Quantified Boolean Bayesian Networks

• QBBN is a probabilistic graphical model that integrates Boolean logic and statistical inference using factor graphs with deterministic and parametric gates.
• It employs message passing and loopy belief propagation to achieve reliable, tractable inference with convergence in under 20 iterations.
• The system uses deterministic slot grammar for zero-ambiguity parsing, enabling verifiable, explainable inference in natural language and formal semantics.

A Quantified Boolean Bayesian Network (QBBN) is a probabilistic graphical model designed to unify logical and statistical reasoning by representing Boolean-propositional and first-order inference as factor graphs with alternating deterministic and parametric gates. QBBNs provide principled support for quantification, logical conjunction/disjunction, negation, and exact or approximate tractable inference, enabling application to natural language inference, information retrieval, and formal semantics with verifiable and interpretable proof traces (Coppola, 12 Feb 2026, Coppola, 2024).

1. Graphical Structure and Formal Specification

At its foundation, a QBBN is a factor graph whose nodes represent Boolean random variables—propositions—and whose factors encode logical connectives and quantifiers. Consider Boolean proposition nodes $p_1,\dots,p_N \in \{0,1\}$ and a set of factors $F$ binding subsets of these nodes. The joint distribution is given by

$$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$

where $Z$ is the normalization constant and factor functions $\Psi_\alpha$ may be deterministic (AND/OR/NEG) or parametric (weighted Noisy-OR).

Variable and factor types include:

• Proposition nodes: Individual Boolean variables.
• Group nodes: Boolean variables representing conjunctions of premises.
• AND factor: Implements deterministic conjunction; $\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$ if $g = p_1 \wedge \cdots \wedge p_n$, else $0$.
• OR factor: Implements (noisy) disjunction; for $p$ supported by groups $g_1,...,g_k$ with weights $F$0:

$F$1

• NEG factor: Couples $F$2 and $F$3 to enforce $F$4.

Quantified rules (e.g., universally quantified Horn clauses) are grounded over all compatible entities, with modal weights for rules mapped to disjunctive factor strengths (e.g., always $F$5, usually $F$6). Quantifiers use parameter-sharing plates: a template $F$7 [premise $F$8 conclusion] gives rise to grounded instances over the entity domain (Coppola, 12 Feb 2026, Coppola, 2024).

2. Inference Procedures and Message Passing

Inference in QBBN is query-driven and bidirectional, constructing finite proposition graphs around the query by backward and forward chaining:

1. Start from query node, backward-expand to premises via AND nodes and register noisy-OR edges from group to query proposition.
2. Forward-expand: for any new proposition, add all rules where it is a premise and link conclusions.
3. Add NEG factors for contrapositive reasoning.
4. Clamp evidence nodes to observed truth.
5. Message-passing iteratively:
   • AND $F$9: premises $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$0 group
   • OR $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$1: group $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$2 conclusion
   • OR $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$3: conclusion $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$4 group
   • AND $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$5: group $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$6 premise
   • Belief update: $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$7
   • Damping: $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$8
   • Convergence: $$P(p_1, ..., p_N) = Z^{-1} \prod_{\alpha \in F} \Psi_\alpha(\cdots)$$9

NEG factors enforce logical consistency and support backward (contrapositive) reasoning: negative evidence on a conclusion propagates backward through OR and AND layers, suppressing the corresponding premise's probability (modus tollens). Loopy belief propagation (LBP) is adopted for tractable approximate inference, with empirical convergence in $Z$0 iterations and per-pass cost $Z$1 for $Z$2 nodes, maximum indegree $Z$3 (Coppola, 12 Feb 2026, Coppola, 2024).

