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

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From Axioms to Rules — a Coalition of Fuzzy, Linear and Substructural Logics 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 – p.1/?? Parties in Nonclassical Logics Modal Logics Default Logic Intermediate Logics (Padova) Basic Logic Paraconsistent Logic Linear Logic Fuzzy Logics Substructural Logics Genova, 21/02/08 – p.2/?? Parties in Nonclassical Logics Modal Logics Default Logic Intermediate Logics (Padova) Basic Logic Paraconsistent Logic Linear Logic Fuzzy Logics Substructural Logics Our aim: Fruitful coalition of the 3 parties Genova, 21/02/08 – p.2/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Genova, 21/02/08 – p.3/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Fuzzy Logics: Standard Completeness ´µ Ä Ã ´Äµ ¼½ Genova, 21/02/08 – p.3/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Fuzzy Logics: Standard Completeness ´µ Ä Ã ´Äµ ¼½ Linear Logic: Cut Elimination Genova, 21/02/08 – p.3/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Fuzzy Logics: Standard Completeness ´µ Ä Ã ´Äµ ¼½ Linear Logic: Cut Elimination A logic without cut elimination is like a car without engine (J.-Y. Girard) Genova, 21/02/08 – p.3/?? 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 – p.4/?? 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 procedure to transform axioms up to level ¼ È È ¿ ( , in the absense of Weakening) into Hyperstructural ¿ Rules in Hypersequent Calculus (Fuzzy Logics). Genova, 21/02/08 – p.4/?? 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 procedure to transform axioms up to level ¼ È È ¿ ( , in the absense of Weakening) into Hyperstructural ¿ Rules in Hypersequent Calculus (Fuzzy Logics). Give a uniform, semantic proof of cut-elimination via DM completion (Substructural Logics) Genova, 21/02/08 – p.4/?? 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 procedure to transform axioms up to level ¼ È È ¿ ( , in the absense of Weakening) into Hyperstructural ¿ Rules in Hypersequent Calculus (Fuzzy Logics). Give a uniform, semantic proof of cut-elimination via DM completion (Substructural Logics) To sum up: Every system of substructural and fuzzy logics ¼ È È defined by ¿ axioms (acyclic , in the absense of ¿ Weakening) admits a cut-admissible hypersequent calculus. Genova, 21/02/08 – p.4/?? Kouan 1: Why uniformity? The Vienna Group once wrote a META-paper (a paper generator) which Genova, 21/02/08 – p.5/?? Kouan 1: Why uniformity? The Vienna Group once wrote a META-paper (a paper generator) which given a sequent calculus Ä as input Genova, 21/02/08 – p.5/?? Kouan 1: Why uniformity? The Vienna Group once wrote a META-paper (a paper generator) which given a sequent calculus Ä as input generates a paper (with introduction and reference) that proves the cut elimination theorem for Ä. Genova, 21/02/08 – p.5/?? Kouan 1: Why uniformity? The Vienna Group once wrote a