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The Epistemology of Nondeterminism

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Abstract

This paper proposes new semantics for propositional dynamic logic (PDL), replacing the standard relational semantics. Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) (Kremer and Mints in Ann Pure Appl Logic 131:133–158, 2005). We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic (Dabrowski et al. in Ann Pure Appl Logic 78:73–110, 1996), and show that the topological interpretation of public announcements as given in Bjorndahl (in: van Ditmarsch and Sandu (eds) Jaakko Hintikka on knowledge and game theoretical semantics, outstanding contributions to Logic, vol 12, Springer, 2018) coincides exactly with a natural interpretation of test programs.

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Notes

  1. Mathematical treatments of nondeterminism have a long history in computer science (see, e.g., (Rabin and Scott, 1959; Dijkstra, 1976; Francez et al., 1979; Søndergaard and Sestoft, 1992)), though this paper focuses specifically on the semantics of PDL. See Troquard and Balbiani (2015) for an overview and brief history of this branch of modal logic.

  2. To be sure, if you had access to a more advanced set of tools than are standardly available, perhaps you could make such a determination. And in this case, thinking of the random number generator as a nondeterministic process loses much of its intuitive appeal. Indeed, any nondeterministic process whatsoever might be viewed as deterministic relative to a sufficiently powerful set of tools (e.g., from God’s perspective). Thus, nondeterminism can be naturally construed as a relative notion that depends on a fixed background set of “feasible measurements”. We make this precise below.

  3. For a standard introduction to topological notions, we refer the reader to Munkres (2000).

  4. Suppose, for a simple example, that you measure your height and obtain a reading of \(180 \pm 2\) cm. If we represent the space of your possible heights using the positive real numbers, \(\mathbb {R}^{+}\), then it is natural to identify this measurement with the open interval (178, 182). And with this measurement in hand, you can safely deduce that you are, for instance, less than 183 cm tall, while remaining uncertain about whether you are, say, taller than 179 cm.

  5. One might wonder about the closure conditions on topologies. Finite intersections can perhaps be accounted for by identifying them with sequences of measurements, but what about unions? One intuition comes by observing that for any set A, \( int (A) = \bigcup \{U \in \mathcal {T}\, : \, U \subseteq A\}\), so \(\int (A)\) is the information state that arises from learning that A is true without learning what particular measurement was taken to ascertain this fact. This idea is formalized in Bjorndahl (2018) using public announcements; we direct the reader to this work for a more detailed discussion of this point.

  6. Of course, once the process \(\textsf{rand}\) has been called, presumably the agent can ascertain its output (e.g., by looking at the screen). In particular, if (say) the number displayed is 1, then this knowable, and therefore it is not the case that \(\Circle _{\textsf{rand}} \Diamond 0\) holds. This shows that the order of the two modalities is crucial in establishing the proper correspondence with nondeterminism. Thanks to an anonymous reviewer for suggesting this clarification.

  7. This claim is made precise and proved in Appendix A.1.

  8. This can be proved using very standard techniques; see, e.g., (Blackburn et al., 2001).

  9. For instance, consider the set \(X = \{a,b\}\) equipped with the topology \(\mathcal {T}= \{\emptyset , \{b\}, X\}\), and the function \(f_{\pi }\) defined by \(f_{\pi }(a) = b\) and \(f_{\pi }(b) = a\). Let \(v(p) = \{a\}\). Then, since \(f_{\pi } \circ f_{\pi } = id\), we have \( cl (f_{\pi ;\pi }^{-1}([\![ p ]\!])) = cl (\{a\}) = \{a\}\), whereas \( cl (f_{\pi }^{-1}( cl (f_{\pi }^{-1}([\![ p ]\!])))) = cl (f_{\pi }^{-1}( cl (\{b\}))) = cl (f_{\pi }^{-1}(X)) = X\).

  10. Typically the concept of openness is applied to total functions, but the definition makes sense for partial functions as well: f is open provided, for all open U, \(f(U) = \{y \in X \, : \, (\exists x \in U)(f(x) = y)\}\) is open.

  11. Or, rather, it coincides with the dual of the definition given in Bjorndahl (2018), but this is not an important difference.

