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Gentzen-Type Proof Systems for Non-Classical Logics Yehonathan

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Tel Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences The Blavatnik School of Computer Science Gentzen-type Proof Systems for Non-classical Logics Thesis submitted for the degree of Doctor of Philosophy by Yehonathan Zohar This work was carried out under the supervision of Prof. Arnon Avron Submitted to the Senate of Tel Aviv University January 2018 i Acknowledgements I consider myself fortunate to have been a student of Prof. Arnon Avron, a truly inspiring logician and mentor. Shortly before starting university, I read his book about G¨odel's theorems, which motivated me to learn much more about logic. At the time, I could only dream that the author of this book would eventually become my advisor. Arnon taught me how to think, investigate, and prove. With his care, dedication and high standards, he inspired me to work harder for better results. Having been his student will always fill me with great pride. This thesis is also the result of joint work with Dr. Ori Lahav, who practically served as a second advisor for me. Ori taught me how to solve problems, and his perspective was always illuminating and fascinating. I thank him for this wonderful and enriching experience. I am thankful to Dr. Anna Zamansky for her academic and spiritual guidance, and for helping me understand what I really want, again and again. I would also like to thank Prof. Nachum Dershowitz and Prof. Alex Rabinovich for their valuable feedback on my work during my years at the logic seminar, and for the interesting and fruitful discussions about what logic is. I thank Prof. Naama Friedmann and Dr. Ilana Arbel for teaching me scientific and critical thinking. Their support along the way was a great compliment, and their belief in me kept me going. I also want to thank Mika Shahar and Anat Amirav for their valuable help. Their kindness and constant willingness to assist is much appreciated. Finally, a big thank you to my wife, Maayan. Your support made this thesis possible. This thesis is dedicated to Meir Shaham (\Dod Meir"). iii Abstract Sequent calculi constitute a prominent proof-theoretic framework, suitable for a vari- ety of logics. These include classical and intuitionistic logic, modal logics, paraconsistent logics, and many-valued logics, including fuzzy logics. The most important property of useful sequent calculi is analyticity, whose most common instance is the subformula prop- erty. When a calculus is analytic, a certain limitation on the search space of derivations is achieved. In the case of propositional sequent calculi, this often leads to a finite bound on the search space, and decidability immediately follows. This limitation may also be useful for proving the consistency of a sequent calculus (the fact that the empty sequent is not derivable). The most standard way to prove analyticity is via cut-admissibility, that is: the redundancy of the cut rule, which is usually the only rule whose premises may include a formula that is unrelated to the conclusion. Indeed, when cuts can be eliminated, and all other rules include in their premises only syntactic material from their conclusions, a simple induction on derivations entails that the end sequent can be derived using only its own syntactic material. Another major motivation for the elimination of the cut rule is the adequacy of cut-free calculi for efficient proof-search procedures. Despite the usefulness of cut-admissibility, relying on it alone for proving analyt- icity leaves out various useful sequent calculi that are analytic, but do not enjoy cut- admissibility. Moreover, even when it is possible to obtain cut-admissibility, it might be easier to prove analyticity directly, rather than to go through complicated and error- prone proofs of cut-admissibility. In addition, a great deal of ingenuity is required for developing efficient proof-search algorithms for cut-free sequent calculi. The main subject of this thesis is the notion of analyticity of sequent calculi discussed above. Our main contribution is a general analysis of it in several wide families of calculi. This analysis includes the following: • We provide several sufficient criteria for analyticity, that are easy to check either \by hand" or in an automated way. These simple criteria can replace complex proofs of analyticity that go through cut-admissibility. In fact, many analyticity results from the literature are obtained as particular instances of this result. The value of these criteria is also demonstrated by several new useful calculi that we introduce in this work. • We study the connection between analyticity and cut-admissibility, and prove that iv in a wide variety of calculi, the two properties are equivalent. Besides