Singular Boolean Expressions (SBEs)

Updated 23 July 2025

Singular Boolean Expressions (SBEs) are Boolean formulas where each variable appears exactly once, ensuring independent and unambiguous condition evaluation.
Their unique structure simplifies combinatorial reasoning, reduces computational complexity, and enables efficient model checking and test case generation.
SBEs support optimized structure graph analysis, probabilistic evaluation, and algebraic simplification, making them pivotal in safety-critical and industrial applications.

Singular Boolean Expressions (SBEs) are Boolean expressions in which each literal or condition variable appears exactly once and only once. This syntactic and semantic “singularity” makes SBEs a central object in the paper of Boolean equation systems, formal verification, probability theory, testing of safety-critical systems, and the algebraic foundations of logic. SBEs possess properties that simplify combinatorial reasoning, reduce complexity in key verification methodologies, and allow the design of optimally efficient test and analysis algorithms. Their prevalence—accounting for approximately 99.7% of conditional decisions in large-scale avionics systems—emphasizes their practical importance (Lee et al., 19 Jul 2025). This article surveys the definition, structural representations, analysis techniques, probabilistic and algebraic aspects, testing strategies, and industrial applications of SBEs.

1. Definition and Foundational Role of Singular Boolean Expressions

A Singular Boolean Expression is a Boolean formula in propositional logic, or a decision predicate in program code, in which each variable (condition) appears precisely once—no repetition (“coupling”) is permitted. Formally, for a variable set $C = \{c_1, ..., c_N\}$ , an SBE $F$ has the property that each $c_i$ appears once (possibly negated) in $F$ . This property has several immediate consequences:

There are no variable self-interactions, which removes ambiguity in reasoning about “independent cause” and enables unique-cause coverage in testing (Lee et al., 19 Jul 2025).
The logical structure of the expression directly encodes dependencies: the influence of each variable is distinct and non-overlapping.
SBEs form a subclass of Boolean expressions commonly found in specifications for safety- and mission-critical systems, as confirmed by empirical studies in avionics and control domains (Lee et al., 19 Jul 2025).

SBEs underpin methods in model checking, formal verification, reliability analysis, and testing—especially when high-assurance guarantees or exhaustive reasoning about condition coverage are required.

2. Structural Analysis and Minimization via Structure Graphs

SBEs can be effectively represented and analyzed using structure graphs, which extend traditional dependency graphs by capturing both variable dependencies and the full logical structure of the formula, including operator hierarchy and free/bound variable distinctions (1002.3222). The translation from an SBE (or a more general Boolean equation system, BES) to its structure graph employs Plotkin-style deduction rules (SOS rules):

Each vertex corresponds to a Boolean subformula or variable.
Vertices are decorated with logical tags: conjunction ( $\land$ ), disjunction ( $\lor$ ), constants ( $\top, \bot$ ), or free variable indicators ( $\nearrow X$ ).
For fixed-point systems, a ranking function $r$ assigns vertices a nesting rank.
Edges encode the decomposition of formulas into their immediate subformulas.

For example, the rule

$\text{\sosrule[(sos:fv\_ax)]{\,}{\,E \vdash X\ \nearrow X}}$

labels free variable nodes, and rules for conjunction flatten nested forms, encoding associativity and commutativity.

A central result is bisimulation equivalence between structure graphs: two SBEs with bisimilar graphs have identical solutions regardless of syntactic differences. Bisimulation minimization identifies and merges redundant vertices (subformulas evaluated identically in all contexts), leading to potentially drastic size reductions in the induced Boolean equation system—thus lowering the computational complexity of subsequent fixed-point evaluation or model checking. In practice, structure graph minimization commonly results in orders-of-magnitude reductions in the number of equations to be solved, particularly valuable in system verification tasks (1002.3222).

3. Probabilistic Evaluation and Dissociation

SBEs are directly relevant in probabilistic analysis, especially where the objective is to calculate the probability of complex Boolean expressions comprising independent random variables. When an expression is not singular (i.e., variables repeat), the dissociation technique transforms it into a singular form by replacing each occurrence of a variable with an independent copy and assigning new probabilities (1105.2813).

The dissociation method:

Converts a general $\phi(x, A)$ into $\phi'(x', A)$ , where each $x_i'$ is a fresh variable.
Assigns probabilities to the independent copies with the goal of providing statically-tight bounds for the original probability: for disjunctive expressions, upper bounds are achieved by setting $p_i' \geq p$ for all copies; symmetric tight bounds are given for both conjunctive and disjunctive cases.
Results in bounds optimal among all methods that ignore dependencies in the background events.