  12. That is, \(\mathcal {A}\) contains the identity function and is closed under composition.

  13. More generally, given a partition \(\mathcal {P}\) of X (perhaps representing a question of interest, such as the value of a certain variable), say that \(\mathcal {P}\) is f-invariant if, for all \(x \in dom (f)\) and each \(C \in \mathcal {P}\), \(x \in C\) implies \(f(x) \in C\) (i.e., f doesn’t change the “answer” to the question). In this case B is f-invariant iff the “yes/no question” is f-invariant.

  14. Specifically, \(x_{1} = \alpha (\emptyset )\) and \((\forall i \ge 2)(x_{i} = \alpha (\pi _{1}, \ldots , \pi _{i-1}))\).

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A Proofs and Details

A Proofs and Details

1.1 A.1 Characterizing Continuity

A dynamic topological frame (over \(\varPi \)) is a tuple \(F = (X, \mathcal {T}, \{f_{\pi }\}_{\pi \in \varPi })\) where \((X,\mathcal {T})\) is a topological space and each \(f_{\pi }: X \rightarrow X\). In other words, a frame is simply a dynamic topological model without a valuation function. A frame F is said to validate a formula \(\varphi \) just in case \(\varphi \) is true at every point of every model of the form (Fv).

Proposition 1

The formula scheme \(\Circle _{\pi } \Box \varphi \rightarrow \Box \Circle _{\pi } \varphi \) defines the class of dynamic topological frames in which \(f_{\pi }\) is continuous: that is, for every dynamic topological frame F, F validates every instance of \(\Circle _{\pi } \Box \varphi \rightarrow \Box \Circle _{\pi } \varphi \) iff \(f_{\pi }\) is continuous.

Proof

First suppose that M is a dynamic topological model in which \(f_{\pi }\) is continuous, and let x be a point in this model satisfying \(\Circle _{\pi } \Box \varphi \). Then \(f_{\pi }(x) \in \textit{int}([\![ \varphi ]\!])\). By continuity, the set \(U = f_{\pi }^{-1}(int([\![ \varphi ]\!]))\) is open. Moreover, it is easy to see that \(x \in U\) and \(U \subseteq [\![ \Circle _{\pi }\varphi ]\!]\), from which it follows that \(x \models \Box \Circle _{\pi } \varphi \).

Conversely, suppose that F is a dynamic topological frame in which \(f_{\pi }\) is not continuous. Let U be an open subset of X such that \(A = f_{\pi }^{-1}(U)\) is not open, and let ; consider a valuation v such that \(v(p) = U\). In the resulting model, since \(f_{\pi }(x) \in U = int (U)\), we have \(x \models \Circle _{\pi } \Box p\). On the other hand, since by definition \(x \notin int (f_{\pi }^{-1}(U))\), we have \(x \not \models \Box \Circle _{\pi } p\). \(\square \)

1.2 A.2 Model Transformation

Our task in this section is to transform an arbitrary serial PDL model into a dynamic topological model in a truth-preserving manner. The intuition for this transformation is fairly straightforward: in a serial PDL model, each state may be nondeterministically compatible with many possible execution paths corresponding to all the possible ways of successively traversing \(R_{\pi }\)-edges. In a dynamic topological model, by contrast, all such execution paths must be differentiated by state—roughly speaking, this means we need to create a new state for each possible execution path in the standard model. Then, to preserve the original notion of nondeterminism, we overlay a topological structure that “remembers” which new states originated from the same state in the standard model by rendering them topologically indistinguishable.

Let \(M = (X, (R_{\pi })_{\pi \in \varPi }, v)\) be a serial PDL model. Let \(\varPi ^{*}\) denote the set of all finite sequences from \(\varPi \). A map \(\alpha : \varPi ^{*} \rightarrow X\) is called a network (through M) provided \((\forall \varvec{\pi } \in \varPi ^{*})(\forall {\pi } \in \varPi )(\alpha (\varvec{\pi })R_{\pi }\alpha (\varvec{\pi }, {\pi }))\). In other words, a network \(\alpha \) must respect \(R_{\pi }\)-edges in the sense that it associates with each sequence \((\pi _{1}, \ldots , \pi _{n})\) a path \((x_{1}, \ldots , x_{n+1})\) through X such that, for each i, \(x_{i}R_{\pi _{i}}x_{i+1}\).Footnote 14 Networks through M constitute the points of the dynamic topological model we are building:

$$\begin{aligned} \tilde{X} = \{\alpha \, : \, \alpha \text { is a network through space } M. \}. \end{aligned}$$

The topology we equip \(\tilde{X}\) with is particularly simple: for each \(x \in X\), let \(U_{x} = \{\alpha \in \tilde{X} \, : \, \alpha (\emptyset ) = x\}\). Clearly the sets \(U_{x}\) partition X and so form a topological basis; let \(\mathcal {T}\) be the topology they generate.