theoretical interest, this result can be used to simplify proofs of cut-admissibility, whenever analyticity has already been established. Using this result, we show that some of the sufficient criteria for analyticity that we propose are also sufficient for cut- admissibility. • We utilize analyticity to construct a uniform decision procedure for a wide family of sequent calculi. Our decision procedure relies only on analyticity, regardless of the admissibility of cut. Moreover, it is based on an efficient reduction to SAT, and thus all heuristic considerations are shifted to the mechanisms of off-the-shelf SAT-solvers. An implementation of this decision procedure is also described. • Finally, we study the framework of non-deterministic matrices (Nmatrices) as a tool for constructing analytic sequent calculi for logics that are already given in some other form. A fundamental operation on Nmatrices is introduced, called rexpansion, and is shown to be a crucial (though so far implicit) ingredient in applying Nmatrices for the construction of analytic sequent calculi for non-classical logics. Contents 1 Introduction 1 1.1 Background . .1 1.2 This Thesis . .3 2 Pure Sequent Calculi 11 2.1 Preliminaries . 12 2.2 What Are Pure Calculi? . 12 2.3 Semantics . 16 2.4 Streamlining, Equivalence, and Gentzen's Axioms . 20 2.5 Analyticity . 22 2.5.1 A Generalized Subformula Property . 23 2.5.2 Sufficient Criterion for Analyticity . 27 2.5.3 Constructing Analytic Calculi . 29 2.5.4 Proof of Theorem 2.5.21 . 33 2.6 Cut-admissibility . 36 2.6.1 Semantics in the Absence of Cut . 36 2.6.2 From Analyticity to Cut-admissibility . 39 2.6.3 Some Applications . 40 2.6.4 Strengthening The Result . 41 2.6.5 Proof of Theorem 2.6.20 . 42 2.7 Single-conclusion Pure Calculi . 44 3 SAT-based Decision Procedure 47 3.1 A Polynomial Reduction from Derivability to UNSAT . 48 3.2 Linear Time Decision Procedure . 50 3.3 Implementation of The Decision Procedure . 52 3.3.1 Features and Usage . 52 3.3.2 Implementation Details . 53 3.3.3 Performance . 54 v vi CONTENTS 4 Extending Pure Calculi with Modal Operators 66 4.1 Impure Rules for Modal Operators . 67 4.2 Semantics . 69 4.3 Analyticity . 76 4.3.1 Proof of Lemma 4.3.12 . 79 4.4 Extending The Decision Procedure . 84 4.4.1 Local Formulas . 84 4.4.2 Extending The Reduction . 86 4.4.3 Implementation . 88 4.5 Equivalence and Admissibility of Modal Rules . 89 4.5.1 Functionality vs. Seriality . 89 4.5.2 On the Admissibility of D-rules . 91 5 Intuitionistic Calculi 93 5.1 Intuitionistic Derivations . 94 5.2 Semantics . 97 5.3 Proof of Theorem 5.1.10 . 104 6 Rexpansions of Non-deterministic Matrices 108 6.1 (N)matrices, Expansions, and Refinements . 110 6.1.1 Logical Matrices . 110 6.1.2 Non-deterministic Matrices . 112 6.1.3 Expansions and Refinements . 113 6.2 Refined Expansions . 115 6.2.1 Combining Expansions and Refinements . 115 6.2.2 Consequence Relations . 118 6.3 Some Examples . 121 6.4 Applications to Sequent Calculi . 123 6.4.1 Three-valued Paraconsistent Logics . 123 6.4.2 Logics of Formal (In)consistency . 127 6.4.3 Conservative Extensions of Sequent Calculi . 132 6.5 Negations for G¨odelLogic . 135 t 6.5.1 The Nmatrix G and Its Refinements . 138 : M 1=2 6.5.2 Two Particular Refinements of G ................ 140 M : t 6.5.3 What is the Cardinality of G 0 < t 1 ?.......... 143 `M : j ≤ 6.5.4 On The Construction of Corresponding Proof Systems . 148 7 Summary and Further Work 151 CONTENTS vii Bibliography 153 List of Figures 2.1 Semantic constraints induced by G .................... 20 L 3 3.1 Examples for inputs to Gen2sat . 53 3.2 A partial class diagram of Gen2sat . 54 3.3 Definition of in MetTeL . 56 T 3.4 Definition of in Gen2sat . 57 S 3.5 Definition of in MetTeL . 58 ST 3.6 Running times on random problems . 60 3.7 Running times on classes 1{2 . 61 3.8 Running times on classes 3{4 . 62 3.9 Easy and hard Rothenberg problems . 65 4.1 Application schemes of sequent rules for modal Operators . 68 6.1 Summary of properties for paraconsistent fuzzy logics . 143 6.2 The hypersequent calculus GLC ....................... 149 6.3 Additional rules for in GRM ....................... 150 : ⊃ viii Chapter 1 Introduction 1.1 Background Gentzen's seminal paper from 1934 [54] begins with the introduction of the first two natural deduction calculi: system NK for classical logic and system NJ for intuitionistic logic.1 The calculus NJ admits the subformula property: whenever a hypothesis can be derived from a set of assumptions, there is a derivation of the same hypothesis that contains only subformulas of the hypothesis and the assumptions. However, this property fails for NK. For the purpose of providing a calculus with the subformula property for both intuitionistic and classical logic, Gentzen introduced two alternative systems: LK and LJ.
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