The intuitionistic temporal logic of dynamical systems

Fernández–Duque, David

doi:10.23638/lmcs-14(3:3)2018

Cited by 1 publication

(1 citation statement)

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“…Modal semantics have also exploited mathematical structures such as: i) Neighborhood models [9], topological models for spatial logics [8], hybrid systems [5], and temporal logics of dynamical systems [20]. ii) Categorical [3], sheaf [28], and pre-sheaf [23] models.…”

Section: Introductionmentioning

confidence: 99%

Constructive Game Logic

Bohrer¹,

Platzer²

2020

Programming Languages and Systems

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Game Logic is an excellent setting to study proofs-aboutprograms via the interpretation of those proofs as programs, because constructive proofs for games correspond to effective winning strategies to follow in response to the opponent's actions. We thus develop Constructive Game Logic, which extends Parikh's Game Logic (GL) with constructivity and with first-order programsà la Pratt's first-order dynamic logic (DL). Our major contributions include: 1. a novel realizability semantics capturing the adversarial dynamics of games, 2. a natural deduction calculus and operational semantics describing the computational meaning of strategies via proof-terms, and 3. theoretical results including soundness of the proof calculus w.r.t. realizability semantics, progress and preservation of the operational semantics of proofs, and Existence Properties on support of the extraction of computational artifacts from game proofs. Together, these results provide the most general account of a Curry-Howard interpretation for any program logic to date, and the first at all for Game Logic.This work is at the intersection of game logic and constructive modal logics. Individually, they have a rich literature, but little work has been done at their intersection. Of these, we are the first for GL and the first with a proofs-asprograms interpretation for a full first-order program logic. Games in Logic.Parikh's propositional GL [38] was followed by coalitional GL [39]. A first-order version of GL is the basis of differential game logic dGL [42] for hybrid games. GL's are unique in their clear delegation of strategy to the proof language rather than the model language, crucially allowing succinct game specifications with sophisticated winning strategies. Succinct specifications are important: specifications are trusted because proving the wrong theorem would not ensure correctness. Relatives without this separation include Strategy Logic [15], Alternating-Time Temporal Logic (ATL) [4], CATL [30], Ghosh's SDGL [24], Ramanujam's structured strategies [44], Dynamic-epistemic logics [7,11,48], evidence logics [10], and Angelic Hoare logic [35]. and terms are rational-valued (Q); we also write B for the set of Boolean values {0, 1} for false and true respectively.

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