Solver-Centric Adjudication Frameworks

• Solver-centric adjudication is an approach that resolves conflicting outputs from multiple solvers by applying formal operators and logic-driven workflows.
• It leverages methods such as Majority Vote, Median, and GLB to enforce algebraic properties ensuring consistency, explainability, and fault-masking in complex decision environments.
• The framework employs calculational synthesis and iterative unsat-core loops to refine decisions, thereby enhancing resilience and verifiability across diverse applications.

Solver-centric adjudication is the formal process of resolving potentially conflicting outputs from multiple computational or algorithmic sources by means of dedicated solver-based mechanisms. This approach is grounded in the theory of N-version programming, distributed decision-making, and formal methods, and is central to applications from dependability engineering to legal automation. Unlike heuristic, majority-only, or purely procedural aggregation, solver-centric adjudication establishes explicit formal operators and logic-driven workflows to guarantee properties such as consistency, explainability, resilience to faults, and verifiability within a solver-ensemble context (Boiten, 2015).

1. Formal Frameworks and Operator Definitions

At the core is the abstract notion of an adjudication operator acting on a multiset (bag) of candidate outputs. Given a nonempty set VALUE of possible result values, the outputs of $n$ solvers are modeled as a bag $b : \mathrm{VALUE} \rightarrow \mathbb{N}$ with $\sum_{x}b(x)\ge 1$. The adjudication operator is a partial function $$\mathrm{ADJOP} : \mathrm{BAG} \rightharpoonup \mathrm{VALUE}$$.

Primary adjudication operators include:

• Majority Vote (MV): Returns $x$ iff $2b(x) > \mathrm{size}(b)$. Enforces strict majority (Boiten, 2015).
• First-Past-The-Post (FPTP): Returns $x$ iff $b(x) > b(y)$ for all $y \neq x$. Selects the output with maximal frequency.
• Greatest Lower Bound (GLB): On a partially ordered VALUE, returns the maximal lower bound shared by all elements in $b$.
• Flat-domain Least Upper Bound (PLUBF): With a designated error value $\omega$, returns $x$ iff all elements equal $x$ or $\omega$.
• Median (M): For totally ordered VALUE, $x$ is the median if both cumulative counts below and above meet the majority threshold.
• Weighted Variants: Replace counts by scores, e.g., $b_w(x) = \sum_{i: S_i = x} w_i$ with MV generalized to $2b_w(x) > \sum_y b_w(y)$.

These operators support dependability both in value diversity and in the presence of possibly faulty or unreliable sources (Boiten, 2015).

2. Axiomatic and Algebraic Properties

Solver-centric adjudication is characterized by strict algebraic properties, ensuring robust aggregation and preventing non-determinism:

• Idempotence: $b \oplus b = b$ (for operator-induced bag-union).
• Commutativity and Associativity: Ordering of bag-union does not affect outcome.
• Unanimity: Consensus returns the unanimous value, i.e., if $b(x) = \mathrm{size}(b)$ then $\mathrm{Adj}(b) = x$.
• Majority Property: If a majority exists, it is selected.
• Permutation-Invariance: Operators act solely on counts, not ordering.
• Consistency: Operator outputs a unique answer for each input bag (partial function semantics).
• Choice Property: For certain operators, adjudication always returns an element present in the bag (holds for FPTP and some totalized MV).

Such algebraic foundations facilitate calculational synthesis of fusion rules tailored to application requirements, supporting partiality (undefined in tie scenarios), error detection/totalization, and resilience (Boiten, 2015).

3. Solver-Centric Adjudication: Methods and Realizations

The solver-centric methodology underpins diverse domains:

• Symbolic Model Checking and Legal Reasoning: SMT-based frameworks formalize argumentation, capture both adversarial narratives, and use unsat-core analysis for iterative self-critique and consistency enforcement, as in the L4M pipeline (Chen et al., 26 Nov 2025). Given fact assertions $F$, statutes $S$, and meta-constraints $M$, the system forms $$\Phi = \bigwedge F \wedge \bigwedge S \wedge \bigwedge M$$, checks satisfiability, and if unsat, uses core extraction and targeted revisions until a coherent (satisfiable) assignment is found.
• Counterfactual-Guided Logic Exploration: Judgment and accountability processes such as trials or review boards are recast as CLEAR loops—interactive queries combining symbolic execution and QF_FPBV-SMT solving to validate factual and counterfactual hypotheses over agent decision traces. Each query builds a formula $\Pi$ encapsulating agent logic under (counter)factual constraints and discharges yes/no questions via SMT isValid/isSat checks. Facts are refined through human-guided selective hypothesis and counterexample loops (Judson et al., 2023).
• Administrative Adjudication with Default Negation: Systems like s(LAW) capture rule, exception, ambiguity, and discretion using logic-programming under stable-model semantics, producing justification forests and supporting scenario-driven what-if reasoning with strong formal semantics (Arias et al., 25 Jan 2024).
• Aggregate Human Annotation (e.g., Coreference): Answer Set Programming encodings express structural constraints (e.g., non-singleton, no nested chains) and minimize divergence via parameterized cost functions, using solver-based optimization (Gringo/Clasp) to ensure output uniqueness and constraint satisfaction (Schüller, 2018).
• Bias- and Consistency-Oriented Arbitration: Hybrid legal-AI architectures (such as SAAP) orchestrate multi-level solver-centric adjudication, combining statistical analysis, pattern mining, and rule-based arbitration to optimize fairness and alignment with procedural norms across jurisdictions (De'Shazer, 6 Feb 2024).

