Bi-Valued Setting in Multi-Domain Research

• Bi-valued settings are frameworks defined by two distinct values or poles, establishing tractable models in diverse fields such as quantum logic and fair division.
• In quantum logic, the framework challenges global bivalence through the Kochen–Specker theorem, allowing only local truth assignments within commutable contexts.
• In allocation, soft computing, and network theory, the two-valued restriction enables concrete algorithms while exposing sharp computational boundaries and dual-valuation structures.

Searching arXiv for recent and foundational uses of “bi-valued” across the relevant literatures. A bi-valued setting is not a single formalism but a family of research frameworks in which the underlying semantics, data, or dynamics are organized around two distinguished values, two poles, or two coefficient systems. In quantum logic, the phrase denotes a classical bivalent valuation $v:\mathcal{P}(H)\to\{0,1\}$ on projection operators and raises the question of whether propositions about a quantum system can be pre-assigned “true” or “false” noncontextually; the Kochen–Specker obstruction shows that, for $\dim H\ge 3$, such a total assignment cannot exist globally (Bolotin, 2017). In algorithmic fair division and random assignment, “bi-valued” usually means that each item has one of two possible positive costs or utilities for each agent, such as $\{1,r\}$ or $\{a_i,b_i\}$ (He et al., 7 Jun 2026, Jin et al., 24 Jul 2025, Aziz et al., 2020). In soft computing, the same expression refers instead to two poles of information, encoded by positive and negative approximations rather than binary truth values (Shabir et al., 2013). In binary network theory and reversible logic, it denotes the specialization to $\{0,1\}$-valued node states or control signals (Ji et al., 2023, Moraga, 2015). In operator-valued bi-free probability, the relevant duality is between $B$-valued and $D$-valued cumulant systems, or between paired expectations $(E,F)$, rather than a two-point numerical codomain (Skoufranis, 2015, Gu et al., 2016).

1. Terminological scope and recurrent formal patterns

The literature exhibits several distinct meanings of “bi-valued,” and the distinction is structural rather than merely terminological. In one class of uses, the codomain itself has two values: $\{0,1\}$ in quantum logic and binary networks, $\{1,r\}$ or $\dim H\ge 3$0 in allocation and auction models, and $\dim H\ge 3$1 in random assignment (Bolotin, 2017, Ji et al., 2023, He et al., 7 Jun 2026, Jin et al., 24 Jul 2025, Aziz et al., 2020, Ben-Zwi et al., 2011). In a second class, the framework is “bi-valued” because it tracks two poles or two coefficient systems: positive/negative approximations in bipolar soft sets, or $\dim H\ge 3$2-valued versus $\dim H\ge 3$3-valued cumulants in bi-free probability (Shabir et al., 2013, Skoufranis, 2015).

Domain	Meaning of “bi-valued”	Canonical formal object
Quantum logic	classical truth values	$\dim H\ge 3$4
Fair division / assignment	two-point utilities or costs	$\dim H\ge 3$5, $\dim H\ge 3$6
Soft sets	positive and negative poles	$\dim H\ge 3$7 with $\dim H\ge 3$8
Binary networks / circuits	Boolean node or control values	$\dim H\ge 3$9, $\{1,r\}$0
Operator-valued bi-free probability	dual cumulant or expectation systems	$\{1,r\}$1, $\{1,r\}$2; or $\{1,r\}$3

This dispersion of meanings matters because results do not transfer automatically across fields. A two-valued truth assignment in quantum foundations is a semantic object; a bi-valued utility profile in allocation is a combinatorial restriction on preferences; a bipolar soft set retains a third hesitation region; and operator-valued bi-free theory uses “valued” in the sense of codomain algebra rather than discrete numerical range.

2. Quantum-logical bivalence and the Kochen–Specker obstruction

In the quantum-logical setting, propositions about a system are represented by projection operators. If a proposition $\{1,r\}$4 is represented by $\{1,r\}$5, the valuation axiom is written

$\{1,r\}$6

and a bi-valued setting requires

$\{1,r\}$7

with $\{1,r\}$8 meaning that $\{1,r\}$9 is true and $\{a_i,b_i\}$0 meaning that $\{a_i,b_i\}$1 is false (Bolotin, 2017). The classical constraints imposed on $\{a_i,b_i\}$2 are noncontextuality and functional compatibility: $\{a_i,b_i\}$3, $\{a_i,b_i\}$4; if $\{a_i,b_i\}$5 are orthogonal then $\{a_i,b_i\}$6; if $\{a_i,b_i\}$7 commute then $\{a_i,b_i\}$8. For any context $\{a_i,b_i\}$9 resolving the identity, $\{0,1\}$0, one further requires

$\{0,1\}$1

so exactly one outcome is true prior to measurement (Bolotin, 2017).

