Blot. Realizability for Peano Arithmetic with Winning Conditions in HON Games

141 Pages·2013·0.58 MB·English

by  Valentin Blot

Realizability for Peano Arithmetic with Winning
Conditions in HON Games
Valentin Blot
ENSdeLyon
April 26, 2013
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 1/23
1 Introduction
2 HON game semantics
3 Winning conditions
4 Realizability
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 2/23
Introduction
1 Introduction
2 HON game semantics
3 Winning conditions
4 Realizability
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 3/23
Realizability for classical logic in presence of references
(cid:73) Realizers live in a model
(cid:73) This model has primitive mechanisms for references (c.f. Abramsky &
Honda & McCusker)
(cid:73) Classical logic is obtained using a CPS translation (c.f. Laird)
Definitions of winningness at the level of interactions
(cid:73) Similar to modified realizability (strategies are typed)
(cid:73) But winningness defined by dissecting rather than observing
Adequacy and extraction for Peano arithmetic
(cid:73) Interpretation of system T terms corresponding to CPS translated
proofs are indeed realizers
(cid:73) Extraction in the spirit of Friedman’s trick
Introduction
Introduction
A realizability based on HON games (c.f. Hyland & Ong, Nickau)
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 4/23
(cid:73) Realizers live in a model
(cid:73) This model has primitive mechanisms for references (c.f. Abramsky &
Honda & McCusker)
(cid:73) Classical logic is obtained using a CPS translation (c.f. Laird)
Definitions of winningness at the level of interactions
(cid:73) Similar to modified realizability (strategies are typed)
(cid:73) But winningness defined by dissecting rather than observing
Adequacy and extraction for Peano arithmetic
(cid:73) Interpretation of system T terms corresponding to CPS translated
proofs are indeed realizers
(cid:73) Extraction in the spirit of Friedman’s trick
Introduction
Introduction
A realizability based on HON games (c.f. Hyland & Ong, Nickau)
Realizability for classical logic in presence of references
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 4/23
(cid:73) This model has primitive mechanisms for references (c.f. Abramsky &
Honda & McCusker)
(cid:73) Classical logic is obtained using a CPS translation (c.f. Laird)
Definitions of winningness at the level of interactions
(cid:73) Similar to modified realizability (strategies are typed)
(cid:73) But winningness defined by dissecting rather than observing
Adequacy and extraction for Peano arithmetic
(cid:73) Interpretation of system T terms corresponding to CPS translated
proofs are indeed realizers
(cid:73) Extraction in the spirit of Friedman’s trick
Introduction
Introduction
A realizability based on HON games (c.f. Hyland & Ong, Nickau)
Realizability for classical logic in presence of references
(cid:73) Realizers live in a model
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 4/23
(cid:73) Classical logic is obtained using a CPS translation (c.f. Laird)
Definitions of winningness at the level of interactions
(cid:73) Similar to modified realizability (strategies are typed)
(cid:73) But winningness defined by dissecting rather than observing
Adequacy and extraction for Peano arithmetic
(cid:73) Interpretation of system T terms corresponding to CPS translated
proofs are indeed realizers
(cid:73) Extraction in the spirit of Friedman’s trick
Introduction
Introduction
A realizability based on HON games (c.f. Hyland & Ong, Nickau)
Realizability for classical logic in presence of references
(cid:73) Realizers live in a model
(cid:73) This model has primitive mechanisms for references (c.f. Abramsky &
Honda & McCusker)
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 4/23
Definitions of winningness at the level of interactions
(cid:73) Similar to modified realizability (strategies are typed)
(cid:73) But winningness defined by dissecting rather than observing
Adequacy and extraction for Peano arithmetic
(cid:73) Interpretation of system T terms corresponding to CPS translated
proofs are indeed realizers
(cid:73) Extraction in the spirit of Friedman’s trick
Introduction
Introduction
A realizability based on HON games (c.f. Hyland & Ong, Nickau)
Realizability for classical logic in presence of references
(cid:73) Realizers live in a model
(cid:73) This model has primitive mechanisms for references (c.f. Abramsky &
Honda & McCusker)
(cid:73) Classical logic is obtained using a CPS translation (c.f. Laird)
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 4/23
(cid:73) Similar to modified realizability (strategies are typed)
(cid:73) But winningness defined by dissecting rather than observing
Adequacy and extraction for Peano arithmetic
(cid:73) Interpretation of system T terms corresponding to CPS translated
proofs are indeed realizers
(cid:73) Extraction in the spirit of Friedman’s trick
Introduction
Introduction
A realizability based on HON games (c.f. Hyland & Ong, Nickau)
Realizability for classical logic in presence of references
(cid:73) Realizers live in a model
(cid:73) This model has primitive mechanisms for references (c.f. Abramsky &
Honda & McCusker)
(cid:73) Classical logic is obtained using a CPS translation (c.f. Laird)
Definitions of winningness at the level of interactions
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 4/23
(cid:73) But winningness defined by dissecting rather than observing
Adequacy and extraction for Peano arithmetic
(cid:73) Interpretation of system T terms corresponding to CPS translated
proofs are indeed realizers
(cid:73) Extraction in the spirit of Friedman’s trick
Introduction
Introduction
A realizability based on HON games (c.f. Hyland & Ong, Nickau)
Realizability for classical logic in presence of references
(cid:73) Realizers live in a model
(cid:73) This model has primitive mechanisms for references (c.f. Abramsky &
Honda & McCusker)
(cid:73) Classical logic is obtained using a CPS translation (c.f. Laird)
Definitions of winningness at the level of interactions
(cid:73) Similar to modified realizability (strategies are typed)
ValentinBlot (ENSdeLyon) RealizabilityinHONgamesemantics April26,2013 4/23