Question Completeness in Formal Systems

• Question Completeness is a multifaceted concept that defines whether a formalism, analysis, dataset, or answer artifact contains enough structure to settle the questions posed.
• It is applied across various domains such as logic, static analysis, and query answering, distinguishing between global/local and binary/quantitative measures.
• Operational methods involve constructing deductive paths, verifying abstraction methods, and ensuring all semantically valid answers are derivable in practice.

Question completeness denotes whether a formalism, analysis, dataset, or answer artifact contains enough structure to settle the questions posed within a specified language, semantics, or use case. Across the cited literature, the notion is explicitly context-relative rather than uniform: in query answering it means that returned answers coincide with all answers that exist in reality; in logic it means that a formula or proof system decides every statement semantically fixed by it; in static analysis it means that an abstraction or algorithm finds the required invariant whenever one exists; and in attributed question answering it means that a gap-free deductive path connects the question to the final answer (Darari et al., 2016, Drabent, 2014, Achilleos, 2016, Constable et al., 2011, Monniaux, 2022, Yan et al., 21 Jan 2026). This suggests that question completeness is best understood as a family of domain-indexed completeness guarantees.

1. Core structure of completeness claims

A recurrent starting point is the distinction between soundness and completeness. In the logical formulation recalled by Monniaux, a proof system is sound with respect to a semantics if it never proves a formula that is false in that semantics, and complete if it can prove every formula that is true in that semantics; a decision procedure is both sound and complete (Monniaux, 2022). The same two notions reappear in static analysis, query answering, modal logic, and proof calculi, but the relevant objects vary: domains, algorithms, formulas, queries, and models.

A second recurrent distinction is between global and local completeness. Monniaux separates completeness of an abstraction/domain from completeness of the analysis procedure that searches for invariants (Monniaux, 2022). Drabent distinguishes completeness of a logic program with respect to a specification from completeness for a query $Q$, where every ground instance required by the specification must be produced as an answer (Drabent, 2014). Achilleos studies completeness of a single modal formula, not of the ambient proof system, by asking whether that formula decides every formula over its own propositional vocabulary (Achilleos, 2016).

A third distinction concerns binary versus quantitative notions. In scenario-based testing for automated driving, completeness of a scenario concept is binary—“a scenario concept is sufficiently complete for a use case if all relevant driving situations are adequately captured”—whereas coverage is “the quantifiable extent to which a set of scenarios or parameters represent a defined ODD or predefined set of scenarios” (Glasmacher et al., 2024). LogicScore adopts the same binary pattern for long-form QA: completeness is $1$ exactly when a minimal sufficient reasoning path exists, and $0$ otherwise (Yan et al., 21 Jan 2026). This suggests that completeness often marks a threshold property, while related notions such as coverage, conciseness, or degree of answer loss provide graded refinements.

2. Proof-theoretic and semantic completeness

In modal logic, a formula $\varphi$ is complete for a logic $l$ when for every $\psi \in L(P(\varphi))$, either $\vdash_l \varphi \rightarrow \psi$ or $\vdash_l \varphi \rightarrow \neg \psi$ (Achilleos, 2016). Model-theoretically, this is equivalent to saying that all pointed $l$-models satisfying $\varphi$ are bisimilar modulo $1$0. The associated decision problem has the same complexity as validity, except for trivial cases: for $1$1, $1$2, $1$3, and $1$4 it is PSPACE-complete, while for $1$5, $1$6, and more generally $1$7, it is coNP-complete; for $1$8 and $1$9 with nonempty propositional vocabulary, satisfiable complete formulas do not exist in general (Achilleos, 2016).

For intuitionistic first-order logic, the relevant semantic notion is uniform validity under BHK semantics. The main theorem for minimal logic states that for any $0$0,

$0$1

and Friedman’s $0$2-transformation lifts this to intuitionistic first-order logic: $0$3 The procedure $0$4 converts a uniform evidence term into a formal proof, and the strongest termination argument uses the Fan Theorem (Constable et al., 2011). Here question completeness is not ordinary validity over all models, but the existence of a single evidence term inhabiting the intersection $0$5.

Graphical calculi use the same syntax–semantics schema. Backens shows that the ZX-calculus is complete for pure state stabilizer quantum mechanics and for the single-qubit Clifford+T group (Backens, 2016). Jeandel, Perdrix, and Vilmart strengthen this line by proving completeness for the Clifford+T fragment, for linear diagrams with Clifford+T constants, and—after adding axiom (A)—for the unrestricted ZX-calculus as a whole (Jeandel et al., 2019). In these settings, completeness means that every semantically valid equality between diagrams is derivable graphically. A plausible implication is that question completeness in proof calculi is the strongest possible form of deductive adequacy: no semantically valid question about equality remains outside the rewrite system.

