From Axioms to Rules A Coalition of Fuzzy, Linear and Substructural Logics

Size: px

Start display at page:

Download "From Axioms to Rules A Coalition of Fuzzy, Linear and Substructural Logics"

Austin Woods
5 years ago
Views:

1 From Axioms to Rules A Coalition of Fuzzy, Linear and Substructural Logics Kazushige Terui National Institute of Informatics, Tokyo Laboratoire d Informatique de Paris Nord (Joint work with Agata Ciabattoni and Nikolaos Galatos) Genova, 21/02/08.1/??

2 Parties in Nonclassical Logics Modal Logics Paraconsistent Logic Intermediate Logics Default Logic (Padova) Basic Logic Linear Logic Fuzzy Logics Substructural Logics Genova, 21/02/08.2/??

3 Parties in Nonclassical Logics Modal Logics Paraconsistent Logic Intermediate Logics Default Logic (Padova) Basic Logic Linear Logic Fuzzy Logics Substructural Logics Our aim: Fruitful coalition of the 3 arties Genova, 21/02/08.2/??

4 Basic Requirements Substractural Logics: Algebraization `L A () j= V(L) A Genova, 21/02/08.3/??

5 Basic Requirements Substractural Logics: Algebraization `L A () j= V(L) A Fuzzy Logics: Standard Comleteness `L A () j= K[0;1] (L) A Genova, 21/02/08.3/??

6 Basic Requirements Substractural Logics: Algebraization `L A () j= V(L) A Fuzzy Logics: Standard Comleteness `L A () j= K[0;1] (L) A Linear Logic: Cut Elimination Genova, 21/02/08.3/??

7 Basic Requirements Substractural Logics: Algebraization `L A () j= V(L) A Fuzzy Logics: Standard Comleteness `L A () j= K[0;1] (L) A Linear Logic: Cut Elimination A logic without cut elimination is like a car without engine (J.-Y. Girard) Genova, 21/02/08.3/??

8 Outcome We classify axioms in Substructural and Fuzzy Logics according to the Substructural Hierarchy, which is defined based on Polarity (Linear Logic). Genova, 21/02/08.4/??

9 Outcome We classify axioms in Substructural and Fuzzy Logics according to the Substructural Hierarchy, which is defined based on Polarity (Linear Logic). Give an automatic rocedure to transform axioms u to level 3 (P 0 3, in the absense of Weakening) into Hyerstructural P Rules in Hyersequent Calculus (Fuzzy Logics). Genova, 21/02/08.4/??

10 Outcome We classify axioms in Substructural and Fuzzy Logics according to the Substructural Hierarchy, which is defined based on Polarity (Linear Logic). Give an automatic rocedure to transform axioms u to level 3 (P 0 3, in the absense of Weakening) into Hyerstructural P Rules in Hyersequent Calculus (Fuzzy Logics). Give a uniform, semantic roof of cut-elimination via DM comletion (Substructural Logics) Genova, 21/02/08.4/??

11 Outcome We classify axioms in Substructural and Fuzzy Logics according to the Substructural Hierarchy, which is defined based on Polarity (Linear Logic). Give an automatic rocedure to transform axioms u to level 3 (P 0 3, in the absense of Weakening) into Hyerstructural P Rules in Hyersequent Calculus (Fuzzy Logics). Give a uniform, semantic roof of cut-elimination via DM comletion (Substructural Logics) To sum u: Every system of substructural and fuzzy logics defined P by 3 axioms P (acyclic 3, in the absense of Weakening) admits a cut-admissible hyersequent calculus. 0 Genova, 21/02/08.4/??

12 Kouan 1: Why uniformity? The Vienna Grou once wrote a META-aer (a aer generator) which Genova, 21/02/08.5/??

13 Kouan 1: Why uniformity? The Vienna Grou once wrote a META-aer (a aer generator) which given a sequent calculus L as inut Genova, 21/02/08.5/??

14 Kouan 1: Why uniformity? The Vienna Grou once wrote a META-aer (a aer generator) which given a sequent L calculus as inut generates a aer (with introduction and reference) that roves the cut elimination theorem for L. Genova, 21/02/08.5/??