Next we define the functions \(f_{\pi }: \tilde{X} \rightarrow \tilde{X}\). Intuitively, \(\alpha \in \tilde{X}\) is a complete record of what paths will be traversed in the original state space X for every sequence of program executions. Therefore, after executing \(\pi \), the updated record \(f_{\pi }(\alpha )\) should simply consist in those paths determined by \(\alpha \) that start with an execution of \(\pi \):

$$\begin{aligned} f_{\varvec{\pi }}(\alpha )({\varvec{\pi }}) = \alpha ({{\pi }},{\varvec{\pi }}). \end{aligned}$$

Finally, define \(\tilde{v}: \textsc {prop} \rightarrow 2^{\tilde{X}}\) by

$$\begin{aligned} \tilde{v}(p) = \{\alpha \in \tilde{X} \, : \, \alpha (\emptyset ) \in v(p)\}. \end{aligned}$$

Let \(\tilde{M} = (\tilde{X}, \mathcal {T}, (f_{\pi })_{\pi \in \varPi }, \tilde{v})\).

Proposition 2

For every \(\varphi \in \mathcal {L}_{PDL}\), for every \(\alpha \in \tilde{X}\), \((\tilde{M},\alpha ) \models \varphi \text { iff } (M,\alpha (\emptyset )) \models \varphi \).

Proof

We proceed by induction on the structure of \(\varphi \). The base case when \(\varphi = p \in \textsc {prop}\) follows directly from the definition of \(\tilde{v}\):

$$\begin{aligned} (\tilde{M},\alpha ) \models p&\text {iff}&\alpha \in \tilde{v}(p)\\&\text {iff}&\alpha (\emptyset ) \in v(p)\\&\text {iff}&(M, \alpha (\emptyset )) \models p. \end{aligned}$$

The inductive steps for the Boolean connectives are straightforward. So suppose inductively that the result holds for \(\varphi \); we wish to show it holds for \(\langle \pi \rangle \varphi \).

Let \(\alpha \in \tilde{X}\) and \(x = \alpha (\emptyset )\). By definition, \((\tilde{M},\alpha ) \models \langle \pi \rangle \varphi \) iff \(\alpha \in cl (f_{\pi }^{-1}([\![ \varphi ]\!]_{\tilde{M}}))\). Since the topology is generated by a partition, we know that \(U_{x}\) is a minimal neighbourhood of \(\alpha \), and therefore the preceding condition is equivalent to:

$$\begin{aligned} U_{x} \cap f_{\pi }^{-1}([\![ \varphi ]\!]_{\tilde{M}}) \ne \emptyset . \end{aligned}$$

This intersection is nonempty just in case there exists an \(\alpha ' \in \tilde{X}\) such that \(\alpha '(\emptyset ) = x\) and \(f_{\pi }(\alpha ') \in [\![ \varphi ]\!]_{\tilde{M}}\). By the induction hypothesis,

$$\begin{aligned} f_{\pi }(\alpha ') \in [\![ \varphi ]\!]_{\tilde{M}}&\text {iff}&(\tilde{M}, f_{\pi }(\alpha ') ) \models \varphi \\&\text {iff}&(M, f_{\pi }(\alpha ')(\emptyset )) \models \varphi \\&\text {iff}&(M, \alpha '(\pi )) \models \varphi . \end{aligned}$$

So to summarize, we have shown that \((\tilde{M},\alpha ) \models \langle \pi \rangle \varphi \) iff there exists an \(\alpha ' \in \tilde{X}\) such that \(\alpha '(\emptyset ) = x\) and \((M, \alpha '(\pi )) \models \varphi \). Since we know that \(\alpha '(\emptyset ) R_{\pi } \alpha '(\pi )\), this is in turn equivalent to \((M,x) \models \langle \pi \rangle \varphi \), which completes the induction. \(\square \)

Moreover, this model transformation method supports the connection between determinism and continuity in the following sense:

Proposition 3

For every \(\pi \in \varPi \) and \(x \in X\), we have \(|R_{\pi }(x)| = 1\) if and only if \(f_{\pi }\) is continuous at all points in \(U_{x}\).

