Public Observation Logic (POL)

• Public Observation Logic (POL) is a dynamic epistemic logic that uses regular-expression-based expectations to drive observation-driven updates in epistemic models.
• It filters possible worlds through observation matching and residuation, ensuring only states with compatible expected observations survive.
• The formal framework integrates automata-theoretic techniques and complexity analyses, offering robust methods for epistemic planning, verification, and protocol modeling.

Public Observation Logic (POL) is a dynamic epistemic logic for reasoning about how agents’ expectations about observations interact with what is publicly observed, and how this interaction influences epistemic reasoning. In POL, each epistemic state carries an explicit expectation about future observations, encoded by an observation expression over a finite alphabet. When an observation is made, agents eliminate possible worlds whose expected observations are incompatible with what was observed, and the surviving worlds update their expectations by residuation. This combination of expectation matching, world elimination, and epistemic restriction gives POL a semantics that is closely related to public announcement update while remaining observation-driven rather than announcement-driven (Chakraborty et al., 2023).

1. Formal framework

The core vocabulary of POL consists of propositional atoms, epistemic modalities, and dynamic observation operators. Observation expressions are regular expressions over a finite alphabet $\Sigma$ of atomic observations or actions. In the star-free fragment $\mathrm{POL}^{-}$, the observation expressions are generated by

$$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$

where $a \in \Sigma$; in full POL, Kleene star is added, yielding

$$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$

The associated language $L(\pi)\subseteq \Sigma^\ast$ is interpreted with the standard regular-language semantics, and the residual $\pi w$ or $\pi \setminus w$ denotes the language of suffixes that remain possible after consuming a word $w$ (Chakraborty et al., 2023).

The formula language appears in dual presentations. One standard presentation uses

$$\varphi ::= \top \mid p \mid \neg \varphi \mid \varphi \land \psi \mid K_i \varphi \mid [\pi]\varphi,$$

with duals $\mathrm{POL}^{-}$0 and $\mathrm{POL}^{-}$1. Another presentation takes epistemic possibility and dynamic diamond as primitive:

$\mathrm{POL}^{-}$2

with $\mathrm{POL}^{-}$3 and $\mathrm{POL}^{-}$4. The intended reading is stable across these variants: $\mathrm{POL}^{-}$5 says that agent $\mathrm{POL}^{-}$6 knows $\mathrm{POL}^{-}$7, while $\mathrm{POL}^{-}$8 or $\mathrm{POL}^{-}$9 says that there exists a compatible observation in $$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$0 after which $$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$1 holds (Ghosh et al., 13 Aug 2025).

Semantically, POL is interpreted on epistemic expectation models. In the star-free presentation, such a model is a quadruple

$$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$2

where $$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$3 is a nonempty set of states, each $$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$4 is an S5-equivalence relation, $$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$5 is a valuation, and $$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$6 assigns to each state a nonempty observation expression. In the full regular-expression presentation the same structure is written as

$$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$7

with the same S5 interpretation of epistemic accessibility (Chakraborty et al., 2022).

2. Observation update, residuation, and epistemic effect

The defining operation of POL is public observation update. If $$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$8 is publicly observed, then the updated model keeps exactly those states whose expectations admit $$\pi ::= \emptyset \mid \epsilon \mid a \mid \pi\cdot\pi \mid \pi + \pi,$$9 as a prefix:

$a \in \Sigma$0

or equivalently, in the full-regular-expression notation,

$a \in \Sigma$1

Epistemic relations are restricted to $a \in \Sigma$2, valuations are restricted to surviving states, and the new expectation at each surviving state is the residual expectation $a \in \Sigma$3 or $a \in \Sigma$4 (Chakraborty et al., 2023).

Truth for the dynamic modality is defined by quantification over compatible observations. In the star-free presentation,

$a \in \Sigma$5

and

$a \in \Sigma$6

In the full presentation,

$a \in \Sigma$7

Thus public observation does not change propositional facts; it changes the domain of live states and the expectation component attached to those states (Ghosh et al., 13 Aug 2025).

