Superficial Self-Locating Uncertainty (SSL)

• SSL is a concept in epistemic logic that handles agents' uncertainty about their own position by using modular, layer-based reasoning.
• It utilizes epistemic splitting to separate programs into bottom and top layers, ensuring that subjective literals do not introduce artificial uncertainty.
• Key theorems validate SSL by demonstrating soundness, uniqueness in stratified programs, and constraint monotonicity, enhancing ASP workflow reliability.

Superficial Self-Locating Uncertainty (SSL) arises in the study of epistemic logic programs, where reasoning involves both objective and subjective literals, and agents must manage uncertainty about their own position in the logical space of possible worlds. The concept is rooted in the modularity properties of epistemic logic programs—specifically, the ability to decompose reasoning into well-defined layers without introducing self-supporting or spurious uncertainty. The Epistemic Separability Principle (ESP), also referred to as “epistemic splitting,” provides a formal criterion for when such separation is justified, and it has significant implications for the proper handling of self-locating uncertainty in this setting. Epistemic splitting characterizes when world views of a program can be constructed by solving lower layers (bottoms) whose atoms are referenced by upper layers (tops) only via subjective modalities such as $K$ (“known”) or $M$ (“may be”). Semantics respecting this property properly manage SSL by isolating genuine epistemic uncertainty from artifacts of the reasoning formalism (Cabalar et al., 2018).

1. Formal Definition of Epistemic Splitting

Epistemic splitting extends the classical splitting property of logic programming to epistemic logic programs, recognizing the role of subjective literals in expressing agent-centric uncertainty.

Given an epistemic program $\Pi$ over atoms $\mathcal{A}$ and a set $U \subseteq \mathcal{A}$, $U$ is an epistemic splitting set if every rule $r \in \Pi$ satisfies one of:

• All atoms of $r$ are in $U$, or
• No head-atom of $r$ is in $M$0 and every reference to atoms from $M$1 in the body appears only within subjective contexts.

The program is then partitioned into bottom $M$2 (rules whose atoms are in $M$3) and top $M$4 (remaining rules), with the body of the top only referring to $M$5 through subjective literals. For a candidate world view $M$6, the subjective reduct $M$7 is constructed by evaluating all subjective literals in $M$8 involving atoms from $M$9 based on whether $\Pi$0 satisfies them, replacing with $\Pi$1 or $\Pi$2 accordingly.

A semantics $\Pi$3 satisfies epistemic splitting if, for every epistemically splittable $\Pi$4 with splitting set $\Pi$5, the $\Pi$6-world views of $\Pi$7 are precisely the disjoint unions $\Pi$8 for some $\Pi$9-world views $\mathcal{A}$0 (of $\mathcal{A}$1) and $\mathcal{A}$2 (of $\mathcal{A}$3 (Cabalar et al., 2018).

2. Key Theorems

Three principal properties arise from epistemic splitting:

1. Soundness (Thm 5.1): The Gelfond 1991 semantics (G91) satisfies epistemic splitting, ensuring that world views can always be constructed via bottom-to-top evaluation when a splitting set exists.
2. Uniqueness in Stratified Programs (Thm 4.5): For any supra-ASP and epistemic splitting semantics, stratified programs (where dependencies only go “downward” in layers) have at most one world view.
3. Constraint Monotonicity (Thm 4.6): For subjective constraints (constraints whose body consists solely of subjective literals), any semantics satisfying epistemic splitting ensures that adding such a constraint only filters out world views violating it, preserving modular reasoning.

These theorems formalize the absence of “artificial” self-locating uncertainty in acyclic epistemic programs under splitting-respecting semantics.

3. Detailed Example: The “College Scholarship” Case

A paradigmatic example illustrates the practical implications of epistemic splitting and the management of SSL:

A college eligibility program for a student $\mathcal{A}$4 is specified with incomplete information: $\mathcal{A}$5

A subjective rule for interview: $\mathcal{A}$6

Defining $\mathcal{A}$7 as the bottom, the program splits accordingly.

