Modal Nonassociative Lambek Calculus with Assumptions: Complexity and Context-Freeness

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1 Modal Nonassociative Lambek Calculus with Assumptions: Complexity and Context-Freeness Zhe Lin Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China Faculty of Mathematics and Computer Science Adam Mickiewicza University, Poznàn, Poland Abstract. We prove that the consequence relation of the Nonassociative Lambek Calculus with S4-modalities (NL S4) is polynomial time decidable and categorial grammars based on NL S4 with finitely many assumptions generate context-free languages. This extends earlier results of Buszkowski [3] for NL and Plummer [16][17] for a weaker version of NL S4 without assumptions. 1 Introduction and Preliminaries Nonassociative Lambek Calculus NL is a type logical system for categorial grammars, introduced in Lambek [10] as a nonassociative version of Syntactic Calculus of Lambek [9]. Both systems are regarded now as basic logics for Type-Logical Grammar (categorial grammar). In a sense, NL is a purely substructural logic, since its sequent system admits no structural rules. Syntactic Calculus, now called the Lambek calculus (L), admits one structural rule (Associativity). Moortgat [12] studies NL with unary modalities, (also several pairs i, i ) as a system which enables one to use certain structure postulates in a controlled way. For instance, one can admit a restricted permutation ( A) B = B ( A) or restricted contraction ( A) ( A) A instead of their unrestricted forms A B, A A A. This follows the usage of exponentials!,? in Linear Logic (Girard [5]). Pentus [14] proves that categorial grammars based on L, (L-grammars), generate precisely all (ε-free) context-free languages. Analogous results for NLgrammars are due to Buszkowski [1] (the product-free NL) and Kandulski [8]. Jäger [7] provides a new proof for NL, which employs a special form of interpolation (of subtrees of the antecedent tree by single formulae). Buszkowski [3] refines Jäger s interpolation to prove that the consequence relation of NL is polynomial time decidable and categorial grammar, based on NL with finitely many assumptions, generate context-free languages. Buszkowski and Farulewki [4] extend the method to Full NL, i.e. NL with additives, with distribution (of respect to, and conversely) and finitely many assumptions, proving the context-freeness. A.-H. Dediu, H. Fernau, and C. Martín-Vide (Eds.): LATA 2010, LNCS 6031, pp , c Springer-Verlag Berlin Heidelberg 2010

2 Modal Nonassociative Lambek Calculus with Assumptions 415 These results are seemingly in conflict with the known facts that the consequence relation for L is undecidable and categorial grammars based on L with assumptions can generate all recursively enumerable languages (Buszkowski [2][3]). Pentus [15] shows that the provability problem for the pure L is NP-complete. Plummer [16][17] employs Jäger s method with refinements of Buszkowski [3] to prove the context-freeness of categorial grammars, based on NL with modalities,, satisfying the axioms: T: A A, 4: A A. He also claims (without proof) the polynomial time decidability of the resulting system. A key idea of Plummer s work is to replace the above axioms with some corresponding structural rules. Here we extend Plummer s result for systems with assumptions. Precisely, we consider the system NL S4 in the sense of Moortgat [12], which admits T, 4, and K: (A B) A B, but our results remain valid for the system with 4, T only (NL S4 in the sense of Plummer [16]). We prove that the consequence relation for NL S4 with finitely many assumptions is polynomial time decidable and categorial grammars based on it generate precisely the (ε-free) context-free