Pregroups and their logics

Transcription

1 University of Chieti May 6-7, Pregroups and their logics Wojciech Buszkowski Adam Mickiewicz University Poznań, Poland

2 University of Chieti May 6-7, Kazimierz Ajdukiewicz ( ) was an excellent Polish philosopher, a prominent representative of the Lvov-Warsaw Philosophical School. Born: December 12, 1890 in Lvov LIFE Studied philosophy, mathematics and physics at Lvov University Teachers: Kazimierz Twardowski, Jan Łukasiewicz, Wacław Sierpiński, Marian Smoluchowski 1912 Ph.D. in Philosophy. Dissertation on Kant s ideas of time and space Studied philosophy and mathematics in Götingen (Husserl, Hilbert) 1920 Teaches philosophy, mathematics and physics in high schools 1920 Married with Maria Twardowska, the daughter of K. Twardowski 1921 Habilitation at Warsaw University, Methodology of Deductive Sciences 1925 Professor of Warsaw University Professor of Lvov University Head of The Chair of Theory and Methodology of Sciences at Poznań University Rector of the Poznań University

3 University of Chieti May 6-7, Head of The Chair of Logic II at the Philosophical Faculty of Warsaw University Head of the Department of Logic at the Institute of Philosophy and Sociology of The Polish Academy of Sciences Since 1952 Member of Polish Academy of Sciences Founder and General Editor of Studia Logica Doctor h.c. of The University Clermont-Ferrand MAIN ACHIEVEMENTS 1. Theory of meaning based on intersubstitutability of expressions in axiomatic and deductive rules of a language, the idea of a semantically open (closed) language. 2. Radical conventionalism: two different semantically closed languages are not mutually translatable, the scientific picture of the world depends on the conceptual apparatus. 3. In Ajdukiewicz turned to empiricism (moderate and radical). 4. Syntactic definitions of metalogical notions: proof, entailment, theorem (1920) which anticipated the work of Tarski. 5. The theory of definitions (nominal definitions). 6. Foundations of erotetic logic (logic of questions and answers), a semantic notion of a presupposition of a question.

4 University of Chieti May 6-7, A method of recognizing grammatical sentences on the basis of types assigned to words. 8. The method of logical paraphrases of philosophical problems. 9. Foundations of inductive reasoning. 10. Methodology of sciences (taxonomy of methods of reasoning, classification of sciences, scientific explanation). Die syntaktische Konnexität, Studia Philosophica 1 (1935), 1-27.

5 University of Chieti May 6-7, PREGROUPS [1] J. Lambek, Type grammars revisited, in: Logical Aspects of Computational Linguistics, LNAI 1582, Springer, 1999, [2] J. Lambek, Type grammars as pregroups, Grammars 4 (2001), [3] C. Casadio and J. Lambek, An algebraic analysis of clitic pronouns in Italian, in: Logical Aspects of Computational Linguistics, LNAI 2099, Springer, 2001, [4] Categorial Grammars. An efficient tool for natural language processing, Proceedings, Montpellier, My papers: [1] W. Buszkowski, Lambek grammars based on pregroups, Logical Aspects of Computational Linguistics, LNAI 2099, Springer, 2001, [2] W. Buszkowski, Pregroups: models and grammars, in: Relational Methods in Computer Science, LNCS 2561, Springer, 2002, [3] W. Buszkowski, Relational models of Lambek logics, in: Theory and Applications of Relational Structures as Knowledge Instruments, LNCS 2929, Springer, 2003, [4] W. Buszkowski, Sequent systems for compact bilinear logic, Mathematical Logic Quarterly 49 (2003),

6 University of Chieti May 6-7, A pregroup is a p.o. monoid G with two additional unary operations ( ) l and ( ) r which satisfy: (ADJ) a l a 1 aa l and aa r 1 a r a, for any a G. (P, ) a finite poset A REWRITING SYSTEM for CBL (Lambek 1999) Atomic types: p (n), p P, n is an integer. We interpret p (n) as p with n l s, if n < 0, and p with n r s, if n 0. We write p (n) q (n) if either p q and n is even, or q p and n is odd. Types: finite sequences of atomic types (X, Y, Z) Rules: (CON) X, p (n), p (n+1), Y X, Y (EXP) X, Y X, p (n+1), p (n), Y (IND) X, p (n), Y X, q (n), Y if p (n) q (n). X Y if there are X 0,..., X n, n 0, such that X = X 0, Y = X n, and X i 1 X i according to the rules, for i = 1,..., n. A derivation X Y is said to be normal, if it is of the form X U Y, the part X U