Proof

First suppose that \(R_{\pi }(x) = \{y\}\). Let \(\alpha \in U_{x}\); we must show that \(f_{\pi }\) is continuous at \(\alpha \). We know that \(\alpha (\emptyset ) = x\) and \(\alpha (\emptyset ) R_{\pi } \alpha (\pi )\), so \(\alpha (\pi ) = y\); thus \(f_{\pi }(\alpha )(\emptyset ) = \alpha (\pi ) = y\), so \(f_{\pi }(\alpha ) \in U_{y}\).

Because \(\mathcal {T}\) is generated by a partition, to show that \(f_{\pi }\) is continuous at \(\alpha \) it suffices to show that \(f_{\pi }(U_{x}) \subseteq U_{y}\), since \(U_{x}\) and \(U_{y}\) are minimal open neighbourhoods of \(\alpha \) and \(f_{\pi }(\alpha )\), respectively. But this is easy to see: given any \(\beta \in U_{x}\), we know that \(\beta (\emptyset ) = x\) and \(\beta (\emptyset ) R_{\pi } \beta (\pi )\), so it must be that \(\beta (\pi ) = y\), thus \(f_{\pi }(\beta )(\emptyset ) = \beta (\pi ) = y\) which means \(f_{\pi }(\beta ) \in U_{y}\), as desired.

For the converse, suppose that \(f_{\pi }\) is continuous at all points of \(U_{x}\). I claim that this implies that there is a \(y \in X\) such that \(f_{\pi }(U_{x}) \subseteq U_{y}\). Indeed, let \(\alpha \in U_{x}\) and set \(y = \alpha (\pi )\). Then we know that \(f_{\pi }(\alpha )(\emptyset ) = \alpha (\pi ) = y\), so \(f_{\pi }(\alpha ) \in U_{y}\); by continuity, then, the minimal neighbourhood of \(\alpha \) must map into \(U_{y}\), i.e., \(f_{\pi }(U_{x}) \subseteq U_{y}\).

Now observe that whenever \(xR_{\pi }z\), this implies there exists a \(\gamma \in \tilde{X}\) with \(\gamma (\emptyset ) = x\) and \(\gamma (\pi ) = z\); in this case we would have \(\gamma \in U_{x}\) so \(f_{\pi }(\gamma ) \in U_{y}\), which yields

$$\begin{aligned} z = \gamma (\pi ) = f_{\pi }(\gamma )(\emptyset ) = y. \end{aligned}$$

This establishes that \(R_{\pi }(x) = \{y\}\), as desired. \(\square \)

1.3 A.3 Sequencing

Lemma

1 Let \((X, \mathcal {T}, \{f_{\pi }\}_{\pi \in \varPi }, v)\) be a dynamic topological model. If \(f_{\pi _{1}}\) is open, then

$$\begin{aligned}{}[\![ \langle \pi _{1};\pi _{2}\rangle \varphi ]\!] = [\![ \langle \pi _{1}\rangle \langle \pi _{2}\rangle \varphi ]\!]. \end{aligned}$$

Proof

It suffices to show that \([\![ \langle \pi _{1};\pi _{2}\rangle \varphi ]\!] \supseteq [\![ \langle \pi _{1}\rangle \langle \pi _{2}\rangle \varphi ]\!]\). So let

$$\begin{aligned} x \in [\![ \langle \pi _{1}\rangle \langle \pi _{2}\rangle \varphi ]\!] = cl(f_{\pi _{1}}^{-1}(cl(f_{\pi _{2}}^{-1}([\![ \varphi ]\!])))); \end{aligned}$$

then for every open neighbourhood U containing x, we know that \(U \cap f_{\pi _{1}}^{-1}(cl(f_{\pi _{2}}^{-1}([\![ \varphi ]\!]))) \ne \emptyset \). This implies that \(f_{\pi _{1}}(U) \cap cl(f_{\pi _{2}}^{-1}([\![ \varphi ]\!])) \ne \emptyset \); since \(f_{\pi _{1}}(U)\) is open, it follows that \(f_{\pi _{1}}(U) \cap f_{\pi _{2}}^{-1}([\![ \varphi ]\!]) \ne \emptyset \) as well. This then implies that \(U \cap f_{\pi _{1}}^{-1}(f_{\pi _{2}}^{-1}([\![ \varphi ]\!])) \ne \emptyset \), and therefore

$$\begin{aligned} x \in cl (f_{\pi _{1}}^{-1}(f_{\pi _{2}}^{-1}([\![ \varphi ]\!]))) = [\![ \langle \pi _{1};\pi _{2}\rangle \varphi ]\!], \end{aligned}$$

as desired. \(\square \)

Say that a dynamic topological model is open if each \(f_{\pi }\) is open.