3. Query-answer completeness in data systems

In incomplete database theory, question completeness is formulated directly as equality between answers over available and ideal data. An incomplete relational database is a pair $0$6 with $0$7, and a query completeness statement $0$8 is satisfied exactly when

$0$9

under the chosen semantics (Razniewski, 2014). The thesis develops metadata-based reasoning from table completeness statements $\varphi$0 to query completeness, reduces key entailment problems to query containment, extends the framework to null values, RDF, spatial data, and process models, and shows that restricted compactness and design-time/runtime verification can be obtained in process settings (Razniewski, 2014).

For RDF under the open-world assumption, completeness is handled via explicit completeness statements $\varphi$1 and entailment $\varphi$2, meaning that every admissible completion $\varphi$3 of the graph that respects $\varphi$4 yields the same answers for $\varphi$5 as $\varphi$6 itself (Darari et al., 2016). The paper defines a transfer operator $\varphi$7, shows that the general completeness-entailment problem is $\varphi$8-complete, and identifies a practically important fragment of SP-statements of the form $\varphi$9 that supports scalable indexing on Wikidata-scale graphs (Darari et al., 2016). Here question completeness is local and entity-centric: one does not close the world globally, but only on the subject–predicate regions explicitly declared complete.

Logic programming gives an especially direct query-level formulation. Drabent defines completeness for a query $l$0 with respect to a specification $l$1 by requiring that for every ground instance $l$2,

$l$3

Coverage, semi-completeness, recurrence, and acceptability provide sufficient conditions, while pruning via cut is analyzed using csSLD-trees and adjustable coverage (Drabent, 2014). In this formulation, question completeness means that the program produces all answers required by the specification for the question actually asked.

Ontology-based data access sharpens the same idea against incomplete reasoners. Glimm, Horrocks, Lutz, and Sattler define a reasoner to be $l$4-complete if, for every ABox $l$5, it detects unsatisfiability whenever $l$6 is unsatisfiable, and otherwise returns all certain answers to $l$7 with respect to $l$8 and $l$9 (Grau et al., 2014). Their framework introduces abstract reasoners, monotonicity and faithfulness conditions, and finite test suites exhaustive for $\psi \in L(P(\varphi))$0-completeness under suitable assumptions. For UCQ-rewritable cases, full or injective instantiations of rewritings yield finite certification suites; for more general recursive settings, first-order reproducible reasoners and datalog$\psi \in L(P(\varphi))$1 rewritings recover positive results, although strong impossibility theorems show that no finite suite can work in full generality (Grau et al., 2014). This is one of the clearest operational readings of question completeness: a reasoner may be incomplete in principle yet complete for the specific questions and ontology used by an application.

4. Operational completeness under approximation

In abstract interpretation, Monniaux distinguishes several levels of completeness: exactness of the abstraction for a specific problem, completeness of the analysis method with respect to a chosen abstract domain, and decidability of the invariant-existence problem for that domain (Monniaux, 2022). A method is complete with respect to its domain when it “would compute always such invariants if they exist.” For finite-height domains this can coincide with ordinary ascending fixpoint iteration; for richer settings, exact solving via quantifier elimination or policy iteration may recover method-level completeness. At the same time, widening is identified as a major source of incompleteness: in the circular-array interval example the ideal interval invariant $\psi \in L(P(\varphi))$2 exists in the domain, yet widening plus narrowing yields $\psi \in L(P(\varphi))$3, which does not prove the assertion (Monniaux, 2022).

Monniaux also exhibits problem-specific completeness. In simplified LRU cache analysis, a chain of exact abstractions—addresses only, then per-set decomposition, then antichain abstraction for “block $\psi \in L(P(\varphi))$4 is present and the set of blocks younger than $\psi \in L(P(\varphi))$5 is $\psi \in L(P(\varphi))$6”—produces a complete/exact analysis of cache behavior with respect to the control-flow-only program model (Monniaux, 2022). This suggests that question completeness in analysis is often modular: exactness can be achieved for a subsystem even when global completeness for full program semantics is abandoned.

