Contribution
CbD 1.0 (LIT-777) asked whether the bunches of a system are compatible with a maximal coupling of each connection. This paper replaces that with a multimaximal coupling, in which every subset of a connection is maximally coupled. It proves that for binary variables this coupling exists, is unique and has a closed form. That gives the binary theory two properties CbD 1.0 lacked: heredity under deleting variables, and a single verdict instead of “partial” and “complete” ones. It also shows that the same definition misbehaves for non-binary variables, which motivates the move to dichotomizations.
Key insight
For binary variables, coupling the copies of a content “as closely as possible” has exactly one answer once every pair is required to be as close as possible. Sort the copies by their probability of a 1. The copies then switch from 1 to 2 one at a time, in that order, like a staircase. With the connection couplings fixed, contextuality is the single question whether the rows (contexts) and the columns (contents) of the system can be marginals of one joint distribution.
Assumptions
- A finite c-c system: each random variable R_q^c is indexed by a content q and a context c. Variables sharing a context are jointly distributed. Variables in different contexts are stochastically unrelated. All variables in one connection have the same set of values.
- Binary variables throughout the definitions and Theorems 1–4. Definition 3 is stated only for binary systems. Section 6 and Theorem 5 treat arbitrary categorical variables.
- Maximal coupling (from Thorisson; LIT-777 Theorem 3.3): a coupling maximizing Pr[T¹ = … = Tᵏ], which for categorical variables is Σ_v min_i Pr[Rⁱ = v].
Key results
- Definition 1. A coupling (T_q^1, …, T_q^k) of a connection is multimaximal if, for every subset of size m > 1, Pr[T_q^{i₁} = … = T_q^{i_m}] is the largest attainable among couplings of that subset.
- Definitions 2–3. A coupling of the whole system is multimaximally connected if its restriction to each connection is multimaximal. A binary system is noncontextual if and only if one exists.
- Theorem 1. For a binary connection with p_i = Pr[R_q^i = 1] sorted p₁ ≤ … ≤ p_k, a coupling is multimaximal if and only if its only nonzero masses are p₁ on 11…1, p_{l+1} − p_l on 2…21…1 (l twos, k − l ones) for l = 1, …, k − 1, and 1 − p_k on 22…2. Sufficiency is checked directly; necessity is by induction on the position of the first 1.
- Corollary 1. A multimaximal coupling of a binary connection exists and is unique.
- Theorem 2 (proof in LIT-836). For a binary connection in that order, a coupling is multimaximal if and only if each adjacent pair (T_q^i, T_q^{i+1}) is a maximal coupling.
- Theorem 3. In a noncontextual binary system, every subsystem (obtained by deleting variables) is noncontextual. The proof deletes the corresponding variable from the coupling; subsets of a multimaximal coupling stay multimaximal.
- Properties stated in Section 4. (a) For consistently connected systems, multimaximal couplings are identity couplings, so the verdict is the traditional one. (b) Binary systems are always completely (non)contextual. (c) For two-element connections, maximal and multimaximal coincide, so the theory of cyclic systems is unchanged. (d) For Kochen–Specker systems with more than two variables per connection (Peres-type proofs), the result “will yield different results” from CbD 1.0. No example is computed.
- Theorem 4. Every binary system has a quasi-coupling (signed masses summing to 1, proper bunch marginals) whose connection marginals are the multimaximal couplings. Among these there is one of least total variation, which is the measure of contextuality. The existence result is LIT-777’s Theorem 6.1.
- Theorem 5. For arbitrary variables, a coupling is multimaximal if and only if every pair is maximally coupled. Proof: if some subset fails, a value v has Pr[all = v] < min_c Pr[T_q^c = v] while every pair attains its pairwise minimum at v. Dichotomizing at v contradicts Theorem 2.