This semantics is often described as “filtering by mismatch.” Expectations at a state are the set $a \in \Sigma$8 of observation strings deemed possible by the protocol at that state. Once an actual observation is seen, worlds whose expected observations are incompatible with that observation are pruned. Because epistemic accessibility is then restricted to the surviving domain, the update immediately affects what agents know and what they still consider possible (Chakraborty et al., 2023).

The automata-theoretic view makes the same mechanism explicit. If $a \in \Sigma$9 is an NFA for $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$0, then residuation by $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$1 replaces the initial state by the set of states reached by $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$2, producing an NFA whose language is exactly the residual language. Survival becomes an emptiness test: a state survives iff the residual NFA still accepts some word (Ghosh et al., 13 Aug 2025).

The worked examples given in the literature illustrate the epistemic force of this update. In one example, after observing a left move $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$3, Bob’s surviving epistemic alternatives only include non-debris states, while Alice’s still include a debris state, so

$$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$4

In another, a surveillance drone initially cannot distinguish between two territories after observing only $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$5, but after a compatible observation in $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$6, the state corresponding to the alternative territory is eliminated and the drone knows $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$7 (Chakraborty et al., 2023).

3. Fragments, proof systems, and algorithmic techniques

A substantial part of the POL literature studies syntactic fragments. The 2022 model-checking work distinguishes the full language, a Star fragment, an Existential fragment, a Star-Existential fragment, and a Word fragment in which $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$8 denotes a fixed word rather than a general regular expression (Chakraborty et al., 2022). The 2023 satisfiability study focuses on a star-free fragment $$\pi ::= \emptyset \mid a \mid \pi + \pi \mid \pi;\pi \mid \pi^\star.$$9 in which observation expressions contain no Kleene star (Chakraborty et al., 2023).

For $L(\pi)\subseteq \Sigma^\ast$0, a tableau proof system combines S5-style modal reasoning with residual computations on observation expressions. Tableau terms have the forms $L(\pi)\subseteq \Sigma^\ast$1, $L(\pi)\subseteq \Sigma^\ast$2, and $L(\pi)\subseteq \Sigma^\ast$3, representing formula truth after projection on a word, survival after projection, and $L(\pi)\subseteq \Sigma^\ast$4-indistinguishability, respectively. The rules include propositional expansion, knowledge and possibility rules for epistemic modalities, and dynamic rules such as diamond concatenation, diamond choice, diamond project for letters, box project via residuals, empty box, constant valuation up, and survival chain. The calculus is sound and complete for $L(\pi)\subseteq \Sigma^\ast$5 satisfiability: if a formula is satisfiable, the tableau initialized with $L(\pi)\subseteq \Sigma^\ast$6 and the corresponding survival and reflexivity terms is open, and conversely any open saturated branch yields a satisfying model (Chakraborty et al., 2023).

The same study gives a truth-preserving translation from the word fragment of POL to Public Announcement Logic (PAL). Fresh atoms $L(\pi)\subseteq \Sigma^\ast$7 encode that the letter $L(\pi)\subseteq \Sigma^\ast$8 is compatible with the current state given that $L(\pi)\subseteq \Sigma^\ast$9 was already observed, and the translation is defined relative to the observation history $\pi w$0, with the key clause

$\pi w$1

The resulting correspondence states that a formula is satisfiable in the word fragment of POL iff its translation is satisfiable in PAL. This makes the word fragment a precise boundary case where observation update can be simulated by public announcement update (Chakraborty et al., 2023).

On the model-checking side, the 2022 paper gives explicit algorithms. The full PSPACE procedure is based on a recursive algorithm mcPOL together with a nondeterministic polynomial-space oracle that guesses a sufficiently long word letter by letter and incrementally updates automata for both the formula observation expression and the expectations at the worlds. For star-existential formulas, a bounded-word nondeterministic procedure WorldsNP is used, supported by ResidueByLetter and AuxOut; for the Word fragment, a deterministic PTIME procedure WorldsP computes the unique update $\pi w$2 and then evaluates the epistemic part bottom-up. The same paper also presents a SAT encoding for the Star-Existential fragment using variables $\pi w$3, $\pi w$4, and $\pi w$5, and reports a prototype implementation in Python3 using pySAT and automata-lib (Chakraborty et al., 2022).