languages. Let us recall the sequent system of NL. Formulae (types) are formed out of atomic types p, q, r...by means of three binary operation symbols (product), \ (right residuation), / (left residuation). Formulae are denoted by A, B, C,... Formula trees (formula-structures) are recursively defined as follows: (i) every formula is a formula-tree, (ii)ifγ, Δ are formula-trees, then (Γ Δ) isaformulatree. Sequents are of the form Γ A such that Γ is a formula tree and A is a formula. One admits the axioms: (Id) A A and the inference rules (\L) Δ A; Γ [B] C Γ [Δ (A\B)] C (\R) A Γ B Γ A\B Γ [A] C; (/L) Δ B (/R) Γ B A Γ [(A/B) Δ] C Γ A/B Γ [A B] C ( L) ( R) Γ A; Δ B Γ [A B] C Γ Δ A B (CUT) Δ A; Γ [A] B Γ [Δ] B The cut-elimination theorem for NL is proved by Lambek [10]. It yields the decidability and the subformula property of NL. However, the cut-elimination theorem does not hold if we affix new non-logical assumptions of the form

3 416 Z. Lin A B. In section 2, we introduce a purely syntactic method based on the cut-elimination theorem in some restricted form to prove a subformula property in some extended form, for systems with finitely many non-logical assumptions. This method is new and ensensially different from the model-theoretic method proposed by Buszkowski [3][4]. We describe the formalism of NL S4. Formulae (types) are formed out of atomic types p, q, r... by means of three binary operation symbols, \, / and two unaryoperation symbols,. Formula trees (formula-structures) are recursively defined as follow: (i) every formula is a formula-tree, (ii) ifγ, Δ are formulatrees, then (Γ Δ) is a formula-tree, (iii) ifγ is a formula-tree, then Γ is a formula-tree. Sequents are of the form Γ A such that Γ is a formula tree and A is a formula. The sequent system of NL S4 is obtained by extending NL with inference rules for the unary modalities and structural rules corresponding to axioms 4, T, K. The following are sequent rules for the unary modalities: ( L) Γ [ A ] B Γ [ A] B Γ A ( R) Γ A Γ [A] B ( L) Γ [ A ] B A ( R) Γ Γ A The following are structural rules corresponding to axioms 4, T, K: (4) Γ [ Δ ] A Γ [ Δ ] A (T) Γ [ Δ ] A Γ [Δ] A (K) Γ [ Δ 1 Δ 2 ] A Γ [ Δ 1 Δ 2 ] A. Our interest in modal postulates and non-logical assumptions can be motivated in various way. First, in linguistic practice, different phenomena may require different sets of structure postulates. For instance, let us consider an NP (noun phrase) such as the man who Mary met today, (an example from Versmissen [18]). In Lambek notation, there is no suitable type assigned to the relative pronoun who in this noun phrase. The solution proposed by Morrill [13] is to assign to who the single type (N\N)/(S/ NP). One also assigns NP/N to the, N to man, NP to Mary, N\(S/NP) to met,ands\s to today. The sequent NP/N N (N\N)/(S/ NP) NP NP\(S/NP) S\S NP, corresponding to the noun phrase the man who Mary met today is derivable in systems enriched with postulate T and B A A B. Moreexamples can be found in Morrill [13]. Second, there are many evidences for the usefulness of non-logical assumptions in linguistics. For example, Lambek [11] uses axioms of the form π i π to express the inclusion of the class of pronouns in i th Person in the class of pronouns. Further, in the Lambek calculus we can not transform S\(S/S) (the type of sentence conjunction) to VP\(VP/VP) (the type of verb phrase conjunction), however, we can add the sequent S\(S/S) VP\(VP/VP) as an assumption.

