Credulous Consequence in Default Theories

• Credulous consequence in default theories is a framework where a proposition is accepted if it appears in at least one acceptable extension of a knowledge base.
• The approach employs default rules and controlled sequent calculus to formalize defeasible inferences and accommodate non-monotonic reasoning.
• Complexity results range from Σ₂^P-complete to tractable fragments, guiding practical algorithm design for automated default reasoning.

Credulous consequence in default theories is a central notion in nonmonotonic reasoning, capturing the idea that a proposition is supported if it holds in at least one acceptable extension of a knowledge base equipped with defaults. In this framework, default rules describe typical, but defeasible, inferences. The credulous (or "brave") consequence relation embodies non-monotonicity and non-determinism characteristic of default logic. This article systematically presents the formal definitions, semantic and proof-theoretic aspects, algorithmic realizations, computational complexity, and selected applications, referencing foundational and recent research.

1. Formal Definitions and Semantics

A default theory is a pair $\Delta = \langle W, D \rangle$, where $W$ is a finite set of propositional formulas (the "hard" facts), and $D$ a finite set of default rules. Each default $d$ has the form:

$d = \frac{\alpha: \beta}{\gamma}$

where $\alpha$ is the prerequisite, $\beta$ the justification, and $\gamma$ the consequent.

Reiter extensions of $\Delta$ model possible sets of accepted beliefs under the action of defaults:

• An extension $E$ satisfies $W \subseteq E$ (contains all facts), is deductively closed ($E=\mathrm{Th}(E)$), and for each default $d$ with $\alpha\in E$ and $\neg\beta\notin E$, it must include $\gamma$.

A formula $\varphi$ is a credulous consequence of $\Delta$, denoted $\Delta \vdash_c \varphi$ (or $\Delta \models_{\text{cred}} \varphi$), if $\varphi$ is contained in at least one extension of $\Delta$:

$$\Delta \vdash_c \varphi \;\; \Leftrightarrow \;\; \exists E \in \mathrm{Ext}(\Delta): \varphi \in E$$

This captures non-monotonicity: new information may eliminate the extensions supporting $\varphi$ (0808.3884, Piazza et al., 21 Dec 2025).

2. Proof-Theoretic Methods: Controlled Sequents

Proof systems for credulous default consequence extend ordinary Gentzen-style sequents by annotating them with control sets, tracking the prerequisites and applicability conditions of defaults. In the framework of controlled sequents, a sequent is enriched as:

$$\langle \Gamma \Rightarrow \Delta; C^+, C^- \rangle$$

• $C^+$ (positive controls): sets whose conjunctions must be derivable from $\Gamma$ for certain defaults to be applicable.
• $C^-$ (negative controls): sets whose disjunctions must not be derivable from $\Gamma$ for justifications to remain unblocked.

Soundness and completeness of this calculus with respect to credulous consequence are established: A formula $\varphi$ is credulously entailed, $\Delta \models_{\text{cred}} \varphi$, if and only if there exists a controlled, cut-free $G4^\delta$-proof of $\langle W \Rightarrow \varphi; C^+, C^-\rangle$ with all sub-sequents satisfying the compatibility conditions. The proof system directly encodes the extension construction and justification checks, tightly linking proof search to the existence of supporting extensions (Piazza et al., 21 Dec 2025).

Bird–Penguin Example

Given:

• $W=\{\mathrm{bird}(\mathrm{Tweety})\}$,
• $$D=\{\mathrm{d}_1: \mathrm{bird}(x):\neg\mathrm{penguin}(x) \Rightarrow \mathrm{fly}(x); \ \mathrm{d}_2: \mathrm{penguin}(x):\Rightarrow \neg\mathrm{fly}(x)\}$$,

there are two extensions: one supports $\mathrm{fly}(\mathrm{Tweety})$ and another supports $\neg\mathrm{fly}(\mathrm{Tweety})$. Thus, both are credulous consequences (Piazza et al., 21 Dec 2025).