The projection lattice supplies the logical background. In the concrete two-qubit model used in “The relation between the Kochen–Specker theorem and bivalence” (Bolotin, 2017), the Hilbert space has dimension $\{0,1\}$2, and the lattice $\{0,1\}$3 is formed by the ranges of projectors, ordered by inclusion, with meet given by intersection and join by span. Inside a context $\{0,1\}$4, orthogonality yields

$\{0,1\}$5

while the join of all context projectors is their sum,

$\{0,1\}$6

Accordingly, tautology and contradiction are represented by $\{0,1\}$7 and $\{0,1\}$8, with $\{0,1\}$9 and $B$0 (Bolotin, 2017).

The Kochen–Specker theorem is the global obstruction. For $B$1, there is no noncontextual total valuation $B$2 satisfying the additivity, product, and context conditions above; equivalently, there is no global two-valued measure or lattice homomorphism $B$3 preserving $B$4 and $B$5 on all projectors (Bolotin, 2017). The paper exhibits this concretely with a set $B$6 of $B$7 projectors partitioned into three contexts $B$8. Within a single context, local bivalence works: for the correlated state

$B$9

one has $D$0 and the remaining three $D$1-context projectors take value $D$2 (Bolotin, 2017).

The contradiction appears when one tries to extend the assignment noncontextually to incompatible contexts. For $D$3, the paper gives explicit range and kernel descriptions and shows that if $D$4, then a bivalent assignment to $D$5 would force

$D$6

which is presented as the contradiction “$D$7” (Bolotin, 2017). The paper’s conclusion is that $D$8 cannot be total on $D$9: $(E,F)$0 At least one associated proposition must therefore be neither true nor false prior to verification. The text explicitly discusses two responses: many-valued semantics, with values in $(E,F)$1, and partial or “gappy” semantics, where some projections have no truth value at all (Bolotin, 2017).

A common misconception is that the Kochen–Specker theorem rules out bivalence everywhere. The paper is more specific: bivalence can hold locally within a single commuting context, but it cannot be extended globally across incompatible contexts by a noncontextual total valuation (Bolotin, 2017).

3. Two-point valuation domains in allocation, assignment, and auctions

In fair division of chores, a bi-valued instance has personalized additive costs in a two-point set. After normalization, the paper “EFX for Additive Chores: Nonexistence, Pareto Incompatibility, and Bi-Valued Existence” uses

$(E,F)$2

where an item is “small” for agent $(E,F)$3 if $(E,F)$4 and “large” if $(E,F)$5 (He et al., 7 Jun 2026). The EFX notion adopted is the “removal from own bundle” version: $(E,F)$6 or equivalently $(E,F)$7 for all $(E,F)$8, where

$(E,F)$9

The paper proves that for every $\{0,1\}$0 and $\{0,1\}$1, there exists a $\{0,1\}$2-bi-valued instance in which every EFX allocation is not Pareto-optimal, and it identifies this as the first incompatibility example with strictly positive costs (He et al., 7 Jun 2026). At the same time, the same paper establishes a positive existence theorem for $\{0,1\}$3: every bi-valued instance admits an EFX allocation, via a constructive decomposition into $\{0,1\}$4, $\{0,1\}$5, and $\{0,1\}$6, using a canonical prefix allocation, a multigraph-orientation step, and an insertion lemma based on an envy graph (He et al., 7 Jun 2026).