Theorem

3SPDL\(_{0}\) + (Seq) is a sound and complete axiomatization of the language \(\mathcal {L}_{PDL}\) with respect to the class of all open dynamic topological models.

Proof

Lemma 1 shows that (Seq) is valid in the class of all open dynamic topological models. For completeness, it suffices to observe that the dynamic topological model \(\tilde{M}\) constructed in Appendix A.2 is itself open: indeed, for each basic open \(U_{x}\), we have

$$\begin{aligned} f_{\pi }(U_{x})= & {} \{\alpha \in \tilde{X} \, : \, xR_{\pi }\alpha (\emptyset )\}\\= & {} \bigcup _{y \in R(x)} U_{y}, \end{aligned}$$

which of course is open. \(\square \)

Proposition 4

The formula scheme \(\Box \Circle _{\pi } \varphi \rightarrow \Circle _{\pi } \Box \varphi \) defines the class of dynamic topological frames in which \(f_{\pi }\) is open: that is, for every dynamic topological frame F, F validates every instance of \(\Box \Circle _{\pi } \varphi \rightarrow \Circle _{\pi } \Box \varphi \) iff \(f_{\pi }\) is open.

Proof

First suppose that M is a dynamic topological model in which \(f_{\pi }\) is open, and let x be a point in this model satisfying \(\Box \Circle _{\pi } \varphi \). Then \(x \in int (f_{\pi }^{-1}([\![ \varphi ]\!]))\). By openness, the set \(V = f( int (f_{\pi }^{-1}([\![ \varphi ]\!])))\) is open. Moreover, it is easy to see that \(f_{\pi }(x) \in V\) and \(V \subseteq [\![ \varphi ]\!]\), from which it follows that \(x \models \Circle _{\pi } \Box \varphi \).

Conversely, suppose that F is a dynamic topological frame in which \(f_{\pi }\) is not open. Let U be an open subset of X such that \(A = f_{\pi }(U)\) is not open, and let \(x \in U\) be such that ; consider a valuation v such that \(v(p) = A\). In the resulting model, since \(x \in U\) and \(f_{\pi }(U) \subseteq [\![ p ]\!]\), we have \(x \models \Box \Circle _{\pi } p\). On the other hand, since by definition \(f_{\pi }(x) \notin int ([\![ p ]\!])\), we have \(x \not \models \Circle _{\pi } \Box p\). \(\square \)

1.4 A.4 Dynamic Topological Epistemic Logic

Theorem

5\(\textsf{DTEL}\) is a sound and complete axiomatization of \(\mathcal {L}_{K,\Box ,{\Circle }}\) with respect to the class of all dynamic topological subset models.

Proof

Soundness has been established already, so we turn our attention to completeness. Let X denote the set of all maximal \(\textsf{DTEL}\)-consistent subsets of \(\mathcal {L}_{K,\Box ,{\Circle }}\). Define a binary relation \(\sim \) on X by

$$\begin{aligned} x \sim y \; \Leftrightarrow \; (\forall \varphi \in \mathcal {L}_{K,\Box ,{\Circle }})(K \varphi \in x \Leftrightarrow K \varphi \in y). \end{aligned}$$

Clearly \(\sim \) is an equivalence relation; let [x] denote the equivalence class of x under \(\sim \). For each \(x \in X\), define

$$\begin{aligned} R(x) = \{y \in X \, : \, (\forall \varphi \in \mathcal {L}_{K,\Box ,{\Circle }})(\Box \varphi \in x \Rightarrow \varphi \in y)\}. \end{aligned}$$