4 Modal Nonassociative Lambek Calculus with Assumptions NL S4 Enriched with Non-logical Assumptions We assume that non-logical assumptions are of the form A B. ForasetΦof formulae A B, NL S4 (Φ) denotes the system of NL S4 with all formulae from Φ as assumptions. Usually, if we can prove the cut-elimination theorem for a system then we immediately get the subformula property: all formulae in a cut-free derivation are subformulae of the endsequent formulae. (CUT) is a legal rule in NL S4 (Φ), and the cut-elimination theorem does not hold for NL S4 (Φ). Hence we can not obtain the standard subformula property for NL S4 (Φ). Here we consider the extended subformula property (see [3]). We introduce a restricted cut rule, Φ- restricted cut, where Φ is the set of assumptions. By Φ-restricted cut, we mean the following rules: (Φ CUT) Γ 2 A Γ 1 [B] C Γ 1 [Γ 2 ] C where A B is an assumption in Φ. We describe another Genzten style presentation of NL S4 (Φ), denoted by NL r S4 (Φ): Axioms of NLr S4 (Φ) are (Id) A A. The inference rules of NLr S4 (Φ) are simply the rules of NL S4 (Φ) together with the Φ-restricted cut given above. By S Γ A, we denote: the sequent Γ A is provable in system S. Lemma 1. If A B Φ then NL r S4 (Φ) A B. Proof: Assume A B Φ. We apply Φ-restricted cut to axioms A A and B B and get A B. Hence NL r S4 (Φ) A B. We provide a proof of the cut-elimination theorem for NL r S4 (Φ). Theorem 2. Every sequent which is provable in NL r S4 (Φ) can be proved also in (Φ) without (CUT). NL r S4 Proof: We must prove: if both premises of (CUT) are provable in NL r S4 (Φ) without (CUT), then the conclusion of (CUT) is provable in NL r S4 (Φ) without (CUT). The proof can be arranged as follows. We apply induction (i): on D(A), the complexity of (CUT) formula A, i.e. the total number of occurrences of symbols in A. For each case, we apply induction (ii): on R(CUT), the rank of (CUT), i.e. the total number of sequents appearing in the proofs of both premises of (CUT). Let us consider one case A = A. Others can be treated similarly. We write the premises of (CUT) as Γ 2 A and Γ 1 [A] B obtained by rules R i (i= 1, 2), respectively. We switch on induction (ii). Three subcases arise. 1. Γ 2 A or Γ 1 [A] B is an axiom (Id). If Γ 2 A is a axiom then Γ 2 = A, and Γ 2 [A] B equals Γ 1 [Γ 2 ] B. IfΓ 1 [A] B is a axiom then Γ 2 A equals Γ 1 [Γ 2 ] B. 2. R 1 R orr 2 L. The thesis follows from the induction hypothesis (ii). We show two typical cases. Others can be treated similarly.

5 418 Z. Lin 2a Let Γ 2 A arise by Φ-restricted cut with premises Γ 2 C and D A. We replace Γ 2 C D A (Φ CUT) Γ 2 A Γ 1 [Γ 2 ] B where C D is an assumption, by... Γ 1 [A] B (CUT).... Γ 2 C D A Γ [ A] B Γ 1 [D] B Γ 1 [Γ 2 ] B (CUT ) (Φ CUT) where C D is an assumption. Clearly R(CUT ) < R(CUT). Hence Γ 1 [D] B is provable without (CUT), by the hypothesis of induction (ii). Then, Φ-restricted cut yields Γ 1 [Γ 2 ] B. Γ 1 [Γ 2 ] B 2b Let Γ 1 [A] B arise by Φ-restricted cut. Similarly, we can first apply (CUT) to Γ A and the premise of Γ 1 [A] C which contains cut formula A. After that, we apply Φ-restricted cut to the conclusion of the new (CUT) and the other premise of Γ 1 [A] B. The thesis follows from the induction hypothesis (ii). 3. R 1 = R andr 2 = L. We replace Γ 2 A Γ 1 [ A ] B Γ 2 A ( R) Γ 1 [ A ] B Γ 1 [Γ 2 ] B where Γ 2 = Γ 2,by ( L) (CUT) Γ 2 A Γ 1 [ A ] B Γ 1 [ Γ 2 ] B (CUT ) where Γ 2 = Γ 2.SinceD(A ) < D(A) then Γ 1 [Γ 2 ] B is provable in NL r S4 (Φ) without (CUT) by the hypothesis of induction (i). Let π be a cut free proof in NL r S4 (Φ). By π+, we mean a proof obtained from π by replacing all occurrences of Φ-restricted cut by two applications of (CUT) as follows: Γ 2 A A B... (CUT) Γ 2 B Γ 1 [B] C (CUT) Γ 1 [Γ 2 ] C where A B is a nonlogical assumption in Φ. Obviously, π + is a proof in NL S4 (Φ). Hence we get the following corollary. Corollary 3. For any sequent Γ A provable in NL S4 (Φ), there exists a proof of Γ A such that all formulae appearing in the proof are subformulae of formulae in Φ or Γ A.