The personalized-goods analogue replaces a common ratio $\{0,1\}$7 by agent-specific ratios $\{0,1\}$8. In “On Pareto-Optimal and Fair Allocations with Personalized Bi-Valued Utilities,” each agent has

$\{0,1\}$9

or after normalization,

$\{1,r\}$0

The paper gives a structural characterization of Pareto improvements for integer ratios via Type I and Type II exchange cycles, implying a polynomial-time algorithm for deciding Pareto-optimality when each $\{1,r\}$1 (Jin et al., 24 Jul 2025). For fractional ratios, the decision problem becomes coNP-complete, even when the ratios take only two rational values (Jin et al., 24 Jul 2025). The same paper proves that an EFX allocation always exists in the personalized bi-valued setting and can be computed in polynomial time through a Match–Modify–Freeze procedure that prioritizes large goods via maximum matchings and bounded freezing times satisfying $\{1,r\}$2 (Jin et al., 24 Jul 2025).

In random assignment, the bi-valued restriction is imposed on utilities of fractional allocations. The model in “Random Assignment Under Bi-Valued Utilities” uses balanced assignments $\{1,r\}$3 with

$\{1,r\}$4

and utilities

$\{1,r\}$5

The paper shows that under bi-valued utilities the Hylland–Zeckhauser rule is equivalent to EPS on the induced dichotomous preferences, to the egalitarian rule of Bogomolnaia–Moulin applied to binary-reduced utilities, to balanced leximin on the binary-reduced instance, and to balanced Nash social welfare on the binary-reduced instance (Aziz et al., 2020). One consequence is invariance under scaling and shifting of the reported values $\{1,r\}$6: HZ depends only on the liked-item sets $\{1,r\}$7. Another is algorithmic: the EPS-based computation yields runtime

$\{1,r\}$8

via network-flow computations (Aziz et al., 2020). The paper also distinguishes HZ from unconstrained CEEI/MNW, which are not strategyproof under general bi-valued utilities, and from Nash bargaining, which fails envy-freeness and strategyproofness even under $\{1,r\}$9–$\dim H\ge 3$00 utilities (Aziz et al., 2020).

In prior-free auctions, a bi-valued domain means bidder values in $\dim H\ge 3$01 for an unlimited-supply digital good. The benchmark is

$\dim H\ge 3$02

where $\dim H\ge 3$03 is the number of $\dim H\ge 3$04-bids (Ben-Zwi et al., 2011). “Optimal Bi-Valued Auctions” constructs an explicit deterministic truthful auction and proves the tight guarantee

$\dim H\ge 3$05

while also showing that for every auction, even randomized and superpolynomial, there exists a bid vector with gap

$\dim H\ge 3$06

The paper argues that this additive-loss formulation is the correct notion of competitiveness in the bi-valued auction setting, because purely multiplicative guarantees fail near the threshold $\dim H\ge 3$07 (Ben-Zwi et al., 2011).

Across these allocation and market-design literatures, the bi-valued restriction serves two roles. It is a tractable special case—yielding equivalences, algorithms, and exact structural lemmas—and it is also a sharp boundary case where impossibility, incompatibility, or hardness can already appear.

4. Binary-valued dynamics, aggregation, and reversible control

In finite-valued network theory, the bi-valued specialization is strictly Boolean: each node takes values in $\dim H\ge 3$08. A transition system $\dim H\ge 3$09 admits an algebraic state space representation

$\dim H\ge 3$10

and in the binary specialization the state is a Boolean vector $\dim H\ge 3$11 (Ji et al., 2023). Quotienting by output equivalence yields a simulation system whose ASSR is

$\dim H\ge 3$12

with

$\dim H\ge 3$13

If the quotient is deterministic, then by Proposition 3.8 it is a bi-simulation, and under Proposition 4.10 replacing the original block by its quotient is lossless for the overall input-output behavior (Ji et al., 2023). If the quotient is non-deterministic, the paper introduces a probabilistic replacement based on the tallied matrix

$\dim H\ge 3$14

thereby converting the block into a probabilistic Boolean network (Ji et al., 2023).

The network-theoretic notion of “bi-valued” is therefore a state-space specialization rather than a semantic one. Its importance lies in the reduction from $\dim H\ge 3$15 internal states of a block to $\dim H\ge 3$16 quotient states when the block exposes only $\dim H\ge 3$17 outputs, together with an explicit tradeoff between reduction and exactness (Ji et al., 2023).