15 Kouan 1: Why uniformity? The Vienna Grou once wrote a META-aer (a aer generator) which given a sequent L calculus as inut generates a aer (with introduction and reference) that roves the cut elimination theorem for L. The generated aer was submitted, and acceted. Genova, 21/02/08.5/??

16 Outline 1. Preliminary: Commutative Full Lambek FLe Calculus and Commutative Residuated Lattices 2. Background: Key concets in Substructural, Fuzzy and Linear Logics 3. Substructural Hierarchy 4. From Axioms to Rules, Uniform Semantic Cut Elimination 5. Conclusion Genova, 21/02/08.6/??

17 A ) A Identity Syntax of FLe = IMALL Formulas: A Ω B, A ffi B, A & B, A Φ B, 1,?, >, 0. Sequents: ) Π ( : multiset of formulas, Π: stou with at most one formula) Inference rules: ) A A; ) Π Cut ; ) Π B; ) Π A; Ω B; ) Π Ωl ) A ) B ; ) A Ω B Ωr A ) A B; ) Π B; ) Π ffil A; ) B ) A ffi B ffir ;A ffi Genova, 21/02/08.7/??

18 ) Π ) Π 1l ) 1 1r? )?l ) )??r 1; Syntax of FLe = IMALL A; ) Π B; ) Π Φl ) i A Φr A 1 Φ A 2 ) 0; ) Π 0l A Φ B; ) Π i ; ) Π A 1 & A 2 ; ) Π &l ) A ) B A ) > >r ) A & B &r Notational Corresondence M Conj M Im A Conj A Disj LinearLogic Ω 1 ffi? & > Φ 0 FuzzyLogics fi e! f ^ > _? Genova, 21/02/08.8/??

19 Commutative Residuated Lattices A (bounded ointed) commutative residuated lattice is P = hp; &; Φ; Ω; ffi; >; 0; 1;?i 1. hp; &; Φ; >; 0i is a lattice with > greatest and 0 least 2. hp; Ω; 1i is a commutative monoid. 3. For any x; y; z 2 P, x Ω y» z () y» x ffi z 4.? 2 P. P j= A if 1» f (A) for any valuation f. CRL: the variety of commutative residuated lattices. Genova, 21/02/08.9/??

20 Algebraization A (commutative) substructural logic L is an extension of FLe with axioms Φ L. V(L): the subvariety of CRL corresonding to L V(L) = fp 2 CRL : P j= A for any A 2 Φ L g Theorem: For every substructural logic L, `L A () j= V(L) A Genova, 21/02/08.10/??

21 Algebraization Is it trivial? Yes, but the consequences are not. Syntax: existential Semantics: universal `L A () 9ß:(ß is a roof of A) j= V(L) A () 8P 2 V(L):(P j= A) Consequence: Semantics mirrors Syntax Syntax Argument Proerty Semantics Argument Proerty Genova, 21/02/08.11/??

22 2 g2 Interoration and Amalgamation Form(X): formulas over variables ff; fi; 2 X. L admits Interolation: Suose A 2 Form(X) and 2 Form(Y ). If`L A ffi B, then there is I 2 Form(X Y ) B such that A ffi I `L I ffi B `L A V class of algebras admits Amalgamation if for any A; B; C 2 V with embeddings f 1 ;f 2, there are D 2 V and embeddings g 1 ;g 2 such that f 1 B g 1 R A D C Genova, 21/02/08.12/??

23 Interoration and Amalgamation Theorem (Maximova): For any intermediate logic L, L admits interolation () V(L) admits amalgamation Extended to substructural logics by Wroński, Kowalski, Galatos-Ono, etc. Syntax is to slit, semantics is to join. Genova, 21/02/08.13/??

24 Basic Requirements Substractural Logics: Algebraization `L A () j= V(L) A Fuzzy Logics: Standard Comleteness `L A () j= K[0;1] (L) A Linear Logic: Cut Elimination Genova, 21/02/08.14/??

25 Linearization Logically, this amounts to adding the axiom of linearity: (lin) (A ffi B) Φ (B ffi A) Examle: Gödel logic = IL + (lin) Comlete w.r.t. the valuations f : Form! [0; 1] s.t. f (>) = 1 f (0) = 0 f (A & B) = min(f (A);f(B)) f (A Φ B) = max(f (A);f(B)) f (A ffi B) = 8 < : (B) if f (A) > f (B) f otherwise 1 Genova, 21/02/08.15/??