6 Proof: above. Modal Nonassociative Lambek Calculus with Assumptions 419 Follows from Lemma 1, Theorem 2, and the construction of π + given Let T be a finite set of formulae, closed under subformulae which contains all formulae appearing in Φ. By a T-sequent, we mean a sequent Γ A such that all formulae appearing in Γ and A belong to T. Corollary 4. For any T -sequent Γ A provable in NL S4 (Φ), there is a proof of Γ A in NL S4 (Φ) such that all sequents appearing in this proof are T -sequents. Proof: Immediate from Corollary 3. 3 Main Results Here after, we assume that Φ is finite. T denotes a finite set of formulae containing all formulae in Φ and closed under subformulae. Let T = { A A T }, T = T T, T = { A A T } and T = T T.Asequentissaidtobe basic if it is a T -sequent of the form A B C, A B, or A B. We describe an effective procedure producing all basic sequents provable in NL S4 (Φ). Let S 0 consist of all T -sequents from Φ, allt -sequents of the form (Id), and all T -sequents of the form: A A, A A, A A. A (A\B) B, (A/B) B A, A B A B. Assume S n has already been defined. S n+1 is S n enriched with all sequents arising by the following rules: (R1) if ( A B) S n and A T then ( A B) S n+1, (R2) if ( A B) S n and B T then (A B) S n+1, (R3) if ( A B C) S n and C T then (A B C) S n+1, (R4) if ( A B) S n then (A B) S n+1, (R5) if (A B C) S n and A B T then (A B C) S n+1, (R6) if (A B C) S n and (A\C) T then (B A\C) S n+1, (R7) if (A B C) S n and (C/B) T then (A C/B) S n+1, (R8) if (A B) S n and ( B C) S n then ( A C) S n+1, (R9) if (A B) S n and (D B C) S n then (D A C) S n+1, (R10) if (A B) S n and (B D C) S n then (A D C) S n+1, (R11) if (Γ B) S n and (B C) S n then (Γ C) S n+1,

7 420 Z. Lin Obviously, S n S n+1, for all n 0. For any n 0, S n is a finite set of basic sequents. S T is defined as the union of all S n. Due to the definition of basic sequents, there are only finitely many basic sequents. Since S T is a set of basic sequents, hence it must be finite. This yields: there exists k 0such that S k = S k+1 and S T = S k. S T is closed under rules (R1)-(R11). The rules (R1), (R2), (R3), (R4), (R5), (R6), (R7) are ( L), ( R),(K),(T),( L), (\R), (/R) restricted to basic sequents, and (R8)-(R11) in fact describe the closure of basic sequents under (CUT). (R5)-(R7) and (R9)-(R11) are the same as in [3]. Lemma 5. S T can be constructed in polynomial time. Proof: Let n denote the cardinality of T. The total number of basic sequents of the form A B, A B, anda B C are no more than n 2, n 2,and n 3 respectively. Therefore there are at most m = n 3 +2 n 2 basic sequents. Hence we can construct S 0 in time O(n 3 ). The construction of S n+1 from S n requires at most 6 (m 2 n) +m 2 +6 m 3 steps. It follows that the time of this construction of S n+1 is O(m 3 ). Since the least k satisfying S T = S k does not exceed m. Thus we can construct S T in polynomial time, in time O(m 4 ). By S(T ), we denote the system whose axioms are all sequents from S T and whose only inference rule is (CUT). Clearly, every proof in S(T ) consists of T -sequents. By S(T ) Γ A we denote: Γ A is provable in S(T ). Lemma 6. Every basic sequent provable in S(T ) belongs to S T. Proof: We proceed by induction on the length of its proof in S(T ). For the base case, the claim is trivial. For the inductive case, we assume that s is a basic sequent provable in S(T ) such that s is obtained from premises s 1 and s 2 by (CUT). Since s is a basic sequent, clearly, s 1 and s 2 must be basic sequents. By the induction hypothesis, s 1 and s 2 belong to S(T ). Hence s belongs to S T, by (R8)-(R11). We prove two interpolation lemmas for S(T ). Lemma 7. If S(T ) Γ [Δ] A then there exists D T such that S(T ) Δ D and S(T ) Γ [D] A. Proof: We proceed by induction on the proofs in S(T ). Base case: Γ [Δ] A belongs to S T. We consider three subcases. First, if Γ = Δ = B then D = A and the claim stands. Second, if Γ = B, Δ = BorΔ= B then D = B or D = A, respectively. Third, if Γ = B C, and either Δ = B, or Δ = C, thend = B or D = C, respectively. Inductive case: Assume Γ [Δ] A is the conclusion of (CUT) whose both premises are Δ B and Γ [B] A such that Γ [Δ] =Γ [Δ ]. Then three cases arise.

