Linear Logic and Noncommutativity in the Calculus of Structures Dissertation zur Erlangung des akademischen Grades Doktor rerum naturalium (Dr. rer. nat.) vorgelegt an der Technischen Universit¨at Dresden Fakult¨at Informatik eingereicht am 26. Mai 2003 von Diplom-Informatiker Lutz Straßburger geboren am 21. April 1975 in Dresden Betreuer: Dr. Alessio Guglielmi Betreuender Hochschullehrer: Prof. Dr. rer. nat. habil. Steffen H¨olldobler Gutachter: Prof. Dr. rer. nat. habil. Steffen H¨olldobler Dr. Fran¸cois Lamarche Prof. Dr. rer. nat. habil. Horst Reichel Disputation am 24. Juli 2003 Acknowledgements Iamparticularlyindebted to Alessio Guglielmi. Whithout his advice and guidance this thesis would not have been written. He awoke my interest for thefascinating field of proof theory and introduced me to the calculus of structures. I benefited from many fruitful and inspiring discussions with him, and in desperate situations he encouraged me to keep going. He also provided me with his TEXmacrosfor typesetting derivations. IamgratefultoSteffenH¨olldobler for accepting the supervision of this thesis and for providing ideal conditions for doing research. He also made many helpful suggestions forimproving the readability. I am obliged to Fran¸cois Lamarche and Horst Reichel for accepting to be referees. Iwouldlike to thank Kai Br¨unnler, Paola Bruscoli, Charles Stewart, and Alwen Tiu for many fruitful discussion during the last three years. In particular, I am thankful to Kai Br¨unnler for struggling himself through a preliminary version and making helpful comments. Jim Lipton and Charles Stewart made valuable suggestions for improving the introduction. Additionally, I would like to thank Claus J¨urgensen for the fun we had in discussing TEXmacros.Inparticular, the (self-explanatory) macro \clap,whichisused on almost every page, came out of such a discussion. This thesis would not exist without the support of my wife Jana. During all the time she has been a continuous source of love and inspiration. This PhD thesis has been written with the financial support of the DFG-Graduierten- kolleg 334 “Spezifikation diskreter Prozesse und Prozeßsysteme durch operationelle Modelle und Logiken”. iii iv Table of Contents Acknowledgements iii Table of Contents v List of Figures vii 1Introduction 1 1.1Proof Theory andDeclarativeProgramming .................. 1 1.2LinearLogic .................................... 5 1.3Noncommutativity ................................ 8 1.4The Calculus of Structures . .......................... 9 1.5 Summary of Results............................... 12 1.6OverviewofContents.............................. 15 2LinearLogic and the Sequent Calculus 17 2.1Formulaeand Sequents . ............................. 17 2.2Rules andDerivations . ............................. 18 2.3Cut Elimination................................. 22 2.4Discussion . .................................... 24 3LinearLogic and the Calculus of Structures 25 3.1Structuresfor Linear Logic........................... 25 3.2Rules andDerivations . ............................. 28 3.3Equivalence to theSequent Calculus System .................. 36 3.4Cut Elimination................................. 41 3.4.1 Splitting .................................. 44 3.4.2Context Reduction ............................ 68 3.4.3Eliminationofthe Up Fragment.................... 72 3.5Discussion . .................................... 80 4TheMultiplicative Exponential Fragment of Linear Logic 83 4.1Sequents, Structures andRules ......................... 83 4.2Permutation of Rules. ............................. 86 4.3Decomposition.................................. 96 4.3.1Chainsand Cycles in Derivations.................... 100 4.3.2SeparationofAbsorptionand Weakening ............... 120 4.4Cut Elimination................................. 129 4.5Interpolation ................................... 139 v vi Table of Contents 4.6Discussion ..................................... 141 5ALocalSystemforLinear Logic 143 5.1Localityvia Atomicity.............................. 143 5.2Rules andCut Elimination........................... 145 5.3Decomposition.................................. 151 5.3.1SeparationofAtomicContraction . .................. 152 5.3.2SeparationofAtomicInteraction .................... 160 5.3.3LazySeparationofThinning . ..................... 167 5.3.4EagerSeparationofAtomicThinning ................. 173 5.4Discussion ..................................... 177 6MixandSwitch 179 6.1Adding theRules Mixand Nullary Mix.................... 179 6.2The Switch Rule . ................................ 185 6.3Discussion ..................................... 195 7ANoncommutative Extension of MELL 197 7.1Structuresand Rules ............................... 198 7.2Decomposition.................................. 