26 (relin) (A ffi B) &1 Φ (B ffi A) &1 Linearization Other Fuzzy Logics: Uninorm Logic FLe + (relin) = Monoidal T-norm Logic FLew + (lin) = Basic Logic FLew + (lin) + (div) = Łukasiewicz Logic FLew + (lin) + (div) + (A = ffi A?? (A & B) ffi (A Ω (A ffi B)) (div) If axioms are added, cut elimination is lost. We need to find corresonding rules. Genova, 21/02/08.16/??

27 G j 1 ; 1 ) Π 1 G j 2 ; 2 ) Π 2 G j 2 ; 1 ) Π 1 j 1 ; 2 ) Π 2 Hyersequent Calculus Hyersequent calculus (Avron 87) Hyersequent: 1 ) Π 1 j j n ) Π n Intuition: (Ω 1 ffi Π 1 ) &1 Φ Φ (Ω n ffi Π n ) &1 HFLe FLe consists of Rules of Ext-Weakening Ext-Contraction G j A; ) B G j ) Π j ) Π j ) A ffi B G Communication Rule: G j ) Π G G j ) Π (com) Genova, 21/02/08.17/??

28 H j 1 ; 1 ) Π 1 H j 2 ; 2 ) Π 2 H j 2 ; 1 ) Π 1 j 1 ; 2 ) Π 2 A ) B j B ) A (com) Hyersequent Calculus HFLe + (com) roves (lin). (com) A ) A B ) B A ffi B j ) B ffi A ( ffir) ) (A ffi B) Φ (B ffi A) j ) (A ffi B) Φ (B ffi A) (Φr) ) (EC) ) (A ffi B) Φ (B ffi A) +(com) = Gödel Logic. Enjoys cut elimination (Avron 92). HIL Similarly for Monoidal T-norm and Uninorm Logics (cf. Metcalfe-Montagna 07) Semantics is to narrow, Syntax is to widen. Genova, 21/02/08.18/??

29 Basic Requirements Substractural Logics: Algebraization `L A () j= V(L) A Fuzzy Logics: Standard Comleteness `L A () j= K[0;1] (L) A Linear Logic: Cut Elimination Genova, 21/02/08.19/??

30 Conservativity Infinitary extension L 1 of L: & i2i A i ) i for 2 A any i ; i I ) A Π for some i 2 I i A i2i & ) & i2i A i ; ) Π L 1 is a conservative extension of L if `L1 A =) `L A for any set [ fag of finite formulas. Theorem: For any substructural logic L L, is a conservative extension of 1 iff any P 2 V(L) can be embedded into a L comlete P algebra 2 V(L). Syntax is to eliminate, Semantics is to enrich. 1 Genova, 21/02/08.20/??

31 Dedekind Comletion of Rationals For any X Q, X B = fy 2 Q : 8x 2 X:x» yg X C = fy 2 Q : 8x 2 X:y» xg X is closed if X = X BC (Q ; +; ) can be embedded into (C(Q ); +; ) with ) = fx Q : X is closedg C(Q Dedekind comletion extends to various ordered algebras (MacNeille). Genova, 21/02/08.21/??

32 X Φ Y = (X [ Y ) BC X Ω Y = fxy : x 2 X; y 2 Y g BC > = P 0 = ; BC Dedekind-MacNeille Comletion Theorem: Every P 2 CRL can be embedded into a comlete P 1 2 CRL, where P 1 = (C(P); &; Φ; Ω; ffi; >; 0; 1;?) X & Y = X Y X ffi Y = fy : 8x 2 Xxy 2 Y g = f1g BC? = f?g BC 1 (Ono 93; cf. Abrusci 90, Sambin 93) Some axioms (eg. distributivity) are not reserved by DM comletion (cf. Kowalski-Litak 07). Genova, 21/02/08.22/??

33 Cut Elimination via Comletion Syntactic argument: elimination rocedure Cut-ful =) Proofs Cut-free Proofs Semantic argument: Quasi-DM comletion (= CRL Intransitive CRL Genova, 21/02/08.23/??