8 Modal Nonassociative Lambek Calculus with Assumptions Δ is a substructure of Δ. Assume Δ = Δ [Δ ], Γ [B] =Γ [Δ [B]]. Hence there exists D T satisfying S(T ) Δ [B] D and S(T ) Γ [D] A by the induction hypothesis. We have S(T ) Δ D from S(T ) Δ B and S(T ) Δ [B] D, by (CUT). 2. Δ is a substructure of Δ. Assume Δ = Δ [Δ] andγ [B] =Γ [Δ [B]]. By the induction hypothesis, it is easy to obtain S(T ) Δ D and S(T ) Δ [D] B for some D T, which yields S(T ) Γ [D] A by (CUT). 3. Δ and Δ do not overlap. Hence Γ [B] mustcontainsδ. Assume Γ [B] = Γ [B,Δ]. By the induction hypothesis, there exists a D T such that S(T ) Γ [B,D] A and S(T ) Δ D. By (CUT), S(T ) Γ [Δ,D] A, which means S(T ) Γ [D] A. Lemma 8. If S(T ) Γ [ Δ ] A, then there exists D T such that S(T ) Δ D and S(T ) Γ [ D] A. Proof: Assume S(T ) Γ [ Δ ] A. By Lemma 7, there exists D T such that S(T ) Γ [D] A and S(T ) Δ D. Again by Lemma 7, we get S(T ) D D and S(T ) Δ D,forsomeD T.Weconsidertwo possibilities. If D T,then D T.Weget S(T ) D D, by Lemma 3.9 and (R1). Since D D and D D belong to S T,weget D D S T. Hence by two applications of (R4), D D S T, which yields S(T ) D D. By applying (CUT) to S(T ) Γ [D] A and S(T ) D D, weget S(T ) Γ [ D ] A. Again S(T ) Δ D and S(T ) D D,so we get S(T ) Δ D. Therefore the claim holds. If D does not belong to T,thenD = D for some D T. Hence S(T ) D Dand S(T ) Δ D. Due to Lemma 6, D Dbelongs to S T. It yields: D D belongs to S (T ),by(r4).hence S(T ) D D. Then, by (CUT), S(T ) Γ [ D ] A. Therefore the claim stands. For any T -sequent Γ A, byγ T A we mean: Γ A has a proof in NL S4 (Φ) consisting of T -sequents only. Lemma 9. For any T -sequent Γ A, Γ T A iff S(T ) Γ A. Proof: The if part is easy. Notice that all T -seuqents which are axioms of NL S4 (Φ) belong to S T. The only if part is proved by showing that all inference rules of NL S4 (Φ), restricted to T -sequents, are admissible in S(T ). The rules (CUT), (\L), (/L) (\R) (/R) ( L) ( R) are settled by Buszkowski[3]. Here we provide full arguments for ( L), ( R), ( L), ( R), (4), (T), (K). 1. For ( L), assume S(T ) Γ [ A ] B and A T. By Lemma 7, there exists D T such that S(T ) Γ [D] B and S(T ) A D. Since S(T ) A D is a basic sequent, then by Lemma 6, A D S T. By (R1), we get A D S T, which yields S(T ) A D. Hence S(T ) Γ [ A] B, by(cut).

