Game Semantics Allison Ramil April 17 2012 Mathematical Logic History Paul Lorenzen Late 1950s Kuno Lorenz Renewed Interest in the mid 1990s Types of Logic Classical Logic Intuitionistic ID: 285511
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Slide1
Lorenzen Game Semantics
Allison Ramil
April 17, 2012
Mathematical LogicSlide2
History
Paul
Lorenzen
Late 1950sKuno LorenzRenewed Interest in the mid 1990’sSlide3
Types of Logic
Classical Logic
Intuitionistic
LogicLinear LogicSlide4
Basics of
Lorenzen
Game Semantics
Meaning of formula drained from dialogue Two charactersProponent
Opponent
Proponent
Proposes formula
Opponent
Denies formulaSlide5
Basics of
Lorenzen
Game Semantics
GamesPropositionsConnectives
Operation on the gamesSlide6
Basics of
Lorenzen
Game Semantics
DialogueStatements made by the Proponent and OpponentProponent 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 ProponentSlide7
Rules of Lorenzen Game SemanticsSlide8
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 onceSlide9
Lorenzen’s D-dialogueSlide10
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
tSlide11
Connectives
Negation
Reverses roles of Proponent and Opponent
DisjunctionAdditiveProponent chooses a or b to defend and abandons the other
Multiplicative
a b
Proponent can switch between a and b until one is wonSlide12
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 blackPlayed on two boards
C
C
Proponent can switch between two boards
C
V
C
Proponent must pick one board to play in the beginningSlide13
Connectives
Conjunction
Additive
MultiplicativeSlide14
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
Slide15
ConnectivesSlide16
Quantifiers
Universal quantifier
Existential quantifier
ExampleP: For every disease, there is a medicine which cures that diseaseO: Names arbitrary disease d
P: names medicine m
P wins if m is a cureSlide17
Applications
Players represent input-output
Opponent move = Input action
Proponent move = Output actionAutomated Verification Tool
First introduced by
Ambramsky
,
Ghica
,
Murawaski
, and
Ong
Advantages: possible to model open terms, internal compositionalitySlide18
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
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