Probabilistic Epistemic Dynamic Agentive Logic

Abstract: I introduce PEDAL -- a probabilistic epistemic logic meant to capture, in propositional dynamic terms, the epistemic state of an agent engaged in checking whether a program meets its specification. Semantically, PEDAL is built `on top of' PDL and uses probability measures defined on the set of possible program valuations of an otherwise-specified PDL-model. A Hilbert system with one infinitary rule is provided and proved to be sound and complete. Near the end, I discuss possible ways to circumvent infinitary proof difficulties.

• The paper introduces PEDAL as a framework combining epistemic, probabilistic, and dynamic modalities to assess program correctness under uncertainty.
• It extends classical Propositional Dynamic Logic by externally imposing probability measures on finite epistemic partitions, capturing agents’ credences effectively.
• The framework derives nontrivial numerical bounds for compound program specifications, offering practical insights for risk assessment in software verification.

Probabilistic Epistemic Dynamic Agentive Logic: Formalization, Axiomatization, and Epistemic Semantics

Introduction

The paper "Probabilistic Epistemic Dynamic Agentive Logic" (2604.22042) introduces PEDAL, a probabilistic epistemic dynamic logic designed to model the epistemic states of agents reasoning about program correctness under uncertainty. Building on classical Propositional Dynamic Logic (PDL), PEDAL incorporates explicit probabilistic operators and an epistemic modal, capturing credences regarding specifications as well as the dynamic behaviors of programs within formal schemes. Unlike existing probabilistic dynamic logics, PEDAL imposes probability measures externally, reflecting agents' uncertainty about program behaviors rather than stochastic properties intrinsic to the program models.

Formal Development and Semantics

PEDAL’s foundation is PDL, which models state transitions and propositional assertions about programs in terms of Kripke semantics. The logic is extended with an epistemic box modal $\Box$ reflecting universal truth within equivalence classes of states, and with probability operators $\Pr$ over ground formulas. Models are constructed as tuples incorporating sets of states, atomic formulas and programs, equivalence relations over states for epistemic closure, partitions of program valuations reflecting the agent's epistemic indifference, and a probability measure $\mu$ encoding epistemic degrees of belief.

Key semantic choices distinguish PEDAL:

• Epistemic Probability Imposition: Probability measures are over partitions of possible program valuations, reflecting the agent’s subjective uncertainty, not objective stochasticity in transition relations. This differs from prior works (e.g., Kozen's Probabilistic PDL) in that the probabilistic structure represents epistemic ignorance rather than intrinsic nondeterminism.
• Finiteness Constraints: The logic models only finitely many epistemic alternatives and assigns probability only to finite sets to match agent limitations and practical program domains.
• Grounding in Universal Validity: Probabilistic assertions about programs are formulated regarding their universal validity in the model, ensuring that epistemic reasoning targets specification satisfaction in all relevant states.

PEDAL’s semantics assign probabilities to formulas via integration over program valuation spaces; agents are modeled as indifferent between program valuations within each partition cell, and probabilities on formulas are computed as the sum over valuations where the formula holds, normalized by epistemic weights.

Motivating Example and Probabilistic Epistemic Reasoning

A central motivating scenario models an agent, Anne, who holds probabilistic credences about properties of two programs and must infer the probability that a compound program meets a compounded specification. Explicitly, with 60% confidence that $A\to[p]B$ and $C\to[q]D$ are universally true, PEDAL calculates—contrary to naive intuition—that Anne should only be 20% confident that $[(?A;p)\cup(?C;q)](B\lor D)$ holds universally. This is formally derivable within PEDAL’s axiomatization and semantics, demonstrating nontrivial interaction between probabilistic beliefs in compositional settings.

This example highlights PEDAL’s ability to rigorously propagate probabilistic judgments about atomic program specifications to more complex, dynamically composed programs, justifying formal epistemic calculations in software verification contexts where full certainty is unattainable.

Axiomatization, Infinitary Rules, and Completeness

PEDAL is axiomatized with propositional tautologies, probability operator axioms (including constraints reflecting Kolmogorov properties), modal axioms for epistemic closure, and crucially, an infinitary rule enabling passage from countably many approximations to real probability bounds. This rule reflects the necessity of reasoning over all rational approximations in the presence of real probabilities, as in classical probability logics.