4. Complexity landscape

The complexity picture of POL is stratified by both the decision problem and the observation language. Model checking and satisfiability have been studied separately, and the shift from star-free observation expressions to full regular expressions with Kleene star changes the upper and lower bounds substantially (Chakraborty et al., 2022).

Setting	Problem	Complexity
Full POL	Model checking	PSPACE-complete
Existential fragment	Model checking	PSPACE-hard
Star fragment	Model checking	PSPACE-hard
Star-Existential fragment	Model checking	NP-complete
Word fragment	Model checking	PTIME
Full multi-agent $\pi w$6	Satisfiability	NEXPTIME-complete
Single-agent $\pi w$7	Satisfiability	PSPACE-hard; in NEXPTIME
Word fragment, single-agent	Satisfiability	NP-complete
Word fragment, multi-agent	Satisfiability	PSPACE-complete
Full POL with Kleene star	Satisfiability	2EXPTIME-complete

For model checking, the principal theorem is that full POL model checking is PSPACE-complete. The upper bound relies on the fact that every update model $\pi w$8 can be represented by a string of polynomial size even though the set of possible updates is exponential, together with a witness-length bound stating that if $\pi w$9, then there exists a witnessing word of length at most

$\pi \setminus w$0

The hardness results isolate two independent sources of intractability: Kleene star in observation expressions and alternation between existential and universal modalities (Chakraborty et al., 2022).

For satisfiability, the 2023 study proves that full multi-agent $\pi \setminus w$1 satisfiability is in NEXPTIME and NEXPTIME-hard, hence NEXPTIME-complete. The upper bound is obtained from the tableau procedure, whose nodes may contain exponentially many words generated by star-free observation expressions, while the lower bound reduces tiling of a $\pi \setminus w$2 square, with $\pi \setminus w$3 in unary, to satisfiability. For the single-agent star-free case, the same paper shows PSPACE-hardness via a reduction from TQBF and leaves open whether the NEXPTIME upper bound is tight. For the word fragment, the translation to PAL yields NP-completeness in the single-agent case and PSPACE-completeness in the multi-agent case (Chakraborty et al., 2023).

The status of full POL with Kleene star changed in 2025. Earlier work had left Kleene-star extensions unaddressed and noted that PAL with Kleene-star is undecidable, while conjecturing that POL with star might remain decidable. The later result proves that satisfiability for full POL is 2EXPTIME-complete. The upper bound uses a finite-model construction via Bubble Transition Structures (BTS) over Hintikka sets and an exponential-time reduction to DPDL; the lower bound reduces acceptance of an alternating Turing machine running in exponential space to POL satisfiability (Ghosh et al., 13 Aug 2025).

5. Relations to PAL, DEL, PDL, and DPDL

POL is routinely positioned relative to Public Announcement Logic, Dynamic Epistemic Logic, and program logics, but its update mechanism is distinct from each of them. In PAL, an update $\pi \setminus w$4 publicly announces a formula and restricts the model to worlds where $\pi \setminus w$5 is true. In POL, the update is driven by matching a publicly observed word or regular pattern against a per-state expectation, followed by residuation of that expectation. The surviving states are those whose expected observations remain compatible with the observed word; valuations remain fixed; epistemic relations are restricted to the surviving domain (Ghosh et al., 13 Aug 2025).

The relation to PAL is tightest on the word fragment. There, fixed-word observation update can be encoded by propositional announcements using fresh atoms that record letter compatibility after a history. This gives immediate complexity transfer from PAL to the word fragment of POL and explains why the word-fragment complexity matches well-known PAL bounds: NP-complete in the single-agent setting and PSPACE-complete in the multi-agent setting (Chakraborty et al., 2023).

At the same time, POL is not merely PAL with strings in place of formulas. The crucial difference is that announcements in PAL are evaluated by truth at the current state, whereas observations in POL are filtered through the expectation component $\pi \setminus w$6. Two worlds with the same valuation may react differently to the same observation because they carry different expected observation languages. This difference is also what separates POL from ordinary PDL-style dynamics: programs in PDL change states along transition relations, whereas observations in POL prune the epistemic model and residuate expectations while leaving valuations unchanged (Chakraborty et al., 2022).