34 Cut Elimination via Comletion Due to (Okada 96). Algebraically reformulated by (Belardinelli-Ono-Jisen 01). MUL = the set of multisets ; ;::: of formulas = the set of sequents ± ) Π SEQ 2 MUL For ± ) Π 2 and ± ) Π iff ; ± ) Π is cut-free rovable in FLe For X MLT and Y SEQ, X B = f(± ) Π) 2 SEQ : 8 2 (± ) Π)g Y C = f 2 MUL : 8(± ) Π) 2 (± ) Π)g Genova, 21/02/08.24/??

35 Cut Elimination via Comletion B; C induce a comlete CRL P cf, and (A) = fx : A 2 X A BC ; X closedg F is a Form! P(P quasi-homomorhism cf ): Λ 2 F (Λ) for Λ 2 f1;?; >; 0g (x)? F (y) F (x? y) for? 2 fω; ffi; &; Φg F If the f (ff) = ff valuation validates A, then A is cut-free rovable in FLe. C Again, not all axioms are reserved by quasi-dm comletion. Which axioms are reserved by quasi-dm comletion? Genova, 21/02/08.25/??

36 Kouan 2: What is Comleteness? Comleteness: to establish a corresondence between syntax and semantics. Gödel `L A () j= Comleteness: A V(L) is trivial in the algebraic setting. Meta-Comleteness: to describe the Syntax-Semantics Mirror corresondence as recisely as ossible. Syntax Interolation Hyersequentialization Conservativity Cut-elimination Semantics Amalgamation Linearization Comletion Quasi-DM comletion Genova, 21/02/08.26/??

37 A =) `cf FLe A `FLe `cf FLe A =) Kouan 3: Is Semantic Cut-Elimination Weak? Proof of Cut-Elimination `FLe A sound =) P cf j= A P cf j= A comlete We have just relaced object cuts with a big META-CUT. Another criticism: it does not give a cut-elimination rocedure. Conjecture: If we eliminate the meta-cut, a concrete (object) cut-elimination rocedure emerges. Genova, 21/02/08.27/??

38 When Cut-Elimination Holds? Cut-Elimination holds FLe when is extended with a natural structural rule: Contraction: A ffi A Ω A A; ) Π A; ) Π A; It fails when extended with an unnatural one: Broccoli: A Ω A ffi A ) Π A; A; ) Π A; What is natural? Genova, 21/02/08.28/??

39 Girard s test (Girard 99) rooses a test for naturality of structural rules. - A structural rule asses Girard s test if, in every hase B; structure C), itroagates from atomic closed (M; sets to all closed sets X. BC fxg If the rule holds for rationals, it also holds for all BC reals. Contraction asses it: 8x 2 M: fxg BC fx xg BC =) 8X : closed (X X Ω X): Broccoli fails it: 8x 2 M: fx xg BC fxg BC 6=) 8X : closed (X Ω X X): Genova, 21/02/08.29/??

40 Irony Broccoli is equivalent to Mingle: A Ω B ffi A Φ B ; ± 1 ; ) Π ; ± 2 ; ) Π ; ± 1 ; ± 2 ; ) Π Mingle asses Girard s test. Genova, 21/02/08.30/??

41 ) ff ff ) ff Φ fi ff ) ff ff ) ff Φ fi ff Broccoli and Mingle Broccoli does not admit cut elimination: Φ fi ) ff Φ fi ff Φ fi; ff Φ fi ) ff Φ fi Ex ff fi ) fi Cut ff; ff Φ fi ) ff Φ fi fi ) ff Φ fi Cut ff; fi ) ff Φ fi When Broccoli is relaced with Mingle, the above cut can be eliminated: fi ) fi fi ) ff Φ fi Min ff; fi ) ff Φ fi Genova, 21/02/08.31/??

42 Characterization of Cut-Elimination Additive structural rules: ; ~ X 1 ; ) C ; ~ X n ; ) C R ; ~ X 0 ; ) C such that f ~ X 1 ;:::; ~ X n g f ~ X 0 g. Theorem (Terui 07): For any additive structural rule (r), + (r) admits cut-elimination iff (r) asses Girard s test. FLe Key (r) fact: asses Girard s =) (r) test is reserved by (quasi-)dm comletion. (Ciabattoni-Terui 06) considerably extends this result. Genova, 21/02/08.32/??