7 University of Chieti May 6-7, does not use (EXP), and the part U Y does not use (CON). The theorem on normal derivations (Lambek 1999). If X Y, then there exists a normal derivation X U Y. Corollary. If X Y and Y is an atomic type or the empty sequence, then there is a derivation of Y from X which does not use (EXP). If X Y and X is an atomic type or an empty sequence, then there is a derivation of Y from X which does not use (CON). Corollary (W.B. 2002). The decision problem for CBL is PTIME. Corollary (W.B. 2001). Languages generated by pregroup grammars are context-free (all ɛ free). Axioms: X X A TWO-SIDE SEQUENT SYSTEM (W.B. 2003) Rules: (LA) X, Y Z X, p (n), p (n+1), Y Z (RA) X Y, Z X Y, p (n+1), p (n), Z (LI) X, q(n), Y Z X, p (n), Y Z (RI) X Y, p(n), Z X Y, q (n), Z if p(n) q (n) (CUT) X Y; Y Z X Z

8 University of Chieti May 6-7, Theorem. The cut-elimination theorem for this system is equivalent to the theorem on normal derivations. X Y holds in the rewriting system iff X Y is derivable in the sequent system (without (CUT)). A ONE-SIDE SEQUENT SYSTEM WITH IMPLICIT ADJOINTS (W.B. 2003) Atomic formulas: atomic types, 1 Formulas: atomic formulas, A B Sequents: Γ such that Γ is a finite sequence of formulas. Adjoints are defined in the metalanguage. Axiom: ɛ Rules (we omit ): 1 l = 1 r = 1, (p (n) ) l = p (n 1), (p (n) ) r = p (n+1), (A B) l = B l A l, (A B) r = B r A r. Γ, Γ, A, B, (I1) (I ) Γ, 1, Γ, A B, Γ, (IA) Γ, p (n+1), p (n), Lemma. The full adjoint-introduction rules are derivable. If Γ,, then both

9 University of Chieti May 6-7, Γ, A, A l, and Γ, A r, A,, for any formula A. Theorem. This system admits cut-elimination. If Γ, A l and A,, then Γ,. If Γ, A and A r,, then Γ,. If Γ, A l, A, then Γ,. If Γ, A, A r, then Γ,. We define Γ iff Γ r iff, Γ l. Theorem. If Γ and Φ then Γ Φ. Corollary. If Γ, are product-free, then Γ holds in the sequent system iff it holds in the rewriting system. A ONE-SIDE SEQUENT SYSTEM WITH EXPLICIT ADJOINTS We present a system which directly handles formulas A l, A r and contains poset rules. Atomic formulas: atomic types, 1 Formulas: atomic formulas, A B, A l, A r Sequents: Γ such that Γ is a finite sequence of formulas In metalanguage we define a formula A (n), for any formula A and integer n. A (0) = A. If n 0, then A (n 1) = (A (n) ) l. If n 0, then A (n+1) = (A (n) ) r. Consequently, if n < 0, then A (n) equals A with n l s, and if n > 0, then A (n) equals A with n r s. Axioms: ɛ

10 University of Chieti May 6-7, Rules (we omit ): (I-A) (A-I) Γ, A (m+n), Γ, (A (m) ) (n), (POS) Γ, p(n), Γ, q (n), if p(n) q (n) Γ, (I-1) Γ, 1 (n), Γ, (where A = p) Γ, A (n+1), A (n), (IPE) Γ, A(2n), B (2n), Γ, (A B) (2n), (IPO) Γ, B(2n+1), A (2n+1), Γ, (A B) (2n+1), (A-I) is the rule of adjoint insertion. Example: from Γ, A lll, infer Γ, A llrll,. First, A lll = (A ll ) l. Since 1 = 1 + ( 2), then we get Γ, ((A ll ) r ) ll,, as required. This rule makes some change if and only if m n < 0. Lemma 1. The full rule (I-A) (for any formula A) is derivable. PROOF. Induction on A. A = p. Nothing to prove. A = 1. Use rule (I-1).

11 University of Chieti May 6-7, A = B l or A = B r. Take A = B (m), m 0. Assume Γ,. By the induction hypothesis, one gets Γ, B (m+n+1), B (m+n),. Then, Γ, A (n+1), A (n),, by (A-I). A = B C. Let n be even. Assume Γ,. By the induction hypothesis, one gets: Γ, C (n+1), C (n),, and Γ, C (n+1), B (n+1), B (n), C (n),. Consequently, Γ, A (n+1), A (n),, by (IPO), (IPE). Q.E.D. Lemma 2. If Γ, and Φ, then Γ, Φ,. Reversing the rules (A-I), (I-1), (IPE), (IPO). Lemma 3. If Γ, (A (m) ) (n), then Γ, A (m+n),. Lemma 4. If Γ, 1 (n),, then Γ,. Lemma 5. If Γ, (A B) (2n), then Γ, A (2n), B (2n),. Lemma 6. If Γ, (B A) (2n+1), then Γ, A (2n+1), B (2n+1),. Key lemma (Contraction Lemma) Lemma 7. If Γ, A (n), A (n+1), then Γ,. Generalization: a sequence Φ for A. PROOF. Induction on A. A = p. We prove more: if Γ, p (n), q (n+1), and p (n) q (n), then Γ,. Induction on derivations (similar to the proof of Lambek s theorem on normal derivations).