9 422 Z. Lin 2. For ( R), assume S(T ) Γ A and A T.Since S(T ) A A, we get S(T ) Γ A, by (CUT). 3. For ( L), assume Γ [A] S(T ) B and A T.Since S(T ) A A, we get S(T ) Γ [ A ] B, by (CUT). 4. For ( R), assume S(T ) Γ B and B T. By Lemma 7, there exists D T such that S(T ) D B and Γ S(T ) D. Then D B S T, by Lemma 6. By (R2), D B S T, which yields D S(T ) B. Hence we get S(T ) Γ B, by (CUT). 5. For (4), assume S(T ) Γ [ Δ ] A. ByLemma8thereexists D T such that S(T ) Γ [ D] A and S(T ) Δ D. Since S(T ) D D, we get S(T ) D D, by (CUT). Hence S(T ) Γ [ Δ ] A, by two applications of (CUT). 6. For (T), assume S(T ) Γ [ Δ ] A. By Lemma 8, there exists D T such that S(T ) Γ [ D] A and S(T ) Δ D. Clearly, S(T ) Γ [Δ] A, by (CUT). 7. For (K), assume S(T ) Γ [ Δ 1 Δ 2 ] A. By Lemma 7, there exists D T such that S(T ) Γ [D] A and S(T ) Δ 1 Δ 2 D. Then, by applying Lemma 7 twice, we get S(T ) Δ 1 D 1, S(T ) Δ 2 D 2 and S(T ) D 1 D 2 D, forsomed 1,D 2 T. By the proof of case 1, we get S(T ) D 1 D 2 D. We consider three possibilities. First, D T. By Lemma 6, we obtain D 1 D 2 D S T. Hence, by (R3), S(T ) D 1 D 2 D. Since S(T ) D D, weget S(T ) D 1 D 2 D, by (CUT). Then, by three applications of (CUT), we get S(T ) Γ [ Δ 1 Δ 2 ] A. Second, D T but D T. Assume D = D,forsomeD T.Since S(T ) D D, by the proof for case 5, we obtain S(T ) D D. Then, due to the proof for case 4, we get S(T ) D D. Hence S(T ) D D, which yields S(T ) D 1 D 2 D. Since D 1 D 1 S T and D 2 D 2 S T, then, by rule (R4), D 1 D 1 S T and D 2 D 2 S T. Hence S(T ) D 1 D 2 D, by (CUT). We get S(T ) Γ [ Δ 1 Δ 2 ] A, by (CUT). Third, D T, but D T. Then, D = D,forsomeD T. Since S(T ) D D, clearly, we get S(T ) D D. Again, S(T ) D 1 D 2 D. Hence S(T ) Γ [ Δ 1 Δ 2 ] A, by several applications of (CUT). We define an operator on formula structure recursively as follows: (i) A = A, for any formula A. (ii) (Γ 1 Γ 2 ) = Γ 1 Γ 2, for any formula structures Γ 1 and Γ 2.(iii) ( Γ ) = (Γ ), for any formula structure Γ. Now we are ready to prove the main results of this paper. Theorem 10. If Φ is finite, then NL S4 (Φ) is decidable in polynomial time. Proof: Let Φ be a finite set of non-logical assumptions and Γ A be a sequent. Clearly, NLS4(Φ) Γ A can be checked in polynomial time if and only

10 Modal Nonassociative Lambek Calculus with Assumptions 423 if NLS4(Φ) Γ A can be checked in polynomial time. Let n be the number of logical constants and atoms occurring in Γ A and in sequents for Φ. The number of subformulae of any formula is equal to the number of logical constants and atoms in it. Hence T can be constructed in time O(n 2 ), and T contains at most n elements. It yields that we can construct T in time O(n 2 ). Since T T, by Corollary 4, NLS4(Φ) Γ A is provable in NL S4 (Φ) iff Γ T A. By Lemma 10, Γ T A iff S(T ) Γ A. SinceΓ A is a basic sequent, we get S(T ) Γ A iff Γ A S T, by Lemma 6. Hence Γ A is provable in NL S4 (Φ) iff Γ A S T. Besides, by Lemma 5, S T can be constructed in polynomial time. Consequently, NLS4(Φ) Γ A can be checked in time polynomial with respect to n. An NL S4 (Φ)-grammar over an alphabet Σ is a pair L, D, wherel, the lexicon, is a finite relation between strings from Σ + and formulae of NL S4 (Φ), and D F is a finite set of designated formulae (types). By s(γ ), we denote a string obtained from a formula structure Γ by dropping all binary and unary operators, and, respectively, and corresponding parentheses (). A language L(G) generated by a NL S4 (Φ)-grammar G = L, D is defined as a set of strings a 1 a n,wherea i Σ +,1 i n, andn 1, satisfying the following condition: there exist formulae A 1,...,A n,s,andformulae structure Γ such that for all 1 i n a i,a i L, S D,and NLS4(Φ) Γ S where s(γ )=A 1 A n. Notice that