43 Questions Which axiom can be transformed into structural rules? Usually checked one by one. Is there a more systematic way? Which structural rules can be transformed into good ones admitting cut-elimination? (eg. =) Broccoli Mingle) How does the situation change if we adot hyersequent calculus? (eg. =) Linearity Communication) Does hyersequent calculus admit semantic cut-elimination? All known roofs are syntactic, tailored to each secific logic (excetion: Metcalfe-Montagna 07), and quite comlicated. Semantic one would lead to a uniform, concetually simler roof. (Ciabattoni-Galatos-Terui 08) =) Genova, 21/02/08.33/??

44 Outline 1. Preliminary: Commutative Full Lambek FLe Calculus and Commutative Residuated Lattices 2. Background: Key concets in Substructural, Fuzzy and Linear Logics 3. Substructural Hierarchy 4. From Axioms to Rules, Uniform Semantic Cut Elimination 5. Conclusion Genova, 21/02/08.34/??

45 Polarity Key notion in Linear Logic since its incetion (Girard 87) Plays a central role in Efficient Proof Search (Andreoli 90), Constructive Classical Logic (Girard 91), Polarized Linear Logic (Laurent), Game Semantics, Ludics. Genova, 21/02/08.35/??

46 ) A 1 & A 2 ) A i (X Y ) BC X BC Y BC Genova, 21/02/08.36/?? Polarity Positive connectives 1; 0; Ω; Φ have invertible left rules: A Ω B; ) Π A; B; ) Π and roagate closure oerator: X BC ffl Y BC (X ffl Y ) BC = X Ω Y Negative connectives >;?; &; ffi have invertible right rules: and distribute closure oerator:

47 Polarity Connectives of the same olarity associate well. Positives: A Ω (B Φ C) = (A Ω B) Φ (A Ω C) Ω 1 = A A Ω 0 = 0 A Φ 0 = A A Negatives: A ffi (B & C) = (A ffi B) & (A ffi C) (A Φ B) ffi C = (A ffi C) & (B ffi C) & > = A A ffi > = > 1 ffi A = A A (olarity reverses on the LHS of an imlication) Genova, 21/02/08.37/??

48 6 P 1 N 6 P 0 N 0 Substructural Hierarchy P 3 N 3 P 2 N 2 The P sets ; N n n of formulas defined by: P (0) = N 0 = 0 the set of atomic formulas N (P1) P n+1 n A; B 2 P (P2) =) A Φ B; A Ω B; 1; 0 2 P n+1 n (N1) P n N n+1 (N2) A; B 2 N n+1 =) A & B;?; > 2 N n+1 (N3) A 2 P n+1 ;B 2 N n+1 =) A ffi B 2 N n+1 Genova, 21/02/08.38/??

49 A 2 N n =) A & 1 2 P 0 n+1 P 1 : Disj of multisets N 1 : Conj of sequents ) Π Substructural Hierarchy Due to lack of P Weakening, n is too strong. It is also convenient to consider a P subclass n P 0 n : B 2 P 0 n+1 =) A Φ B; A Ω B; 1; 0 2 P0 n+1 A; Intuition: P 0 : Formulas : Conj of hyersequents S 1 j j S 0 P n 2 2 : Conj of structural rules N 0 3 : Conj of hyerstructural rules P Genova, 21/02/08.39/??

50 Examles N 2 ff? ff ffi ffi ff ffi Ω ff ff ffi ff ff ff Ω ffi Ωff m (n; Ωff m ff? (ff ) &? 2 ff P Φ ff (fi ffi ffi ff) fi) Φ (ff ((ff ffi P fi) 0 & 1) Φ ((fi ffi ff) & 1) 3 Class Axiom Name 1, weakening contraction exansion knotted axioms 0) weak contraction excluded middle linearity relinearity P 3 ff? Φ ff?? weak excluded middle L k i=0 ( i ffi L j6=i j) Krike models with width» k A? = A ffi? Genova, 21/02/08.40/??

51 Examles Axioms N in 3 : Łukasiewicz ((ff ffi fi) ffi fi) ffi ((fi ffi ff) ffi ff) axiom (ff & (fi Φ fl)) ffi ((ff & fi) Φ (ff & fl)) Distributivity (ff & fi) ffi (ff Ω (ff ffi fi)) Divisibility Genova, 21/02/08.41/??