Reversible logic uses the same binary codomain but in a circuit-synthesis context. In “Mixed polarity reversible Peres gates,” all control signals are binary-valued, $\dim H\ge 3$18, and mixed polarity is encoded by a polarity vector $\dim H\ge 3$19 with effective controls

$\dim H\ge 3$20

A multi-control Peres gate fires when

$\dim H\ge 3$21

and the target transformation is

$\dim H\ge 3$22

For $\dim H\ge 3$23 controls, the target-line cascade uses $\dim H\ge 3$24th roots of NOT with

$\dim H\ge 3$25

and the synthesis employs $\dim H\ge 3$26 elementary controlled-root gates on the target, together with $\dim H\ge 3$27 CNOT updates, without ancillary lines (Moraga, 2015). The total quantum cost is

$\dim H\ge 3$28

and conversion to a Toffoli gate yields

$\dim H\ge 3$29

Here the bi-valued setting is the Boolean substrate that makes mixed-polarity control and ancilla-free synthesis possible (Moraga, 2015).

5. Two-pole information in bipolar soft sets

Bipolar soft sets use “bi-valued” in a different sense: not two truth values, but two poles of parameterized information. A bipolar soft set over a universe $\dim H\ge 3$30 is a triplet

$\dim H\ge 3$31

where $\dim H\ge 3$32, $\dim H\ge 3$33, $\dim H\ge 3$34, and

$\dim H\ge 3$35

The positive approximation $\dim H\ge 3$36 and the negative approximation $\dim H\ge 3$37 are accompanied by a hesitation region

$\dim H\ge 3$38

Thus the formalism is bi-polar but not exhaustively binary, because elements may belong to neither side (Shabir et al., 2013).

The algebra of the formalism is explicit. Complement exchanges the poles: $\dim H\ge 3$39 Binary operations are defined both over product parameter sets and over overlapping parameter sets. For example,

$\dim H\ge 3$40

with

$\dim H\ge 3$41

while

$\dim H\ge 3$42

uses unions on the positive side and intersections on the negative side (Shabir et al., 2013). The paper proves De Morgan laws, associativity and commutativity for the extended and restricted operations, and bounded distributive lattice structures. For fixed $\dim H\ge 3$43, $\dim H\ge 3$44 is a bounded distributive lattice with least element $\dim H\ge 3$45 and greatest element $\dim H\ge 3$46, and with complement it becomes a De Morgan algebra (Shabir et al., 2013).

The decision-theoretic encoding makes the two-pole semantics operational. A bipolar soft set can be tabulated by entries

$\dim H\ge 3$47

where $\dim H\ge 3$48 denotes positive membership, $\dim H\ge 3$49 negative membership, and $\dim H\ge 3$50 hesitation. The unweighted decision value is

$\dim H\ge 3$51

and the weighted variant replaces $\dim H\ge 3$52 by

$\dim H\ge 3$53

The paper’s algorithms then use indiscernibility, dispensability, and reduct computation to choose an optimal object (Shabir et al., 2013).

A plausible implication is that bipolar soft sets occupy an intermediate position between binary classification and genuinely many-valued uncertainty formalisms: they preserve a two-pole structure, but they do so together with an explicit unassigned region.

6. Dual-valued cumulant systems in operator-valued bi-free probability

In operator-valued bi-free probability, the relevant duality is algebraic. A $\dim H\ge 3$54-noncommutative probability space $\dim H\ge 3$55 has $\dim H\ge 3$56-valued moments and cumulants, but once a unital subalgebra $\dim H\ge 3$57 and a conditional expectation $\dim H\ge 3$58 are introduced, one obtains parallel $\dim H\ge 3$59-valued and $\dim H\ge 3$60-valued cumulant systems (Skoufranis, 2015). The central characterization in “On Operator-Valued Bi-Free Distributions” states that, assuming a faithfulness condition on $\dim H\ge 3$61, a two-faced family $\dim H\ge 3$62 is bi-free from $\dim H\ge 3$63 over $\dim H\ge 3$64 if and only if

$\dim H\ge 3$65

equivalently,

$\dim H\ge 3$66

The same paper proves that a two-faced family of matrices is $\dim H\ge 3$67-cyclic if and only if it is bi-free from the scalar matrices over the scalar diagonal matrices (Skoufranis, 2015).

