 2 are provable in LK (b) V (D) V (Γ 1, 1 ) V (Γ 2, 2 ); (ii) otherwise, either Γ 1 1 or Γ 2 2 is provable in LK. For more precise proof through this paper, see [5]. Acknowledgements. The author would like to thank Professor Hiroakira Ono for his many helpful suggestions and advices. The author is also grateful to Professor Mitio Takano for his valuable comments and discussions.

8 154 Takahiro Seki References [1] W. Craig, Linear reasoning. A new form of the Herbrand-Gentzen theorem, The Journal of Symbolic Logic 22 (1957), pp [2] W. Craig, Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory, The Journal of Symbolic Logic 22 (1957), pp [3] S. Maehara, Craig no interpolation theorem (in Japanese), Suugaku 12 (1960), pp [4] H. Ono, Proof-theoretic methods in nonclassical logic an introduction, in Theories of Types and Proofs, edited by M. Takahashi, M. Okada and M. Dezani-Ciancaglini, MSJ Memoirs 2, Mathematical Society of Japan, (1998), pp [5] T. Seki, On the proof of the interpolation theorem without using constants, Research Report IS-RR , Japan Advanced Institute of Science and Technology, School of Information Science Japan Advanced Institute of Science and Technology Tatsunokuchi, Ishikawa, , Japan tseki@jaist.ac.jp

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