52 1 ) Ψ 1 n ) Ψ n 0 ) Ψ 0 G j 0 1 ) Ψ0 1 G j 0 n ) Ψ 0 n From Axioms to Rules A structural rule is where i ; Ψ i are sets of metavariables (jψ i j» 1). A hyerstructural rule is j 1 ) Ψ 1 j j m ) Ψ m G Theorem: 1. N Any 2 -axiom is equivalent to (a set of) structural rules in FLe. 2. P Any 3-axiom is equivalent to hyerstructural rules in HFLe. 3. Any 0 3 -axiom is equivalent to hyerstructural rules in HFLew. P Genova, 21/02/08.42/??

53 (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Examle Weak nilotent minimum (Esteva-Godo 01): Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to Genova, 21/02/08.43/??

54 (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Examle Weak nilotent minimum (Esteva-Godo 01): Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j ) (ff Ω fi) ffi? j ) ff & fi ffi ff Ω fi Genova, 21/02/08.43/??

55 (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Examle Weak nilotent minimum (Esteva-Godo 01): Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j ff Ω fi )? j ) ff & fi ffi ff Ω fi Genova, 21/02/08.43/??

56 (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Examle Weak nilotent minimum (Esteva-Godo 01): Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j ff; fi )? j ) ff & fi ffi ff Ω fi Genova, 21/02/08.43/??

57 (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Examle Weak nilotent minimum (Esteva-Godo 01): Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j ff; fi ) j ) ff & fi ffi ff Ω fi Genova, 21/02/08.43/??

58 (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Examle Weak nilotent minimum (Esteva-Godo 01): Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j ff; fi ) j ff & fi ) ff Ω fi Genova, 21/02/08.43/??

59 Examle Weak nilotent minimum (Esteva-Godo 01): (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j fl ) ff & fi G j ff; fi ) j fl ) ff Ω fi Genova, 21/02/08.43/??

60 Examle Weak nilotent minimum (Esteva-Godo 01): (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j fl ) ff G j fl ) fi G j ff; fi ) j fl ) ff Ω fi Genova, 21/02/08.43/??

61 Examle Weak nilotent minimum (Esteva-Godo 01): (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j fl ) ff G j fl ) fi G j ff Ω fi ) ffi G j ff; fi ) j fl ) ffi Genova, 21/02/08.43/??

62 (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) Examle Weak nilotent minimum (Esteva-Godo 01): Its P 0 3 version: (ff Ω fi)? &1 Φ (ff & fi ffi ff Ω fi) &1 is equivalent to G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi The rocedure is automatic (in contrast to the usual ractice). Similarity with rincile of reflection (automatic derivation of inference rules for a logical connective from its defining equation; Sambin-Battilotti-Faggian 00). Genova, 21/02/08.43/??

63 ) ff fi fl ) fi ff; Towards Cut Elimination Not all rules admit cut elimination. They have to be comleted. In absence of Weakening, cyclic rules are roblematic: ff; fl ) fi Theorem: 1. Any acyclic hyerstructural rule can be transformed into an equivalent one HFLe in that enjoys cut elimination. 2. Any hyerstructural rule can be transformed into an equivalent one HFLew in that enjoys cut elimination. Genova, 21/02/08.44/??

64 ) ff fi fl ) fi ff; Towards Cut Elimination Not all rules admit cut elimination. They have to be comleted. In absence of Weakening, cyclic rules are roblematic: ff; fl ) fi Theorem: 1. Any acyclic hyerstructural rule can be transformed into an equivalent one HFLe in that enjoys cut elimination. 2. Any hyerstructural rule can be transformed into an equivalent one HFLew in that enjoys cut elimination. Genova, 21/02/08.44/??

65 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to Genova, 21/02/08.45/??

66 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ) ff G j ; fi ) j fl ) ffi Genova, 21/02/08.45/??

67 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ) ff G j ) fi G j ; ) j fl ) ffi Genova, 21/02/08.45/??

68 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ) ff G j ) fi G j Λ ) fl G j ; ) j Λ ) ffi Genova, 21/02/08.45/??

69 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ) ff G j ) fi G j Λ ) fl G j ffi; ± ) Π G j ; ) j Λ; ± ) Π Genova, 21/02/08.45/??

70 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to G j fl ) ff G j fl ) fi G j ff; fi; ± ) Π G j ) ff G j ) fi G j Λ ) fl G j ; ) j Λ; ± ) Π Genova, 21/02/08.45/??