4. Dependability, Fault Models, and Reliability Amplification

Solver-centric adjudication is tightly coupled to dependability theory:

• Byzantine and Weighted Fault Masking: For up to $f < n/2$ faulty solvers in $n$-member ensembles, MV ensures correctness; in weighted settings, trust can be allocated to maintain fault-tolerance as long as faulty weight $< 1/2$ (Boiten, 2015).
• Error Filtering: PLUBF and related error-aware schemes enable the masking or flagging of singleton failures, with double-failure detection heuristics. These are essential when the cost of undetected faults exceeds that of raising alarms.
• Statistical Reliability Amplification: Analytical bounds demonstrate that, for independent failure rate $p < 1/2$, the probability $P_{>n/2}$ that the majority is wrong decays exponentially in $n$, providing arbitrarily strong assurance by scaling solver ensemble size (Boiten, 2015).

These properties are central to rigorous assurance in autonomous systems, safety-critical applications, and legal/administrative adjudication.

5. Calculational Synthesis and Algorithmic Frameworks

The design of solver-centric adjudication operators proceeds via an explicit calculational template:

1. Axiom Selection: Identify non-negotiable properties (e.g., unanimity, totality).
2. Relational Specification: Abstract as $$\mathrm{Adj} : \mathrm{BAG} \leftrightarrow \mathrm{VALUE}$$.
3. Axiom Intersection: Refine the relation by imposing properties as set-theoretic conditions (e.g., enforce majority, permutation invariance).
4. Partial-to-Total Extension: Address under-specification (tie-breaking) by introducing choice or error totalization.
5. Closure/Proof: Demonstrate equivalence between specification and explicit PFN implementation; e.g., majority vote function is intersection of majority and valid-value sets (Boiten, 2015).

Algorithmically, this extends to specialized loops:

• Unsat-Core Loop (Autoformalization/Iterative Self-Critique): Invoke an SMT/ASP solver, extract minimal conflicting subsets on unsat, localize responsibility, trigger targeted revision, and iterate to convergence (Chen et al., 26 Nov 2025).
• Interactive Counterfactual Querying: Alternate between factual and “would/might” counterfactual queries by symbolic execution, SMT encoding, and model validation, building an ever-growing fact base explicable in legal or policy terms (Judson et al., 2023).

6. Practical Applications, Benchmark Results, and Comparative Impact

The solver-centric approach is broadly deployed:

• Legal AI and Administrative Law: L4M achieves a valid sentencing output ratio of 94.12%, outperforming LLM-based and hybrid AI alternatives by margins of 3–5 percentage points and reducing sentencing error by several months compared to leading LLM baselines (Chen et al., 26 Nov 2025). s(LAW) delivers human-readable justifications, with execution times an order of magnitude lower than bottom-up systems as $N$ grows, scaling linearly for 100+ cases (Arias et al., 25 Jan 2024).
• Algorithmic Accountability: CLEAR-based adjudication classifies algorithmic behaviors (e.g., driving agents) as standard, reckless, or pathological by analyzing SMT oracle answers to factual and systematically generated counterfactuals. Query times remain practical for real-world scenarios (Judson et al., 2023).
• Semi-Automated Arbitration: SAAP achieves a 25% reduction in logical inconsistency and a 30% improvement in alignment across legal systems, explicitly quantifying bias and cross-jurisdictional drift (De'Shazer, 6 Feb 2024).
• Data Adjudication in NLP: CaspR, an ASP-driven coreference adjudication tool, precisely manages hard and soft constraints, balancing solution optimality and resource use via architectural and solver choices (Schüller, 2018).

Solver-centric adjudication thus operationalizes a principled, performance-validated, and extensible model for aggregating, verifying, and justifying decisions from ensembles of computational agents under both logical and statistical uncertainty.