The 2025 upper-bound proof establishes a second important connection, namely to DPDL. Full POL satisfiability is reduced to satisfiability of a single DPDL formula $\pi \setminus w$7 of exponential size. The translation uses labels for filtered worlds, atoms of the form $\pi \setminus w$8 to assert that $\pi \setminus w$9 holds at a label, relation atoms $w$0, and survival atoms $w$1, together with semantic clauses such as

$w$2

and

$w$3

Because DPDL satisfiability is EXPTIME-complete, the exponential blow-up in the translation yields the 2EXPTIME upper bound for full POL (Ghosh et al., 13 Aug 2025).

The literature also treats POL as a DEL-adjacent formalism rather than a general special case of DEL. DEL uses event or action models and product update, may allow valuations to change, and typically formulates action preconditions as formulas. POL instead assumes perfect recall, keeps valuations static under observation, and encodes dynamics through deterministic regular-expression-based observations combined with expectation residuation. No general translation from DEL into POL or vice versa is given (Ghosh et al., 13 Aug 2025).

6. Applications, protocol modeling, and open directions

POL has been used to model verification and planning tasks in systems whose behavior is constrained by explicit protocols. The 2022 model-checking study gives a traffic-light example in which a traveler uses expectations about signal sequences to determine whether she is in France, a message-passing example with Sender, Receiver, and Attacker where Receiver can infer that a decision has been taken after observing $w$4 while the Attacker remains ignorant, and an interactive farming-drone example with behaviors such as searching for water, searching for power, and patrolling. The same paper formulates plan verification, epistemic planning, bounded planning, and mixed necessity/possibility properties using Word, Existential, Star-Existential, Star, and full-language formulas (Chakraborty et al., 2022).

A later example uses a surveillance drone with expectations

$w$5

This model illustrates two characteristic patterns. First, partial observations such as those in $w$6 can preserve uncertainty, because both states survive and the agent continues to consider both territories possible. Second, an informative observation in $w$7 eliminates the incompatible state and yields knowledge of the current territory. The example shows how regular-expression expectations support fine-grained reasoning about when an observation is epistemically informative and when it is not (Ghosh et al., 13 Aug 2025).

POL has also been adapted to concurrent epistemic protocols. The 2023 muddy-children study reformulates update in action-based terms, using the father’s repeated question $w$8 as the sole observable action, and extends the setting to multiple sessions running in parallel. Each session is modeled by an expectation model, and the global parallel model is their Cartesian product. When an action in one session leads a shared agent to know whether they are muddy, this fact is propagated to other sessions by restricting those session models to worlds consistent with the learned value. In the three-session example with

$w$9

parallel propagation reduces the total number of father’s questions from eight to four. A two-session worked example likewise reduces the total from five to four (Macovei, 2023).

Several limitations and open problems remain explicit in the literature. The 2022 work identifies private observations, dynamic expectations, richer expectation languages such as context-free grammars, symbolic model checking, and satisfiability as open directions. The 2025 full-POL complexity result leaves open the problem of complete axiomatics for POL and points to natural fragment and extension questions, including expectations given by context-free grammars, Epistemic Protocol Logic (EPL), and further comparisons with temporal-epistemic formalisms such as $$\varphi ::= \top \mid p \mid \neg \varphi \mid \varphi \land \psi \mid K_i \varphi \mid [\pi]\varphi,$$0 and CTLK (Chakraborty et al., 2022). Earlier work also left the exact complexity of single-agent star-free satisfiability unresolved; the known bounds are PSPACE-hardness and a general NEXPTIME upper bound (Chakraborty et al., 2023).

A recurring misconception is to identify POL with ordinary public announcement update. The formal developments do not support that identification. POL is announcement-adjacent, and on the word fragment it can be translated into PAL, but in general it reasons about observation compatibility relative to per-state regular-expression expectations, not about truth of an announced formula. The dynamic content lies in matching, pruning, and residuation. That mechanism is what allows POL to capture protocol-sensitive knowledge change, from fixed-word verification and epistemic planning to iterative regular observations and parallel session updates.