META-paper (a paper generator) which given a sequent calculus Ä as input generates a paper (with introduction and reference) that proves the cut elimination theorem for Ä. The generated paper was submitted, and accepted. Genova, 21/02/08 – p.5/?? Outline 1. Preliminary: Commutative Full Lambek Calculus Ä and Commutative Residuated Lattices 2. Background: Key concepts in Substructural, Fuzzy and Linear Logics 3. Substructural Hierarchy 4. From Axioms to Rules, Uniform Semantic Cut Elimination 5. Conclusion Genova, 21/02/08 – p.6/?? Syntax of Ä ÁÅÄÄ Formulas: ª , Æ , ² , ¨ , ½, , , ¼. Sequents: µ ¥ ( : multiset of formulas, ¥: stoup with at most one formula) Inference rules: µ ¡ µ ¥ Á ÒØ ØÝ ÙØ ¡ µ ¥ µ µ ¥ µ ¡ µ ªÖ ªÐ ª µ ¥ ¡ µ ª µ ¡ µ ¥ µ ÆÖ ÆÐ Æ ¡ µ ¥ µ Æ Genova, 21/02/08 – p.7/?? Syntax of Ä ÁÅÄÄ µ ¥ µ ¥ µ ¨Ö ¨Ð ¼Ð ¨ µ ¥ µ ¨ ¼ µ ¥ ½ ¾ µ ¥ µ µ ²Ð ²Ö Ö ² µ ¥ µ ² µ ½ ¾ µ µ ¥ Ð Ö ½Ð ½Ö ½ µ ¥ µ ½ µ µ Notational Correspondence Å ÓÒ Å ÁÑÔ ÓÒ × Ä ÒÖ ÄÓ ª ½ Æ ² ¨ ¼ ÙÞÞÝÄÓ × ¬ Genova, 21/02/08 – p.8/?? Commutative Residuated Lattices A (bounded pointed) commutative residuated lattice is È È ² ¨ ª Æ ¼ ½ 1. È ² ¨ ¼ is a lattice with greatest and ¼ least 2. È ª ½ is a commutative monoid. 3. For any Ü Ý Þ ¾ È , Ü ª Ý Þ ´µ Ý Ü Æ Þ 4. ¾ È . È if ½ ´µ for any valuation . ÊÄ: the variety of commutative residuated lattices. Genova, 21/02/08 – p.9/?? Algebraization A (commutative) substructural logic Ä is an extension of Ä ¨ with axioms Ä . Î ´Äµ: the subvariety of ÊÄ corresponding to Ä Î ´Äµ È ¾ ÊÄ È ¾ ¨ for any Ä Theorem: For every substructural logic Ä, ´µ Ä Î ´Äµ Genova, 21/02/08 – p.10/?? Algebraization Is it trivial? Yes, but the consequences are not. Syntax: existential ´µ ´ µ Ä is a proof of Semantics: universal ´µ È ¾ Î ´Äµ´È µ Î ´Äµ Consequence: Semantics mirrors Syntax Syntax Semantics Argument Argument Property Property Genova, 21/02/08 – p.11/?? Interporation and Amalgamation ÓÖÑ´ µ: formulas over variables « ¬ ¡¡¡ ¾ . Ä admits Interpolation: Suppose ¾ ÓÖÑ´ µ and ¾ ÓÖÑ´ µ Æ Á ¾ ÓÖÑ´ µ .If Ä , then there is such that Æ Á Á Æ Ä Ä A class Î of algebras admits Amalgamation if for any ¾ Î ¾ Î with embeddings ½ ¾ , there are and embeddings ½ ¾ such that Ô ½ ½ Ô Ô Ô Ê Ô Ô Ô Ê Ô ¾ ¾ Genova, 21/02/08 – p.12/?? Interporation and Amalgamation Theorem (Maximova): For any intermediate logic Ä, Ä admits interpolation ´µ Î ´Äµ admits amalgamation Extended to substructural logics by Wronski,´ Kowalski, Galatos-Ono, etc. Syntax is to split, semantics is to join. Genova, 21/02/08 – p.13/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Fuzzy Logics: Standard Completeness ´µ Ä Ã ´Äµ ¼½ Linear Logic: Cut Elimination Genova, 21/02/08 – p.14/?? Linearization Logically, this amounts to adding the axiom of linearity: ´ÐÒµ ´ Æ µ ¨ ´ Æ µ Example: Gödel logic = ÁÄ · ´ÐÒµ Complete w.r.t. the valuations ÓÖÑ ¼ ½ s.t. ´µ ½ ´¼µ ¼ ´ ² µ ÑÒ´ ´µ´ µµ ´ ¨ µ ÑÜ´ ´µ´ µµ ´ µ if ´µ ´ µ ´ Æ µ ½ otherwise Genova, 21/02/08 – p.15/?? Linearization Other Fuzzy Logics: Uninorm Logic = Ä · ´ÔÖ Ð Òµ Monoidal T-norm Logic = ÄÛ · ´ÐÒµ Basic Logic = ÄÛ · ´ÐÒµ · ´Ú µ Łukasiewicz Logic = ÄÛ · ´ÐÒµ · ´Ú µ · ´ Æ ´ÔÖ Ð Òµ ´ Æ µ ¨ ´ Æ µ ²½ ²½ ´Ú µ ´ ² µ Æ ´ ª ´ Æ µµ If axioms are added, cut elimination is lost. We need to find corresponding rules. Genova, 21/02/08 – p.16/?? Hypersequent Calculus Hypersequent calculus (Avron 87) µ ¥ ¡¡¡ µ ¥ Hypersequent: ½ ½ Ò Ò ´ª Æ ¥ µ ¨ ¡¡¡ ¨ ´ª Æ ¥ µ ½ ½ Ò Ò Intuition: ²½ ²½ ÀÄ consists of Rules of Ä Ext-Weakening Ext-Contraction µ µ ¥ µ ¥ µ Æ µ ¥ µ ¥ Communication Rule: ¡ µ ¥ ¡ µ ¥ ½ ½ ½ ¾ ¾ ¾ ´Óѵ ¡ µ ¥ ¡ µ ¥ ¾ ½ ½ ½ ¾ ¾ Genova, 21/02/08 – p.17/?? Hypersequent Calculus ÀÄ · ´Óѵ proves ´ÐÒµ. À ¡ µ ¥ À ¡ µ ¥ ½ ½ ½ ¾ ¾ ¾ ´Óѵ À ¡ µ ¥ ¡ µ ¥ ¾ ½ ½ ½ ¾ ¾ µ µ ´Óѵ µ µ ´ ÆÖ µ µ Æ µ Æ ´¨Ö µ µ ´ Æ µ ¨ ´ Æ µ µ ´ Æ µ ¨ ´ Æ µ ´ µ µ ´ Æ µ ¨ ´ Æ µ ÀÁÄ ·´Óѵ = Gödel Logic. Enjoys cut elimination (Avron 92). Similarly for Monoidal T-norm and Uninorm Logics (cf. Metcalfe-Montagna 07) Semantics is to narrow, Syntax is to widen. Genova, 21/02/08 – p.18/?? Basic Requirements Substractural Logics: Algebraization ´µ Ä Î ´Äµ Fuzzy Logics: Standard Completeness ´µ Ä Ã ´Äµ ¼½ Linear Logic: Cut Elimination Genova, 21/02/08 – p.19/?? Conservativity ½ Ä Ä ² Infinitary extension of : ¾Á µ ¾ Á µ ¥ ¾ Á for any for some µ ² ² µ ¥ ¾Á ¾Á ½ Ä is a conservative extension of Ä if ½ µ Ä Ä for any set of finite formulas. ½ Theorem: For any substructural logic Ä, Ä is a conservative extension of Ä iff any È ¾ Î ´Äµ can be embedded into a ½ complete algebra È ¾ Î ´Äµ. Syntax is to eliminate, Semantics is to enrich. Genova, 21/02/08 – p.20/?? Dedekind Completion of Rationals For any É , Ý ¾ É Ü ¾ Ü Ý Ý ¾ É Ü ¾ Ý Ü is closed if ´É · ¡µ can be embedded into ´ ´É µ · ¡ µ with ´É µ É is closed Dedekind completion extends to various ordered algebras (MacNeille). Genova, 21/02/08 – p.21/?? Dedekind-MacNeille Completion Theorem: Every È ¾ ÊÄ can be embedded into a complete ½ ½ È ¾ ÊÄ, where È ´ ´Èµ ² ¨ ª Æ ¼ ½ µ ² ¨ ´ µ ª ÜÝ Ü ¾ Ý ¾ Æ Ý Ü ¾ ÜÝ ¾ È ¼ ½ ½ (Ono 93; cf. Abrusci 90, Sambin 93) Some axioms (eg. distributivity) are not preserved by DM completion (cf. Kowalski-Litak 07). Genova, 21/02/08 – p.22/?? Cut Elimination via Completion Syntactic argument: elimination procedure Cut-ful Proofs µ Cut-free Proofs Semantic argument: Quasi-DM completion CRL ´ ‘Intransitive’ CRL Genova, 21/02/08 – p.23/?? Cut Elimination via Completion Due to (Okada 96). Algebraically reformulated by (Belardinelli-Ono-Jipsen 01). ÅÍÄ the set of multisets ¡ of formulas ËÉ the set of sequents ¦ µ ¥ For ¾ ÅÍÄ and ¦ µ ¥ ¾ ËÉ, ¦ µ ¥ iff ¦ µ ¥ is cut-free provable in Ä For ÅÄÌ and ËÉ, ´¦ µ ¥µ ¾ ËÉ ¾ ´¦ µ ¥µ ¾ ÅÍÄ ´¦ µ ¥µ ¾ ´¦ µ ¥µ Genova, 21/02/08 – p.24/?? Cut Elimination via Completion È induce a complete CRL , and ´µ ¾ closed ÓÖÑ È ´È µ is a quasi-homomorphism : £ ¾ ´£µ for £ ¾ ½ ¼ ´Üµ ´Ý µ ´Ü Ý µ for ¾ ª Æ ² ¨ If the valuation ´« µ « validates , then is cut-free provable in Ä .
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