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LIT-843: Combining contextuality and causality: a game semantics approach

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Properties6
StatusActive — read in full 2026-10-09 (NOTE-646); worth reading as the version of sheaf-theoretic contextuality in which measurements are performed in a causal order and the choice of the next one may depend on earlier outcomes. A causal measurement scenario adds an enabling relation (Kahn and Plotkin's information matrices renamed); Nature's deterministic strategies over it replace the event sheaf, and the Abramsky–Brandenburger definitions of empirical model and contextuality then go through unchanged. Flat scenarios and Gogioso and Pinzani's Bell scenarios with a causal order come out as special cases, and Experimenter strategies express adaptive measurement, as in the Anders–Browne GHZ OR-gate. Its sharpest point is that the strategy presheaf is not a sheaf: deterministic, compatible local strategies can fail to glue (Example 7.2) or glue non-uniquely (Example 7.3). A short paper; several results are deferred to an extended version.
Tagscontextuality, causality, quantum-foundations, mathematics, logic, game-theory
ExtendsLIT-016 — The sheaf-theoretic structure of non-locality and contextuality, LIT-813 — The Sheaf-Theoretic Structure of Definite Causality
SupportsCLAIM-038 — A communicative object is a context-indexed system of empirical constraints over a cover of contexts, and these need not extend to one globally realizable interpretation
Cited as “paper” byNOTE-646 — Combining contextuality and causality
Cited as “source” byTHEORY-171 — When the causal constraints on events depend on context, deterministic assignments that are causal in each context can fail to glue into one causal assignment, so causal structure is itself a source of contextuality

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Samson Abramsky, Rui Soares Barbosa and Amy Searle (2024), Philosophical Transactions of the Royal Society A 382(2268):20230002, theme issue “Quantum contextuality, causality and freedom of choice” — ARXIV-2307.04786, DOI-10.1098/rsta.2023.0002, PMC10822710

Key takeaways

  • Causal measurement scenarios. A tuple (X, O, ⊢): measurements, outcome sets, and an enabling relation s ⊢ x saying x may be performed after the events in s. Because s includes outcomes, which measurement comes next can depend on what was observed. Histories are the consistent event sets reachable from ∅. The structure is Kahn and Plotkin’s information matrix (ca. 1975) under new names.
  • Contextuality as a game. The Experimenter’s moves are measurements and Nature’s are outcomes. A strategy for Nature is a down-closed, deterministic, total set of histories: a hidden variable whose answer may depend on the causal past. Strategies form a presheaf Γ, and an empirical model is a compatible family of distributions on Γ over a cover. It is causally noncontextual when it extends to a global distribution, and the noncontextual models form a polytope checkable by linear programming.
  • Special cases. With everything enabled from ∅, Γ is the event sheaf and the theory is Abramsky and Brandenburger’s (LIT-016). With a causal order on the sites of a Bell scenario it reproduces Gogioso and Pinzani’s sheaf of sections (LIT-813), for the two-party example worked in §6b. The general equivalence is deferred to an extended version.
  • Not a sheaf. Deterministic local strategies that agree on overlaps can have no gluing (Example 7.2: z enabled by x = 0 or by y = 0, with local strategies answering z differently) or several (Example 7.3). Unions of compatible strategies are always deterministic (Proposition 7.1); totality is what fails. A note added in proof says the sheaf property holds for covers of causally secured sets, with the proof deferred to a sequel.
  • Two sources of causality. Nature-imposed causal background changes which models count as classical; Experimenter-imposed causality is an E-strategy (adaptive measurement choice) played against a model, as in the Anders–Browne construction of OR from GHZ Pauli measurements.

Standing in the record

Filed on 2026-10-09 at the owner’s direct request, from the PubMed Central link https://pmc.ncbi.nlm.nih.gov/articles/PMC10822710/. It is not from the manuscript bibliography: the owner asked for it directly on 2026-10-09. It joins the record’s causal-contextuality line beside Gogioso and Pinzani’s trilogy (LIT-800, LIT-808, LIT-788) and the withdrawn paper it builds on (LIT-813), and it gives a second, independent instance of THEORY-171’s deterministic non-gluing. NOTE-646 says how it bears on the manuscript’s argument on line pragmatic-transport.

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