12 University of Chieti May 6-7, A = 1. Use lemma 4. A = B l or A = B r. Take A = B (m), m 0. Assume Γ, A (n), A (n+1),. By lemma 3, one gets Γ, B (m+n), B (m+n+1),. By the induction hypothesis, Γ,. A = B C. Use lemma 5, lemma 6 and the induction hypothesis. Q.E.D. Cut-Elimination Theorem (first form). If Γ, A l and A,, then Γ,. If Γ, A and A r,, then Γ,. Generalization: as above. PROOF. Assume Γ, A l and A,. By lemma 2, Γ, A l, A,, whence Γ,, by lemma 7. Q.E.D. Lemma 8. Γ r, iff, Γ l. Definition. Γ iff, Γ l. Cut-Elimination Theorem (second form). If Γ and Φ, then Γ Φ. PROOF. Assume Γ and Φ. Then, Φ, l and, Γ l. By the first form of cut-elimination, we get Φ, Γ l, which means Γ Φ. Q.E.D. One can prove a completeness theorem: Γ iff the sequent is true in all pregroups under assignments satisfying the poset conditions.

13 University of Chieti May 6-7, Sequents: Γ. Meaning: f (Γ) 1. THE DUAL SYSTEM Axiomatization: exactly as above except that in metalanguage we now set A (n 1) = (A (n) ) r, for n 0, A (n+1) = (A (n) ) l, for n 0. The same lemmata with the same proofs. This also holds for theorems, if one writes A (n), A (n+1) in them. Sequents: Γ Axioms: Γ Γ Rules: (LI-A) THE TWO-SIDE SYSTEM Γ 1, Γ 2 Γ 1, A (n) A (n+1), Γ 2 (RI-A) Γ 1, 2 Γ 1, A (n+1), A (n), 2 and rules (A-I), (I-1), (POS), (IPE), (IPO) on each side of the sequent. Theorem. Γ is derivable in the one-side system iff it is derivable in the two-side system. Corollary. The two-side system admits cut-elimination (second form). Corollary (the normalization theorem). If Γ is derivable, then there is a sequence U such that Γ U is derivable by means of left rules only, and U is derivable by means of right rules only.

14 University of Chieti May 6-7, A REWRITING SYSTEM WITH PRODUCT AND ADJOINTS Types: sequences of formulas. Contraction rules: (CON) Γ, A (n) A (n+1), Γ, (COL) Γ, (A (m) ) (n), Γ, A (m+n), (PEE) Γ, (A B) (2n), Γ, A (2n), B (2n), (PEO) Γ, (B A) (2n+1), Γ, A (2n+1), B (2n+1), Expansion rules: (EXP) Γ, Γ, A (n+1), A (n), (INS) Γ, A (m+n), Γ, (A (m) ) (n), (PIE) Γ, A (2n), B (2n), Γ, (A B) (2n), (PIO) Γ, B (2n+1), A (2n+1), Γ, (A B) (2n+1), Poset rules (IND) Γ, p (n), Γ, q (n), if p (n) q (n). Theorem. Γ in this rewriting system iff Γ is derivable in the sequent system.

15 University of Chieti May 6-7, Theorem on normal derivations. If Γ in this rewriting system, then there is U such that Γ U without expansion rules and U without contraction rules. 1. Finite model property. OTHER TOPICS Theorem (W>B> 2001,2003). Every finite pregroup is a group with the trivial ordering (identity). Consequently, CBL does not possess the finite model property. It possesses FMP if one admits partial models. One of the consequences of cut-elimination is: derivable sequents are true not only in total models but also in some partial models. 2. Interpretation in the Lambek calculus. For any formula A we define a formula I(A) in the language of the Lambek calculus: I(p) = p, I(p (n+1) ) = I(p (n) )\1, I(p (n 1) ) = 1/I(p (n) ), I(1 (n) ) = 1 I((A B) (2n) ) = I(A (2n) ) I(B (2n) ), and similarly for 2n + 1 I((A (m) ) (n) ) = I(A (m+n) ) Theorem. A 1,..., A n 1 is provable in CBL iff I(A 1 ),..., I(A n ) 1 is provable in the Lambek calculus.