Let \(\mathcal {B}= \{R(x) \, : \, x \in X\}\), and let \(\mathcal {T}\) be the topology generated by \(\mathcal {B}\). It is easy to check that \(\mathcal {B}\) is a basis for \(\mathcal {T}\), and each R(x) is a minimal neighbourhood about x (see, e.g., (Aiello et al., 2003)). Given \(x \in X\), define

$$\begin{aligned} f_{\pi }(x) = \left\{ \begin{array}{ll} \{\varphi \, : \, \Circle _{\pi }\varphi \in x\} &{} \; \text { if }\Circle _{\pi }\top \in x\\ \text {undefined} &{} \; \text {otherwise.} \end{array} \right. \end{aligned}$$

The following lemmas establishes that each \(f_{\pi }\) is a partial, open function \(X \rightharpoonup X\). \(\square \)

Lemma 2

Whenever \(f_{\pi }(x)\) is defined, it is maximal \(\textsf{DTEL}\)-consistent.

Proof

First, suppose for contradiction that \(\varphi _{1}, \ldots , \varphi _{k} \in f_{\pi }(x)\) are inconsistent. Then \(\Circle _{\pi } \varphi _{1}, \ldots , \Circle _{\pi }\varphi _{k} \in x\), so it follows that \(\Circle _{\pi } \varphi _{1} \wedge \cdots \wedge \Circle _{\pi }\varphi _{k} \in x\), and therefore (using (\(\wedge \)-C\(_{\pi }\))) \(\Circle _{\pi }(\varphi _{1} \wedge \cdots \wedge \varphi _{k}) \in x\). Now we know by assumption that

$$\begin{aligned} \vdash _{\textsf{DTEL}} (\varphi _{1} \wedge \cdots \wedge \varphi _{k}) \rightarrow \lnot \top , \end{aligned}$$

so by (Mon\(_{\pi }\))

$$\begin{aligned} \vdash _{\textsf{DTEL}} \Circle _{\pi }(\varphi _{1} \wedge \cdots \wedge \varphi _{k}) \rightarrow \Circle _{\pi }\lnot \top , \end{aligned}$$

which implies that \(\Circle _{\pi }\lnot \top \in x\). Now (\(\lnot \)-PC\(_{\pi }\)) implies that \(\lnot \Circle _{\pi } \top \wedge \Circle _{\pi } \top \in x\), contradicting consistency of x. This shows that \(f_{\pi }(x)\) is consistent.

Next suppose for contradiction that \(\varphi \notin f_{\pi }(x)\) and \(\lnot \varphi \notin f_{\pi }(x)\). Then \(\Circle _{\pi } \varphi \notin x\) and \(\Circle _{\pi } \lnot \varphi \notin x\), so \(\lnot \Circle _{\pi } \varphi \in x\) and \(\lnot \Circle _{\pi } \lnot \varphi \in x\). Since \(f_{\pi }(x)\) is defined we know that \(\Circle _{\pi }\top \in x\), so we have \(\lnot \Circle _{\pi } \varphi \wedge \Circle _{\pi }\top \in x\) and \(\lnot \Circle _{\pi } \lnot \varphi \wedge \Circle _{\pi }\top \in x\). Therefore, by (\(\lnot \)-PC\(_{\pi }\)), we have \(\Circle _{\pi } \lnot \varphi \in x\) and \(\Circle _{\pi } \lnot \lnot \varphi \in x\). But this implies that \(\lnot \varphi \in f_{\pi }(x)\) and \(\lnot \lnot \varphi \in f_{\pi }(x)\), contradicting consistency and thereby establishing maximality. \(\square \)

Lemma 3

For each \(x \in X\), \(f_{\pi }(R(x)) \in \mathcal {T}\).

Proof

Let \(z \in f_{\pi }(R(x))\) and let \(z' \in R(z)\); it suffices to show that \(z' \in f_{\pi }(R(x))\). By choice of z, there exists \(y \in R(x)\) with \(z = f_{\pi }(y) = \{\varphi \, : \, \Circle _{\pi } \varphi \in y\}\). We wish to find a \(y' \in R(x)\) with \(f_{\pi }(y') = z'\). To this end, consider the set

$$\begin{aligned} \varGamma = \{\psi \, : \, \Box \psi \in x\} \cup \{\Circle _{\pi }\chi \, : \, \chi \in z'\}. \end{aligned}$$

It is not hard to see that if \(y' \supseteq \varGamma \) then \(y'\) has the desired properties; by Lindenbaum’s lemma, we will therefore be done if we can show that \(\varGamma \) is \(\textsf{DTEL}\)-consistent.