3. Computational Complexity and Post’s Lattice Classification

The decision problem for credulous consequence—whether $\Delta \vdash_c \varphi$—is computationally intractable in general. For unrestricted propositional default logic (functionally complete connectives), the problem is $\Sigma_2^P$-complete, placing it at the second level of the polynomial hierarchy (0808.3884).

Beyersdorff et al. (2009) refine the complexity landscape by classifying the problem under restrictions on the allowed Boolean connectives (cf. Post's lattice). Table 1 summarizes the complexity jumps as the base set $B$ of connectives varies:

Clone $[B]$	Base Connectives (example)	Complexity of $\CRED(B)$
$\CloneS_{1}\dots\CloneBF$	$\{\land,\neg\}$	$\Sigma_2^P$-complete
$\CloneS_{11}\dots\CloneM$	monotone Horn-like	$\Delta_2^P$-complete
$X\subseteq[B]\subseteq \CloneR_1$	$S_{00},S_{10},D_2$	coNP-complete
$\CloneN,\CloneN_2,\ldots$	one-var./affine	NP-complete
$\CloneV_2\dots\CloneV$,$[B]\in\{\CloneL_1,\CloneL_2\}$	OR-like, parity+TRUE	P-complete
$\CloneI_2\dots\CloneI$	projections only	NL-complete

The complexity drops to P or NL when only Horn, dual-Horn, or projection connectives are used. In these cases, extension construction and checking are reducible to tractable problems—iterative propagation, graph reachability, or solving systems of equations (0808.3884).

4. Argumentation-Theoretic Characterization

A transformation exists from any propositional default theory $\Delta$ to an assumption-based argumentation system $A = (L, A, R, \bar{\cdot})$. For each default, fresh "assumption atoms" represent its precondition, justification, and conclusion. The strict rule base includes implications reflecting the semantics of defaults and the contrary mapping encodes attack relations between assumptions.

Key correspondence: Reiter extensions of $\Delta$ correspond bijectively to stable extensions of $A$. The credulous consequence $\Delta \vdash_c \varphi$ holds if and only if $A$ has a stable extension supporting $\varphi$. The transformation is linear in theory size, although credulous entailment checking is exponential in the worst case, with decision procedures requiring enumeration of candidate sets of assumptions and propositional-derivability checks [0207065].

5. Algorithmic Aspects and Practical Considerations

The embedding into argumentation and the controlled sequent calculus both provide algorithms for credulous consequence. For general propositional default logic, the worst-case complexity is $\Sigma_2^P$-complete. However, tractable strategies exist for fragments:

• NL-complete fragments: Reduction to reachability in directed graphs, applicable when formulas are restricted to projections (0808.3884).
• P-complete fragments: Iterative consequence propagation in Horn or dual-Horn settings, or solving affine equations for certain clones (0808.3884).
• SAT-based approaches: For harder fragments, decision procedures involve SAT-solving and careful extension enumeration.

Modern default-logic engines can exploit these complexity results by analyzing the Post clone of input rule sets and selecting dedicated reasoning routines accordingly (0808.3884). An example pseudocode for credulous consequence decision, based on the argumentation embedding, is given in [0207065].

6. Theoretical and Practical Implications

Credulous reasoning in default theories offers a robust, expressive formalism for non-monotonic reasoning. The strong completeness of controlled sequent calculi with respect to credulous consequence ensures direct alignment of proof theory and semantics, facilitating both conceptual analysis and protocol implementation (Piazza et al., 21 Dec 2025). The complexity-theoretic landscape guides the design of efficient, application-driven systems by constraining knowledge base expressivity to tractable fragments when necessary (0808.3884).

A plausible implication is that, beyond pure default reasoning, these methods extend to normative and deontic systems, as shown in work leveraging controlled sequents for dynamic, context-sensitive, and even conflicting norms (Piazza et al., 21 Dec 2025). The embedding into argumentation further connects default logic to extensive work in assumption-based and abstract argumentation, supporting transfer of automated reasoning techniques [0207065].