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Lorenzen Game Semantics

Lorenzen Game Semantics. Allison Ramil April 17, 2012 Mathematical Logic. History. Paul Lorenzen Late 1950s Kuno Lorenz Renewed Interest in the mid 1990’s. Types of Logic. Classical Logic Intuitionistic Logic Linear Logic. Basics of Lorenzen Game Semantics.

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Lorenzen Game Semantics

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  1. Lorenzen Game Semantics Allison Ramil April 17, 2012 Mathematical Logic

  2. History • Paul Lorenzen • Late 1950s • Kuno Lorenz • Renewed Interest in the mid 1990’s

  3. Types of Logic • Classical Logic • Intuitionistic Logic • Linear Logic

  4. Basics of Lorenzen Game Semantics • Meaning of formula drained from dialogue • Two characters • Proponent • Opponent • Proponent • Proposes formula • Opponent • Denies formula

  5. Basics of Lorenzen Game Semantics • Games • Propositions • Connectives • Operation on the games

  6. Basics of Lorenzen Game Semantics • Dialogue • Statements made by the Proponent and Opponent • Proponent proposes formula • Opponent attacks • Play continues until one cannot make a move • Player who makes the last move wins • A formula is valid if there is a winning strategy for the Proponent

  7. Rules of Lorenzen Game Semantics

  8. Rules of Lorenzen Game Semantics • P may assert an atomic formula only after it has been asserted by O • If there are more than one attacks left to be answered by P , then the only one that can be answered is the most recent • An attack must be answered at most once • An assertion made by P may be attacked at most once

  9. Lorenzen’s D-dialogue

  10. Extensions of Lorenzen • Flesher • Opponent can react only upon the immediately preceding claim of P • Blass • Allowed for infinite games • Defined games as ordered triple (M, s, G) • Defined strategy as a function t

  11. Connectives • Negation • Reverses roles of Proponent and Opponent • Disjunction • Additive • Proponent chooses a or b to defend and abandons the other • Multiplicative a b • Proponent can switch between a and b until one is won

  12. Connectives • Disjunction Example • C = game of chess in which Proponent plays white and wins within at most 100 moves • C = game of chess in which Proponent does not lose within 100 moves playing black • Played on two boards • C C • Proponent can switch between two boards • C V C • Proponent must pick one board to play in the beginning

  13. Connectives • Conjunction • Additive • Multiplicative

  14. Connectives • Conjunction Example • If you have $1, then you can get 1,000 Russian Rubles (RR) • If you have $1, then you get 1,000,000 Georgian coupons (GC) • means having the option to convert it either into A or into B • A B means have both A and B • Having $1 implies 1,000 RR 1,000,000 GC but not 1,000 RR 1,000,000 GC

  15. Connectives

  16. Quantifiers • Universal quantifier • Existential quantifier • Example • P: For every disease, there is a medicine which cures that disease • O: Names arbitrary disease d • P: names medicine m • P wins if m is a cure

  17. Applications • Players represent input-output • Opponent move = Input action • Proponent move = Output action • Automated Verification Tool • First introduced by Ambramsky, Ghica, Murawaski, and Ong • Advantages: possible to model open terms, internal compositionality

  18. References • http://en.wikipedia.org/wiki/Paul_Lorenzen • Lorenzen’s Games and Linear Logic by Rafael Accorsi and Dr. Johan van Benthem • A1 Mathematical Logic: Lorenzen Games for Full Intuitionistic Linear Logic by Valeria de Paiva • A Constructive Game Semantics for the Language of Linear Logic by GiorgiJaparidze • Applications of Game Semantics: From Program Analysis to Hardware Synthesis by Dan Ghica • Towards using Game Semantics for Crypto Protocol Verification: Lorenzen Games by Jan Jurjens

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