Suppose not. Then there are \(\psi _{1}, \ldots , \psi _{n} \in \{\psi \, : \, \Box \psi \in x\}\) and \(\Circle _{\pi }\chi _{1}, \ldots , \Circle _{\pi }\chi _{m} \in \{\Circle _{\pi }\chi \, : \, \chi \in z'\}\) such that

$$\begin{aligned} \vdash _{\textsf{DTEL}} (\psi _{1} \wedge \cdots \wedge \psi _{n}) \rightarrow \lnot (\Circle _{\pi }\chi _{1} \wedge \cdots \wedge \Circle _{\pi }\chi _{m}), \end{aligned}$$

so by (\(\wedge \)-C\(_{\pi }\)),

$$\begin{aligned} \vdash _{\textsf{DTEL}} (\psi _{1} \wedge \cdots \wedge \psi _{n}) \rightarrow \lnot \Circle _{\pi }(\chi _{1} \wedge \cdots \wedge \chi _{m}), \end{aligned}$$

hence (using \(\textsf{S4}_{\Box }\))

$$\begin{aligned} \vdash _{\textsf{DTEL}} (\Box \psi _{1} \wedge \cdots \wedge \Box \psi _{n}) \rightarrow \Box \lnot \Circle _{\pi }(\chi _{1} \wedge \cdots \wedge \chi _{m}). \end{aligned}$$

Now each \(\Box \psi _{i}\) is contained in x, and therefore also in y; it follows that \(\Box \lnot \Circle _{\pi }(\chi _{1} \wedge \cdots \wedge \chi _{m}) \in y\) as well; moreover, since \(f_{\pi }(y) = z\), we have \(\Circle _{\pi }\top \in y\), so by (O\(_{\pi }\)) we obtain \(\Circle _{\pi }\Box \lnot (\chi _{1} \wedge \cdots \wedge \chi _{m}) \in y\) and therefore \(\Box \lnot (\chi _{1} \wedge \cdots \wedge \chi _{m}) \in z\), hence \(\lnot (\chi _{1} \wedge \cdots \wedge \chi _{m}) \in z'\), a contradiction. \(\square \)

For each \(p \in \textsc {prop}\), set \(v(p) {:}{=}\{x \in X \, : \, p \in x\}\). Let \(\mathcal {X}= (X, \mathcal {T}, \{f_{\pi }\}_{\pi \in \varPi }, v)\). Clearly \(\mathcal {X}\) is a dynamic topological subset model.

Lemma 4

(Truth Lemma) For every \(\varphi \in \mathcal {L}_{K,\Box ,{\Circle }}\), for all \(x \in X\), \(\varphi \in x\) iff \((\mathcal {X}, x, [x]) \models \varphi \).

Proof

First we show that \(y \in R(x) \Rightarrow y \sim x\). Let \(y \in R(x)\). If \(K\psi \in x\) then \(KK\psi \in x\), so \(\Box K \psi \in x\) and therefore \(K\psi \in y\); conversely, if \(\lnot K\psi \in x\) then \(K\lnot K\psi \in x\), so \(\Box \lnot K\psi \in x\) and therefore \(\lnot K\psi \in y\). Note that this implies that \([x] \in \mathcal {T}\), so (x, [x]) is indeed an epistemic scenario.

The proof proceeds by induction on the structure of \(\varphi \). The base case holds by definition of v, and the inductive steps for the Boolean connectives are straightforward. The inductive step for K mirrors exactly the corresponding step in (Bjorndahl 2018, Theorem 1).

So suppose inductively the result holds for \(\varphi \) and let us show it holds for \(\Box \varphi \). If \(\Box \varphi \in x\) then by definition of R we know that for every \(y \in R(x)\), \(\varphi \in y\). By the inductive hypothesis, this implies that \((\forall y \in R(x))(y,[y]) \models \varphi \); since \(y \in R(x) \Rightarrow y \sim x\), this is equivalent to \((\forall y \in R(x))((y,[x]) \models \varphi )\); since R(x) is an open neighbourhood of x, this yields \((x,[x]) \models \Box \varphi \).