Scenario-based testing for automated vehicles frames the same issue in an open context. A scenario concept is sufficiently complete for a use case if all relevant driving situations are adequately captured, whereas coverage is the quantifiable extent to which scenarios or parameters represent the ODD or a predefined scenario set (Glasmacher et al., 2024). The methodology uses Goal Structuring Notation, counter-hypotheses, knowledge-based evidence, and data-driven evidence. In the inD case study, rules derived from the concept detected 59,253 base scenarios, and “there is no second in the recordings where no base scenario is assigned” at the chosen abstraction level (Glasmacher et al., 2024). The paper therefore argues not for absolute completeness of an open traffic world, but for sufficient completeness relative to a use case, an ODD, and an abstraction boundary.

5. Completeness in attributed question answering

LogicScore operationalizes question completeness for long-form attributed QA as a property of the reasoning chain rather than of isolated factual statements. Given a question $\psi \in L(P(\varphi))$7, retrieved documents $\psi \in L(P(\varphi))$8, a long-form answer $\psi \in L(P(\varphi))$9, and a short answer $\vdash_l \varphi \rightarrow \psi$0, the framework transforms $\vdash_l \varphi \rightarrow \psi$1 into atomic propositions $\vdash_l \varphi \rightarrow \psi$2 and then applies backward verification from the gold answer entity $\vdash_l \varphi \rightarrow \psi$3 to the question entity $\vdash_l \varphi \rightarrow \psi$4 (Yan et al., 21 Jan 2026). If a connected chain $\vdash_l \varphi \rightarrow \psi$5 exists, it is the minimal sufficient set $\vdash_l \varphi \rightarrow \psi$6; otherwise $\vdash_l \varphi \rightarrow \psi$7. Completeness is then defined by

$\vdash_l \varphi \rightarrow \psi$8

This formulation is explicitly global. It is not enough that individual sentences be supported by citations; the propositions must form a gap-free deductive path from question to answer. Conciseness and Determinateness are then layered on top: $\vdash_l \varphi \rightarrow \psi$9 The empirical results expose a large attribution–reasoning gap. Over more than 20 LLMs and three multi-hop datasets, leading systems can achieve high attribution precision while remaining weak on global logic; the paper reports 92.85\% precision for Gemini-3 Pro but only 35.11\% Conciseness (Yan et al., 21 Jan 2026). Completeness on MusiQue remains around 60% even for top proprietary models, with Gemini-3-Pro at 59.21, GPT-5.1 at 60.11, GPT-o3 at 60.83, and Claude-4.5 at 63.88 (Yan et al., 21 Jan 2026). Human evaluation yields a Jaccard similarity of about 94.39\% between LogicScore completeness labels and human labels (Yan et al., 21 Jan 2026). In this setting, question completeness is a binary structural property of reasoning continuity, not a synonym for citation correctness.

6. Open theories, causal models, and the limits of completeness

In the philosophy of physics, the paper on quantum theory defines completeness of a physical theory through the existence of a formal (continuous) causal model whose laws are complete, consistent, and reality conformal (Diel, 2015). A formal causal model is a collection of conditional state-transition rules

$\vdash_l \varphi \rightarrow \neg \psi$0

possibly using $\vdash_l \varphi \rightarrow \neg \psi$1 for nondeterministic transitions. A physical theory is complete if such a model can be constructed for the theory (Diel, 2015).

Within that framework, “question completeness” becomes the demand that every physically meaningful question about state evolution, including measurement, be answerable by the model using only theory-internal predicates and operations. The paper argues that standard quantum theory is not complete in this sense because the measurement problem, the interference-collapse rule, and the lack of a fully specified causal account of interactions prevent construction of a complete formal causal model (Diel, 2015). Questions such as exactly when a measurement occurs or exactly when interference is lost cannot be mapped to the $\vdash_l \varphi \rightarrow \neg \psi$2 schema without extra-theoretical notions such as “observer” or “capable of determining” (Diel, 2015).

A plausible implication, reinforced by the static-analysis and scenario-testing literature, is that question completeness is frequently bounded by undecidability, open-world assumptions, or irreducible modeling choices. Monniaux notes undecidability of invariant existence for sufficiently rich abstract domains and leaves the convex-polyhedral case as an open problem (Monniaux, 2022). Scenario-based testing replaces absolute completeness by sufficiently complete concepts relative to an ODD (Glasmacher et al., 2024). Quantum theory, in the analyzed formulation, lacks a fully internal causal account of some of its own central questions (Diel, 2015). Across these cases, completeness remains a target of formalization, but not always an attainable global property.