- Examples 1–3. (1) Three 3-valued variables with masses (0, ½, ½), (½, 0, ½), (½, ½, 0): pairwise maximality forces three mutually exclusive events of probability ½ each, so no multimaximal coupling exists. (2) Three 6-valued variables have two distinct multimaximal couplings. (3) Lumping i with i′ in Example 2 gives Example 1, so a system with a multimaximal coupling (noncontextual) becomes one without (contextual) under coarse-graining.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A binary connection has exactly one multimaximal coupling, of the staircase form | strong (proof) | Theorem 1, Corollary 1 |
| C2 | Noncontextuality in CbD 2.0 is hereditary for binary systems | strong (proof) | Theorem 3 |
| C3 | CbD 1.0 noncontextuality is not hereditary | asserted, no example | Section 1 |
| C4 | CbD 2.0 agrees with CbD 1.0 on cyclic systems and on any system whose connections have two variables, and with the traditional definition under consistent connectedness | strong (immediate from the definitions) | Section 4 |
| C5 | Multimaximality is equivalent to pairwise maximality for arbitrary variables | strong (proof, relying on Theorem 2) | Theorem 5 |
| C6 | For non-binary variables, multimaximal couplings may not exist, may not be unique, and can be destroyed by coarse-graining | strong (counterexamples) | Examples 1–3 |
| C7 | CbD 2.0 and 1.0 give different verdicts on Peres-type Kochen–Specker systems | asserted, not computed | Section 4 |
Concepts
- conteXt / conteNt. The paper’s typography for context and content.
- connection. All variables sharing a content. They are pairwise stochastically unrelated.
- C-coupling; partially / completely C-(non)contextual. For a property C of connection couplings, a system is partially noncontextual if the bunches are compatible with some combination of C-couplings, and completely noncontextual if they are compatible with every combination. The adjectives are new here. They coincide when C-couplings are unique.
- multimaximal coupling. Every subset maximally coupled (Definition 1). By Theorem 5, the same as every pair maximally coupled.
- CbD 1.0 / CbD 2.0. The authors’ names for the maximal-coupling and multimaximal-coupling versions.
Connections
It revises LIT-777’s Definition 3.4 and keeps that paper’s principles, its linear-programming test and its Section 6 measure. Its Theorem 2 is proved in the companion LIT-836, which in turn cites this paper for the existence and uniqueness theorem. The two papers cite each other for their halves. Its closing proposal, to replace every non-binary variable by all its dichotomizations, is the programme of the canonical-systems paper LIT-790. The maximal-coupling theorem is Thorisson’s. The quasi-coupling idea is credited to de Barros and Oas.
Bearing on the record
- LIT-777 and NOTE-600. NOTE-600 reported from footnote 10 that the definition was being superseded and that the binary cyclic results were unchanged. This reading confirms both, from the source: for two-variable connections, maximal and multimaximal coincide. NOTE-600’s statement that CbD agrees with the sheaf criterion under consistent connectedness also holds for 2.0, since multimaximal couplings of identically distributed variables are identity couplings.
- THEORY-013. Its source’s systems (question order, matching, the Bruza word pairs) are binary and cyclic, so the switch to 2.0 does not change any verdict THEORY-013 reports.
- THEORY-014. Theorem 4 has that theory’s shape: a signed coupling always exists, and classicality is having a nonnegative one. It is LIT-777’s Theorem 6.1 applied to multimaximal couplings, so it is not a new instance.
- The manuscript’s claim CLAIM-069. The paper supports it. Different verdicts arise only for contents measured in more than two contexts. The cyclic case and the measure are unchanged. But the paper states the CbD 1.0 failure of heredity without an example, and it does not compute any system on which 1.0 and 2.0 disagree.
- It produces, with the canonical-systems paper, THEORY-168.
- No instruction for machine-learning practice.
Limitations
- Binary variables only. For non-binary ones the paper gives only problems and a proposal.
- The motivating claim, that CbD 1.0 noncontextuality is not hereditary, has no worked example.
- Theorem 2, which Theorem 5 depends on, is proved elsewhere.
- No example system is computed under both versions.
Open questions
- A concrete inconsistently connected binary system that is CbD 1.0-noncontextual with a contextual subsystem would make C3 precise.
- How large the disagreement between 1.0 and 2.0 is on Peres-type Kochen–Specker systems with realistic inconsistent connectedness.
- Whether the dichotomization programme is “feasible”, the paper’s own closing question. LIT-790 answers it for a single pair of categorical variables.