3. Logical Semantics and Expressivity

QBBN semantics are defined by a typed logical language with three expressiveness tiers:

• Tier 1: First-order quantification (Horn clauses, negation).
  • Example: always $Z$4 sparrow(theme:$Z$5) $Z$6 bird(theme:$Z$7)
• Tier 2: Propositions as arguments (sentential type $Z$8).
  • Example: should(content: apologize(agent:$Z$9))
• Tier 3: Predicate quantification, $\Psi_\alpha$0-abstraction (second-order, not yet empirically tested).
  • Example: $\Psi_\alpha$1

Predicates have explicit role labels (e.g., man{theme:e}, trust{agent:e, patient:e}) and are grounded into the model for inference. Quantified rules use modalifiers (always, usually, sometimes, never) that directly map to probabilistic weights on OR factors, yielding fine-grained control over logical uncertainty (Coppola, 12 Feb 2026).

4. Syntactic Parsing and Compilation

QBBN incorporates a deterministic, type-driven slot grammar to map natural language input into logical form with zero ambiguity. Parsing proceeds as:

• Input is preprocessed and disambiguated by an LLM (lemmatization, sense disambiguation, type assignment).
• Typed slot grammar rules deterministically compile each sentence to a single logical form; ambiguity is eliminated on tested suites (33/33 accuracy).
• LLM-based reranking is used where multiple parse candidates exist, but direct dependency parsing by LLMs attains low accuracy (UAS 12.4%).

Sample patterns include:

• Fact: "Jack trusts Jill" $\Psi_\alpha$2 trust(agent:jack, patient:jill)
• Rule: "All men are mortal" $\Psi_\alpha$3 always[$\Psi_\alpha$4] man(theme:$\Psi_\alpha$5) $\Psi_\alpha$6 mortal(theme:$\Psi_\alpha$7) (Coppola, 12 Feb 2026).

5. System Architecture and End-to-End Pipeline

The system architecture supports hallucination-free, explainable natural language inference via the following pipeline:

$\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$8

Components:

1. LLM handles preprocessing (disambiguation, POS tagging).
2. Type-driven slot grammar deterministically parses text.
3. LLM reranks multiple parses (if needed).
4. QBBN grounds rules, constructs factor graph around the query, clamps evidence, and runs belief propagation to compute $\Psi_\alpha$8.

This architecture enables exact, explanation-bearing inference and supports integration with large pretrained LLMs, leveraging their annotation capabilities while reserving structured, logical deduction for the QBBN (Coppola, 12 Feb 2026).

6. Experimental Findings and Evaluations

QBBN systems demonstrate:

• 44/44 inference test cases solved (covering 22 distinct reasoning patterns: conjunction, disjunction, contrapositive, causation, conditionals, etc.).
• Convergence achieved within $\Psi_\alpha$9 iterations (typically 2–3).
• Deterministic slot grammar parsing evaluated at 33/33 with zero ambiguity.
• Negation and contrapositive reasoning operational via the NEG factor.
• LLM preprocessing and reranking achieve high POS tagging and PP-attachment accuracy (95%, GPT-4 zero-shot); full dependency parse (UAS) is low, underscoring grammar necessity.
• Example: Given "All men are mortal" and "Zeus is not mortal," a query on man(theme:zeus) triggers backward reasoning, yielding $\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$0 (contraposition).

Synthetic experiments confirm theoretical runtime: per-pass $\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$1 (tiny $\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$2), stable convergence in $\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$3 iterations on graphs of size $\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$4; empirical runtimes outperform MCMC and traditional Markov logic inference, while preserving logical soundness and traceability (Coppola, 12 Feb 2026, Coppola, 2024).

7. Logical Completeness and Theoretical Properties

QBBN’s AND/OR alternating bipartite architecture implements the forward and backward fragments of natural deduction (Prawitz, 1965), supporting implication-introduction, $\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$5/$\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$6-elimination, and contrapositive reasoning. Completeness is sketched by reduction to message-passing over bipartite graphs that mirror inference rules, guaranteeing that all valid entailments in first-order logic can be reproduced by the network. Consistency is enforced structurally: NEG factors guarantee $\Psi_\text{and}(g\mid p_1,\dots,p_n) = 1$7 at all times. This construction provides a formal bridge between standard logical deduction and statistical graphical models (Coppola, 12 Feb 2026, Coppola, 2024).