71 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to G j Λ ) ff G j Λ ) fi G j ff;fi;± ) Π G j ) ff G j ) fi G j ; ) j Λ; ± ) Π Genova, 21/02/08.45/??

72 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to G j ; fi; ± ) Π G j Λ; fi; ± ) Π G j Λ ) fi G j ) fi G j ; ) j Λ; ± ) Π Genova, 21/02/08.45/??

73 Examle Weak nilotent minimum: (ff Ω fi)? Φ (ff & fi ffi ff Ω fi) We have obtained: G j fl ) ff G j fl ) fi G j ff; fi ) ffi G j ff; fi ) j fl ) ffi It is equivalent to G j ; ; ± ) Π G j ; Λ; ± ) Π G j Λ; ; ± ) Π G j Λ; Λ; ± ) Π G j ; ) j Λ; ± ) Π Genova, 21/02/08.45/??

74 Uniform Cut Elimination The resulting rule G j ; ; ± ) Π G j ; Λ; ± ) Π G j Λ; ; ± ) Π G j Λ; Λ; ± ) Π G j ; ) j Λ; ± ) Π satisfies Strong subformula roerty: Any formula occuring on the LHS (res. RHS) of a remise also occurs on the LHS (res. RHS) of the conclusion Conclusion-linearity: No metavariable occurs in the conclusion twice. Couling:... Genova, 21/02/08.46/??

75 Uniform Cut Elimination Theorem: If a hyerstructural (r) rule satisfies the above conditions, HFLe + (r) then admits cut-elimination. Proof: The above are sufficient conditions for a rule to be reserved by quasi-dm comletion. Semantics allows for a uniform roof. Genova, 21/02/08.47/??

76 Main Results Theorem: 1. Any N acyclic 2 -axiom is equivalent FLe in to (a set of) structural rules enjoying cut-elimination. 2. N Any 2 -axiom is equivalent FLew in to (a set of) structural rules enjoying cut-elimination. 3. Any P acyclic 3-axiom is equivalent in HFLe to hyerstructural rules enjoying cut-elimination P Any 3 -axiom is equivalent HFLew in to hyerstructural rules enjoying cut-elimination. Our results automatically yield: Esteva-Godo s logic FLew + (linearity) + (weak nilotent minimum) admits a cut-admissible hyersequent calculus. Genova, 21/02/08.48/??

77 Examles N 2 ff? ff ffi ffi ff ffi Ω ff ff ffi ff ff ff Ω ffi Ωff m (n; Ωff m ff? (ff ) &? 2 ff P Φ ff (fi ffi ffi ff) fi) Φ (ff ((ff ffi P fi) 0 & 1) Φ ((fi ffi ff) & 1) 3 Class Axiom Name 1, weakening contraction exansion knotted axioms 0) weak contraction excluded middle linearity relinearity P 3 ff? Φ ff?? weak excluded middle L k i=0 ( i ffi L j6=i j) Krike models with width» k A? = A ffi? Genova, 21/02/08.49/??

78 Conclusion Our coalition successfully combines various ideas: Polarity, Girard s test from Linear Logic Hyersequent calculus from Fuzzy Logic DM comletion from Substructural Logic to establish uniform cut-elimination for extensions FLe of with 3 axioms. P Research directions: Comutational meaning of axioms in Fuzzy Logic. Eg. Peirce s law corresonds to call/cc in Functional Programming. What about (ff ffi fi) Φ (fi ffi Linearity ff)? Better understanding of Syntax-Semantics Mirror Genova, 21/02/08.50/??

79 6 P 0 N 0 Conclusion 6 6 P 3 N 3 P 2 N 2 P 1 N 1 Uniform treatment of axioms N in 3 : Łukasiewicz axiom, divisibility, cancellativity, distributivity, etc. (Known calculi are tailord to each secific logic; cf. Metcalfe-Olivietti-Gabbay 04) How high can we climb u? Genova, 21/02/08.51/??

How and when axioms can be transformed into good structural rules

How and when axioms can be transformed into good structural rules Kazushige Terui National Institute of Informatics, Tokyo Laboratoire d Informatique de Paris Nord (Joint work with Agata Ciabattoni and