Applications include probabilistic databases (safe query evaluation), Bayesian network approximations, and static reliability estimation. In all such settings, the reduction to singular (or “dissociated”) Boolean expressions simplifies reasoning and allows scalable approximation with guarantees on the quality of bounds (1105.2813).

4. Automated Testing and Coverage for SBEs in Safety-Critical Systems

Testing Singular Boolean Expressions is notably tractable and efficient. In the context of structural coverage, notably Modified Condition/Decision Coverage (MC/DC), SBEs play a central role because each condition can influence the result independently—ensuring unambiguous cause-effect relationships (Badhera et al., 2012). Unique-Cause MC/DC requires that, for each condition, there exist a pair of test cases differing only in that condition, such that the overall decision outcome toggles.

Robin’s Rule is a deterministic, multi-phase algorithm designed to generate minimal N+1 test cases that guarantee 100% Unique-Cause MC/DC for an SBE with N conditions, entirely circumventing the need to enumerate the full $2^N$ truth table (Lee et al., 19 Jul 2025). Its phases include normalization (to ensure canonical form), relation table generation (reflecting operator structure), and direct construction of test vectors based on minimal required patterns. Validation against industry standards has demonstrated both the correctness (full coverage) and optimality (minimum test set size) of the approach, outperforming commercial test generators on avionics specification benchmarks.

Empirical studies, particularly those surveying avionics control systems such as TCAS-II, confirm that the vast majority of Boolean decisions are SBEs, further emphasizing the utility and impact of specialized minimal test generation algorithms (Lee et al., 19 Jul 2025, Badhera et al., 2012).

5. Algebraic and Model-Theoretic Perspectives

SBEs are also prominent in algebraic and logical formulations of Boolean reasoning. Finite free Boolean algebras provide a canonical structure for representing SBEs, wherein each singular expression corresponds to a Boolean vector generated by the action of a finite set of independent vector generators (Mijajlovic et al., 2013). This framework enables:

Efficient computation and enumeration of models of logical theories translated into propositional logic.
Parallelization of model construction using blockwise operations on bit-vectors (supported by GPU/CPU architectures).
Direct interpretation of SBEs as atoms or minimal elements in the algebra—allowing combinatorial object counting and solution space exploration.

In more advanced algebraic settings, such as Clifford algebras $Cl(\mathbb{R}^{n,n})$ , SBEs correspond to primitive idempotents, providing an explicit bridge between logical and geometric representations. This perspective supports the reformulation of SAT problems in terms of covering continuous spaces (e.g., the orthogonal group $O(n)$ ), highlighting further theoretical connections and potential applications (Budinich, 2021).

6. Expression Simplification and Algorithmic Optimization

The simplification of Boolean expressions—including SBEs—benefits from focused, congruence-based methods that avoid brute-force enumeration of equivalent forms. Frameworks employing “collections of structures” and valuation-driven application of Boolean axioms (associativity, commutativity, distributivity, etc.) can systematically reduce SBEs to canonical or minimal forms (Charlier, 2020). The approach involves:

Encoding structure as normalized equivalence classes.
Iteratively applying axioms via focused valuation mappings, only where the substitution is relevant to parts of the original expression.
Employing variants and heuristics to control exploration depth, avoid fixpoint traps, and manage computational resources efficiently.

Empirical evaluations indicate that, for randomly constructed or adversarial SBEs, such methods can yield near-optimal simplification with manageable computational cost, especially when appropriately parameterized for expression size and variable alphabet (Charlier, 2020).

While SBEs are highly impactful for propositional systems, recent results demonstrate that in modal logics, operator choice can exponentially affect formula succinctness. However, in propositional logic, all complete Boolean bases (including those using SBEs) remain polynomially equivalent in expressiveness and formula size (Berkholz et al., 21 Aug 2024). This underlines that, within propositional frameworks, SBEs do not incur formula size penalties across different operator sets, maintaining their central role in efficient logic synthesis and analysis.

In summary, Singular Boolean Expressions constitute a foundational class of Boolean formulas, offering unique advantages in model-theoretic representation, computational tractability, automated testing, probabilistic approximation, and algebraic analysis. Their singularity property underpins deterministic and optimal techniques in safety-critical testing (Unique-Cause MC/DC), efficient model computation, focused simplification, and operator basis independence. These features explain their widespread presence in real-world specifications and their continuing prominence in theoretical and applied logic research.