for NL S4 (Φ)-grammars the definition of L(G) can be modified by assuming that Γ does not contain. ForifΓ A is provable in NL S4 (Φ), then Γ B is provable in NL S4 (Φ), where Γ arises from Γ by dropping all. Theorem 11. Every language generated by an NL S4 (Φ)-grammar is contextfree. Proof: Let Φ be a finite set of sequents of the form A B, G 1 = L, D be an NL S4 (Φ)-grammar, and T be the set of all subformulae of formulae in D and all subformulae of formulae appearing in L. WeconstructT as above. Now we construct an equivalent CFG (context-free grammar) G 2, in the following way. The terminal elements of G 2 are lexical items of G 1. The non-terminals are all types from T and a fresh non-terminal S. Productions are {A B B A S T } {A B, C B C S T } {A v v, A L} {S A A D}. If v 1...v m is generated by G 1, then there is a NL S4 (Φ)-derivable sequent Γ B where B is a designated type, s(γ )=A 1 A m,and v i,a i Lfor 1 i m. WegetS G 2 B by the construction of G 2. Due to Lemma 9 and the construction of G 2,weobtainS G 2 A 1 A m which leads to S G 2 v 1...v m. Hence v 1 v m is generated by G 2. Now suppose v 1 v m is generated by G 2,whichmeansS G 2 v 1 v n. Then, there exists a B Dsuch that B G 2 A 1 A n where v i,a i L, 1 i m. Hence, by the construction of G 2, there exists a formula structure Γ such that s(γ )=A 1 A n and S(T ) Γ B. By Lemma 9, NLS4(Φ) Γ B. Therefore v 1 v m is generated by G 1.

11 424 Z. Lin Obviously, we can easily obtain the same results for systems without K (NL S4 in the sense of Plummer [16][17]). The inclusion of the class of ε-free context free languages in the class of NL S4 (Φ)-recognizable languages can be easily established. Every context-free language is generated by some NL-grammars (see [8]). Since neither the lexicon nor designated formulae contain modal operators, by Corollary 4, these NL-grammars can be conceived of as an NL S4 (Φ)-grammars, where Φ is empty. Hence NL S4 (Φ)-grammars generate exactly the ε-free contextfree languages. 4 Variants In [6], grammars based on L enriched with structural rules (K 1 ), (K 2 )are shown to surpass the context-free languages. Plummer [16][17] conjectures that structural rules (K 1 ), (K 2 ) extend the generative capacity of NL. However the situation is different for system with 4 and T. We consider a system NL S4,which admits 4, T, and K 1 : (A B) A B K 2 : (A B) A B The sequent system of NL S4 is obtained by extending NL S4 without (K) with the following rules corresponding to axioms K 1 K 2 (K 1 ) Γ [ Δ 1 Δ 2 ] A Γ [ Δ 1 Δ 2 ] A, (K 2) Γ [Δ 1 Δ 2 ] A Γ [ Δ 1 Δ 2 ] A The main results in section 3 can be easily extended to NL S4 (Φ), NL S4 (Φ) with finitely many non-logical assumptions Φ, utilizing the technique presented in section 3. We outline the proof as follows. It is easy to prove the analogues of theorem 2 for NL S4 (Φ). Then we modify the construction of S(T ). We replace rule (R3) by the following two rules: (R3.1) if A B C S n and C T then A B C S n +1, (R3.2) if A B C S n and C T then A B C S n +1. TheproofoftheanalogueofLemma9issimilartotheproofofLemma9in section 3, except for replacing Δ 1 Δ 2 by Δ 1 Δ 2 or Δ 1 Δ 2 and D 1 D 2 by D 1 D 2 or D 1 D 2. The remainder of the proof goes without changes. 5 Conclusion This article shows that the consequence relation of NL S4 or NL S4 are polynomial time decidable, and the categorial grammars based on NL S4 (Φ) or NL S4 (Φ) generate context-free languages. The basic idea of proof is adapted from Buszkowski [3]. However the proof of the extended subformula property is different. We introduce the Φ-restricted cut and prove the cut-elimination theorem for systems enriched with finitely many assumptions.

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