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 Meaning of formula drained from dialogue Two characters Proponent Opponent Proponent Proposes formula Opponent Denies formula

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

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

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

Lorenzen’s D-dialogue

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 

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

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

Connectives Conjunction Additive Multiplicative

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

Connectives

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

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

References 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 Giorgi Japaridze 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