The combinatorial basis for this theory is developed in “Combinatorics of Bi-Freeness with Amalgamation,” which constructs operator-valued bi-free moment and cumulant functions indexed by bi-noncrossing partitions $\dim H\ge 3$68 (Charlesworth et al., 2014). The cumulants are defined by Möbius inversion,

$\dim H\ge 3$69

and the main criterion is that a family of pairs of $\dim H\ge 3$70-faces is bi-free with amalgamation over $\dim H\ge 3$71 if and only if all mixed operator-valued bi-free cumulants vanish (Charlesworth et al., 2014). The same paper also gives a products-of-variables formula: $\dim H\ge 3$72 which replaces the cleaner scalar Kreweras-complement factorization that generally fails in the operator-valued setting (Charlesworth et al., 2014).

The conditional extension introduces a second expectation. In “Conditionally Bi-Free Independence with Amalgamation,” a quadruple $\dim H\ge 3$73 carries both a $\dim H\ge 3$74-valued expectation $\dim H\ge 3$75 and a $\dim H\ge 3$76-valued expectation $\dim H\ge 3$77, and c-bi-free independence is characterized by the simultaneous vanishing of mixed operator-valued bi-free cumulants and mixed operator-valued conditionally bi-free cumulants (Gu et al., 2016). The paper constructs operator-valued conditionally bi-multiplicative moment and cumulant pairs $\dim H\ge 3$78 and $\dim H\ge 3$79, together with a conditionally bi-free partial $\dim H\ge 3$80-transform (Gu et al., 2016).

The analytical version with completely positive maps pushes the dual-valued framework further. “Bi-Free Entropy with Respect to Completely Positive Maps” develops conjugate variables, Fisher information, and non-microstate bi-free entropy relative to CP maps $\dim H\ge 3$81 (Katsimpas et al., 2021). The left conjugate variable is characterized by cumulant conditions such as

$\dim H\ge 3$82

with higher mixed cumulants vanishing, and the Fisher information is the sum of squared $\dim H\ge 3$83-norms of left and right conjugate variables (Katsimpas et al., 2021). The entropy is defined by integrating the difference between a semicircular reference term and the $\dim H\ge 3$84-bi-free Fisher information along a bi-semicircular perturbation flow, and the paper proves minima of the Fisher information and maxima of the entropy at bi-$\dim H\ge 3$85-diagonal elements (Katsimpas et al., 2021).

This operator-valued literature therefore uses a “bi-valued” architecture in a precise but non-discrete sense: two coefficient systems, two expectations, or two cumulant calculi coexist, and independence is expressed as compatibility or vanishing across both layers.

7. Comparative perspective

Across these fields, the common thread is the attempt to impose a two-part structure on a problem that is otherwise more complex. In quantum logic, the attempt is a total noncontextual truth assignment, and the result is a no-go theorem forcing partiality or many-valuedness (Bolotin, 2017). In fair division and auctions, the restriction to two values sharpens the tractability frontier: it enables constructive EFX procedures for $\dim H\ge 3$86-agent bi-valued chores, polynomial-time HZ computation via EPS on binary-reduced preferences, and tight additive-competitive auction design, but it does not remove Pareto incompatibility or computational hardness in all regimes (He et al., 7 Jun 2026, Aziz et al., 2020, Ben-Zwi et al., 2011). In personalized bi-valued goods, integer ratios admit a cycle-based characterization of Pareto improvements, whereas fractional ratios already induce coNP-completeness (Jin et al., 24 Jul 2025).

The same comparison clarifies important non-equivalences. A bipolar soft set is not merely a binary classifier because it includes a hesitation region (Shabir et al., 2013). A binary network quotient is not merely a truth-value reduction because it is defined by simulation and output equivalence (Ji et al., 2023). A mixed-polarity Peres gate is bi-valued because its controls are Boolean, not because it is semantically bivalent (Moraga, 2015). Operator-valued bi-free probability is “bi-valued” only in the sense of dual algebra-valued analytic structures (Skoufranis, 2015, Gu et al., 2016, Katsimpas et al., 2021).

A plausible unifying description is therefore the following: a bi-valued setting is any framework in which the primitive formal objects are constrained by a two-level, two-pole, or dual-valued architecture, and where the main research question is whether that architecture is expressive enough, stable enough, or computationally advantageous enough for the problem at hand. The cited literature shows that the answer depends sharply on context. In some domains the bi-valued restriction yields exact equivalences and polynomial algorithms; in others it exposes impossibility, contextuality, or the necessity of partial or richer semantics.