Steps:

1. The world view of the bottom $\mathcal{A}$8 yields two models: $\mathcal{A}$9 and $U \subseteq \mathcal{A}$0.
2. The subjective reduct of the top, based on $U \subseteq \mathcal{A}$1, becomes a simple program with a unique world view containing $U \subseteq \mathcal{A}$2.
3. The final world view $U \subseteq \mathcal{A}$3 includes all epistemically justifiable assignments, matching stratified expectations (Cabalar et al., 2018).

This demonstrates how splitting semantics preclude spurious or superficial self-locating uncertainty in epistemic logic programs.

4. Consequences for ASP Workflows and Conformant Planning

In answer set programming (ASP), the “generate–define–test” workflow is a standard pattern. Epistemic splitting allows this methodology to be lifted to the epistemic setting:

• Generation: The bottom encodes nondeterministic choices (e.g., $U \subseteq \mathcal{A}$4 for actions).
• Definition: The domain’s laws are encoded in the middle layer.
• Testing: Subjective constraints (e.g., $U \subseteq \mathcal{A}$5) encode goals or forbidden states.

Splitting on the choice atoms allows sequential resolution—first generating candidate world views, then propagating through definitions, and finally filtering with constraints. This modular approach is only possible when SSL—arising from violations of splitting—does not contaminate world view construction (Cabalar et al., 2018).

5. Counterexamples: Failure Modes and the Origin of SSL

Many recent epistemic ASP semantics violate epistemic splitting, leading to pathological behaviors evocative of SSL—where a reasoning agent’s “self-location” in the program’s world view structure becomes ill-defined or artificially constrained. Representative counterexamples include:

• Gelfond 2011 (G11): For $U \subseteq \mathcal{A}$6, G11 semantics admit a world view $U \subseteq \mathcal{A}$7 when splitting would rule out all world views.
• Kahl et al. 2015 (K15), Shen & Eiter 2017 (S17): These semantics inherit failures from G11 for programs with positive $U \subseteq \mathcal{A}$8-literals.
• Farinas del Cerro & Herzig 2015 (F15): For $U \subseteq \mathcal{A}$9, F15 inappropriately allows $U$0 as a world view.

The source of SSL in these cases is the inability to modularly construct world views from bottom to top, as demanded by epistemic splitting. Consequently, such semantics may allow world views to persist or arise solely due to improper entanglement of subjective and objective reasoning layers (Cabalar et al., 2018).

6. Comparison of Semantics: Which Respect Epistemic Splitting

A survey of prominent epistemic ASP semantics, summarized below, reveals that only G91 (Gelfond 1991) and its “founded” refinement C19 (Cabalar & Fandinno 2019) fully satisfy epistemic splitting and its associated properties (subjective constraint monotonicity, supra-ASP behavior), thus robustly precluding SSL.

Semantics	Epistemic Splitting	Subjective Constraint Monotonicity	Foundedness
G91 (Gelfond 1991)	Yes	Yes	No
C19 (Cabalar & Fandinno 2019)	Yes	Yes	Yes
G11 (Gelfond 2011)	No	Yes	No
K15, S17	No	No	No
F15	No	No	No

Failure to satisfy splitting is directly correlated with the emergence of superficial, ill-founded self-locating uncertainty in the epistemic reasoning process (Cabalar et al., 2018).

7. The Role of Epistemic Splitting in Managing SSL

Epistemic splitting is identified as a minimal and essential criterion to guarantee modular, stratified evaluation, ensuring the absence of SSL and the correct treatment of subjective constraints. Under splitting-respecting semantics, stratified uses of epistemic modalities ($U$1) act precisely as a querying layer atop a standard ASP base. Subjective constraints act monotonically, preserving world views except where explicitly ruled out. Attempts to address self-support in cycles by other means have generally violated splitting, introducing SSL even in non-cyclic (acyclic) programs. Only G91 and its founded extension C19 retain the desired separability and thus manage self-locating uncertainty in an epistemically transparent fashion (Cabalar et al., 2018).