Soundness and completeness proofs are presented by canonical model constructions, encoding agent epistemic states as partitions and verifying the axioms against semantics. The canonical model leverages bijective sections over sets of maximal consistent formula sets, ensuring correspondence between syntactic derivability and semantic truth.

Alternative approaches to circumventing infinitary rules are discussed: restricting probability ranges to finite subsets of rationals (yielding finitary axiomatizations, cf. Fattorosi-Barnaba), bounding model state spaces, or employing schematic variables. These approaches promise implementability and tractability in automated reasoning and verification contexts.

Numerical Results and Contradictory Claims

The paper demonstrates strong numerical results in epistemic probability propagation:

• Upper Bound Derivation: PEDAL proves that, under specified credences (0.6 for two atomic specifications), the probability for the compound specification cannot exceed 0.2. This is derived via formal manipulations on probability measures over partitions, modeling program composition and epistemic uncertainty.
• Negative Result: Any claim of confidence exceeding 0.2 for the compound specification must be justified by additional epistemic information; PEDAL models warrant no further increase in credence.

These results contradict naive intuition that probabilistic credences can simply sum or multiply; epistemic independence and interaction can reduce compounded credence substantially, exposing pitfalls in informal probabilistic reasoning about program correctness.

Practical and Theoretical Implications

PEDAL offers a rigorous framework for formal epistemic reasoning in software verification, especially under conditions of partial knowledge or probabilistic assurance. Practical implications include:

• Risk Assessment in Verification: PEDAL provides quantitative lower bounds on specification satisfaction, vital for security evaluation, certification, and risk management (connecting to Common Criteria EALs and the cost of poor software quality [Krasner 2022]).
• Formalization of Testing and Review: PEDAL models the epistemic state after incomplete verification procedures like testing or code review, supporting structured reasoning about residual uncertainty.
• Compositional Reasoning: The logic supports formal derivation of probabilities for composite programs, avoiding informal combinatorial errors.

Theoretically, PEDAL contributes to modal logics by integrating epistemic, dynamic, and probabilistic modalities, and by proposing flexible model constructions reflecting agent uncertainty. The epistemic imposition of probability distinguishes PEDAL from stochastic and belief-based logics (cf. [Fagin et al. 2003], [Kozen 1983]), opening avenues for logics of agentive epistemic probability and potentially for multi-agent scenarios.

Speculation on Future Developments

Future research directions include:

• Finitary Axiomatizations: Developing implementable, finitary proof systems by restricting the probability value range, enabling realistic automated reasoning tools.
• Bounded-State Models: Constraining model state sizes for more efficient reasoning, following approaches similar to Abadi-Halpern [abadi1994decidability].
• Schematic Variable Approaches: Incorporating restricted quantification to approximate infinitary rules within manageable proof theories.
• Extension to Multi-Agent Settings: Adapting PEDAL to model interacting agents with differential epistemic partitions, possibly linking to dynamic epistemic logics [Van Ditmarsch et al. 2007].

The adoption of PEDAL in formal verification environments may facilitate more nuanced certification and risk estimation, particularly when full formal verification is infeasible.

Conclusion

PEDAL, as introduced in the paper, formalizes probabilistic epistemic reasoning about program specifications, enabling agents to rigorously assess their credences concerning complex program behaviors. By externally imposing probability measures over epistemic partitions of program valuations, PEDAL models realistic scenarios of incomplete knowledge and reasoning under uncertainty. The logic’s axiomatization, completeness and soundness proofs, and its capacity to derive nontrivial numerical bounds on specification satisfaction, position it as a robust tool for epistemic probability in verification and reasoning.

PEDAL’s architecture, while currently dependent on infinitary inference, suggests promising directions for practical, finitary deduction systems and future extensions. Its formal rigor supports accurate, compositional reasoning about epistemic probabilities in program verification contexts, helping to sharpen both theoretical and practical instincts where informal reasoning fails.