The Observer-Situation Lattice: A Unified Formal Basis for Perspective-Aware Cognition

Saad Alqithami

cs.AIMar 2, 2026

The Observer-Situation Lattice: A Unified Formal Basis for Perspective-Aware Cognition

Authors: Saad Alqithami

Organizations: Computer Science Department, Al-Baha University, Albaha 65779, Saudi Arabia

Abstract

Autonomous agents operating in complex, multi-agent environments must reason about what is true from multiple perspectives. Existing approaches often struggle to integrate the reasoning of different agents, at different times, and in different contexts, typically handling these dimensions in separate, specialized modules. This fragmentation leads to a brittle and incomplete reasoning process, particularly when agents must understand the beliefs of others (Theory of Mind). We introduce the Observer-Situation Lattice (OSL), a unified mathematical structure that provides a single, coherent semantic space for perspective-aware cognition. OSL is a finite complete lattice where each element represents a unique observer-situation pair, allowing for a principled and scalable approach to belief management. We present two key algorithms that operate on this lattice: (i) Relativized Belief Propagation, an incremental update algorithm that efficiently propagates new information, and (ii) Minimal Contradiction Decomposition, a graph-based procedure that identifies and isolates contradiction components. We prove the theoretical soundness of our framework and demonstrate its practical utility through a series of benchmarks, including classic Theory of Mind tasks and a comparison with established paradigms such as assumption-based truth maintenance systems. Our results show that OSL provides a computationally efficient and expressive foundation for building robust, perspective-aware autonomous agents.

Figures & tables

Panel of Figure 1 — Figure 1. OSL-based agent architecture. Solid arrows indicate primary data/control flow, dashed arrows show feedback loops. All modules communicate exclusively via the OSL or the Global Workspace broadcast mechanism.

Panel of Figure 2 — Figure 2. OSL cognitive cycle with nominal 100ms period. Solid arrows show the sense-plan-act pathway, dashed arrows indicate meta-cognitive shortcuts for adaptive behavior modification.

Figure 3. Runtime and memory comparison showing OSL’s competitive performance against established truth maintenance systems. OSL balances efficiency with perspective-aware reasoning capabilities.

System	Runtime (ms)	Mem. (MB)	Proc. Rate	Elem.	Conv.	Capabilities
OSL	6.87 ± 0.22	0.06	3494 ops/s	24	100%	Perspective-aware
ATMS	11.65 ± 0.81	0.36	2060 ops/s	154	100%	Assumption-based
DTMS	5.34 ± 0.37	0.09	4494 ops/s	154	100%	Dependency tracking
MEPK	67.72 ± 9.66	0.42	354 ops/s	120	100%	Probabilistic

Table 1. Performance comparison across baseline systems (24-element lattices, 5 trials).

Scalability Analysis					Ablation Study
Size	Runtime	Memory	Iter	Elem	Config	Runtime	Consist.	Cov.	Bel.
4	0.96ms	0.01MB	20	3.6	Full	7.60ms	-5.49	1.0	142
8	1.62ms	0.00MB	20	7.4	No MCC	5.06ms	-5.49	1.0	142
16	3.90ms	0.04MB	20	15.4	Limited RBP	4.39ms	-5.49	1.0	142
32	13.19ms	0.05MB	20	32.0	No Propagation	1.01ms	-6.89	0.0	72
64	34.82ms	0.17MB	19.4	64.0	Minimal	0.001ms	-6.89	0.0	72

Table 2. Scalability results and ablation study across configs.

Scenario	OSL result	Expected	Confidence
Sally–Anne (basic)	PASS	PASS	1.000
Sally–Anne with distractor	PASS	PASS	1.000
Nested belief (Level 2)	PASS	PASS	0.950
Multiple objects	PASS	PASS	1.000
Temporal belief change	PASS	PASS	0.975
False photograph	PASS	PASS	1.000

Table 3. Theory-of-mind test results across scenarios.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Lattice	Observers	Situations	Balanced	Avg Time	Std Dev	Affected	Max	Trials
Size				( $\mu$ s)	( $\mu$ s)	Elements	Affected
9	3	3	No	8.31	7.07	2.5 $\pm$ 2.3	8	30
16	4	4	No	9.79	7.84	3.5 $\pm$ 3.1	11	30
30	5	6	No	21.21	19.01	7.1 $\pm$ 6.9	23	30
56	7	8	No	13.91	13.57	4.8 $\pm$ 5.4	17	30
100	10	10	Yes	9.25	9.53	3.2 $\pm$ 3.7	19	30

Appendix

Table 4. RBP scaling performance results across all tested lattice sizes. “Avg Time” and “Std Dev” are computed over the

30

trials for each lattice configuration (units:

\mu

s). “Affected Elements” reports the mean

\pm

standard deviation of the number of lattice elements updated by RBP in a trial, and “Max Affected” reports the maximum updated count observed across the trials.

Test Scenario	OSL Result	Expected	Confidence
Sally-Anne Basic	PASS	PASS	1.000
Sally-Anne with Distractor	PASS	PASS	1.000
Nested Belief (Level 2)	PASS	PASS	0.950
Multiple Objects	PASS	PASS	1.000
Temporal Belief Change	PASS	PASS	0.975
False Photograph Task	PASS	PASS	1.000

Appendix

Table 5. Theory-of-mind test results across multiple scenarios. PASS indicates that the produced result matches the expected outcome for the corresponding scenario.

Panel of Figure 4 — Figure 4. Illustrative examples of observer–situation lattice structures. Nodes are labeled by $(o,\sigma)$ pairs; edges depict order relations.

Lattice	Observers	Situations	Balance	Balanced	Avg Time	Std Dev	Affected	Max	Trials
Size $n$	$\|O\|$	$\|\Sigma\|$	Factor $\kappa$		( $\mu$ s)	( $\mu$ s)	Elements	Affected
9	3	3	1.00	Yes	8.31	7.07	2.5 $\pm$ 2.3	8	30
16	4	4	1.00	Yes	9.79	7.84	3.5 $\pm$ 3.1	11	30
30	5	6	1.20	Yes	21.21	19.01	7.1 $\pm$ 6.9	23	30
56	7	8	1.14	Yes	13.91	13.57	4.8 $\pm$ 5.4	17	30
100	10	10	1.00	Yes	9.25	9.53	3.2 $\pm$ 3.7	19	30

Appendix

Table 6. Extended RBP scaling performance results with balance factors. All values are from measured runs of our implementation; the number of trials per configuration is reported in the Trials column.

Method	Avg Time ( $\mu$ s)	Memory (MB)	Empirical scaling (fit)	Max Size
OSL (Ours)	55.94 $\pm$ 94.57	12.3	$n^{0.336}$	100,000+
ATMS	847.2 $\pm$ 234.1	28.7	$n^{1.2}$	5,000
Distributed TMS	167.3 $\pm$ 89.4	49.6	$n^{0.8}$	10,000
MEPK Planner	2341.7 $\pm$ 1456.2	156.4	(rapid growth)	50 agents

Appendix

Table 7. Baseline comparison results on lattices up to

n\leq 2000

elements. “Empirical scaling” summarizes fitted behavior over the evaluated range (not a theoretical complexity bound).

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