202 7.2.1Permutation of Rules.......................... 204 7.2.2CyclesinDerivations ........................... 209 7.3Cut Elimination................................. 215 7.3.1 Splitting .................................. 215 7.3.2Context Reduction ............................ 224 7.3.3Eliminationofthe Up Fragment.................... 226 7.4 The Undecidability of System NEL . ............... 231 7.4.1Two CounterMachines ......................... 232 7.4.2EncodingTwo CounterMachinesinNEL Structures . ........ 233 7.4.3Completenessofthe Encoding ..................... 235 7.4.4SomeFacts aboutSystemBV...................... 236 7.4.5Soundness of the Encoding . ............... 241 7.5Discussion ..................................... 248 8OpenProblems 249 8.1Quantifiers.................................... 249 8.2AGeneralRecipe forDesigning Rules ..................... 250 8.3The Relation between Decompositionand CutElimination......... 251 8.4 Controlling the Nondeterminism . ............... 252 8.5The Equivalencebetween System BV andPomsetLogic ........... 252 8.6 The Decidability of MELL . ......................... 253 8.7The EquivalenceofProofsinthe Calculus of Structures ........... 253 Bibliography 255 Index 263 List of Figures 1.1Overviewoflogicalsystems discussedinthisthesis.............. 13 2.1 System LL in thesequent calculus . ...................... 20 3.1 Basic equations for the syntactic congruence for LS structures ........ 26 3.2 System SLS .................................... 35 3.3 System LS . .................................... 36 3.4 System LS .................................... 46 4.1 System MELL in thesequent calculus ...................... 84 4.2 Basic equations for the syntactic congruence of ELS structures . ..... 85 4.3 System SELS ................................... 86 4.4 System ELS .................................... 86 4.5 Possible interferences of redex and contractum of two consecutive rules . 88 4.6Examplesfor interferences between twoconsecutive rules. ......... 89 4.7 Permuting b↑ up and b↓ down ......................... 99 4.8Connectionof!-links and?-links ........................ 102 4.9 A cycle χ with n(χ)=2 ............................. 106 4.10 A promotion cycle χ with n(χ)=3....................... 107 4.11 A pure cycle χ with n(χ)=2 .......................... 109 4.12 Example (with n(χ)=3) forthe markinginside∆ .............. 110 4.13 Cut elimination for system SELS ∪{1↓} . ................... 132 4.14 Proof of the interpolation theorem for system SELS .............. 141 5.1 System SLLS ................................... 146 5.2 System LLS .................................... 152 6.1 Basic equations for the syntactic congruence of ELS◦ structures . ..... 181 6.2 System SELS◦ ................................... 182 6.3 System ELS◦ ................................... 182 6.4 System SS◦ .................................... 183 6.5 System S◦ . .................................... 183 7.1 Basic equations for the syntactic congruence of NEL structures . ..... 198 7.2 System SNEL ................................... 199 7.3 System NEL .................................... 200 7.4 System SBV .................................... 201 7.5 System BV .................................... 201 vii viii List of Figures 1 Introduction 1.1 Proof Theory and Declarative Programming Proof theory is the area of mathematics which studies the concepts of mathematical proof and mathematical provability [Bus98]. It is mainly concerned with the formal syntax of logical formulae and the syntactic presentations of proofs, and can therefore be regarded as “logic from the syntactic viewpoint” [Gir87b]. An important topic of research in proof theory is the relation between finite and infinite objects. In other words, proof theory investigates how infinite mathematical objects are denoted by finite syntactic constructions, andhow facts concerning infinite structures are proved by finite proofs. Another important question of proof theoretical research is the investigation of intuitive proofs [Kre68]. More precisely, not only the study of a given formal system is of interest, but also the analysis of the intuitive proofs and the choice of the formal systems needs attention. These general questions of research are not the only ones that are investigated in proof theory, but they arethemostimportantonesfortheapplication of proof theory in computer science in general, and in declarative programming in particular. By restricting itself to finitary methods, proof theory studies the objects that computers (which are, per se, finite) can deal with. In declarative programming the intention is to describe what the user wants to achieve, rather than how the machine is accomplishing it. By concerning itself with