For the converse, suppose that \(\Box \varphi \notin x\). Then

$$\begin{aligned} \{\psi \, : \, \Box \psi \in x\} \cup \{\lnot \varphi \} \end{aligned}$$

is consistent, for if not there are \(\psi _{1}, \ldots , \psi _{m} \in \{\psi \, : \, \Box \psi \in x\}\) such that

$$\begin{aligned} \vdash _{\textsf{DTEL}} \psi _{1} \wedge \cdots \wedge \psi _{m} \rightarrow \varphi , \end{aligned}$$

from which it follows that

$$\begin{aligned} \vdash _{\textsf{DTEL}} \Box \psi _{1} \wedge \cdots \wedge \Box \psi _{m} \rightarrow \Box \varphi , \end{aligned}$$

which implies \(\Box \varphi \in x\), a contradiction. By Lindenbaum’s lemma, we therefore obtain a point \(y \in X\) with \(y \in R(x)\) and \(\varphi \notin y\). This latter fact, by the inductive hypothesis, yields \((y,[y]) \not \models \varphi \) and thus \((y,[x]) \not \models \varphi \) (since \(y \sim x\)), which in turn yields \((x,[x]) \not \models \Box \varphi \) since R(x) is a minimal neighbourhood of x. \(\square \)

For the last inductive step a lemma will be useful.

Lemma 5

For all \(x \in X\), if \(f_{\pi }(x)\) is defined then \(f_{\pi }([x]) = [f_{\pi }(x)]\).

Proof

First suppose that \(z \in f_{\pi }([x])\), so there is some \(y \sim x\) such that \(f_{\pi }(y) = z\). We need to show that \(z \sim f_{\pi }(x)\). If \(K\psi \in z\) then so is \(KK\psi \), so \(\Circle _{\pi }KK\psi \in y\); since \(f_{\pi }(y)\) is defined, certainly \(\Circle _{\pi }\top \in y\), and therefore by (K-PC\(_{\pi }\)) we can deduce that \(K(\Circle _{\pi }\top \rightarrow \Circle _{\pi }K\psi ) \in y\). Since \(y \sim x\), this implies that \(\Circle _{\pi }\top \rightarrow \Circle _{\pi }K\psi \in x\), and therefore (since \(f_{\pi }(x)\) is defined), \(\Circle _{\pi }K\psi \in x\), hence \(K\psi \in f_{\pi }(x)\). An analogous argument shows that if \(\lnot K\psi \in z\) then \(\lnot K \psi \in f_{\pi }(x)\), which establishes that \(z \sim f_{\pi }(x)\).

Conversely, suppose that \(z \sim f_{\pi }(x)\); we wish to show that there exists a \(y \in [x]\) such that \(f_{\pi }(y) = z\). Using Lindenbaum’s lemma, we will be done if we can show that the set

$$\begin{aligned} \{K\psi \, : \, K \psi \in x\} \cup \{\Circle _{\pi }\chi \, : \, \chi \in z\} \end{aligned}$$

is \(\textsf{DTEL}\)-consistent. So suppose not; then there are \(K\psi _{1}, \ldots , K\psi _{n} \in \{K\psi \, : \, K \psi \in x\}\) and \(\Circle _{\pi }\chi _{1}, \ldots , \Circle _{\pi }\chi _{m}, \Circle _{\pi }\top \in \{\Circle _{\pi }\chi \, : \, \chi \in z\}\) such that

$$\begin{aligned} \vdash _{\textsf{DTEL}} (K\psi _{1} \wedge \cdots \wedge K\psi _{n}) \rightarrow (\Circle _{\pi }\top \rightarrow \lnot (\Circle _{\pi }\chi _{1} \wedge \cdots \wedge \Circle _{\pi }\chi _{m})), \end{aligned}$$

from which it follows (using \(\textsf{S5}_{K}\)) that

$$\begin{aligned} \vdash _{\textsf{DTEL}} (K\psi _{1} \wedge \cdots \wedge K\psi _{n}) \rightarrow K(\Circle _{\pi }\top \rightarrow \lnot \Circle _{\pi }(\chi _{1} \wedge \cdots \wedge \chi _{m})). \end{aligned}$$

Observe that \(\Circle _{\pi }\top \rightarrow \lnot \Circle _{\pi } \chi \) is propositionally equivalent to \(\Circle _{\pi }\top \rightarrow (\lnot \Circle _{\pi } \chi \wedge \Circle _{\pi }\top )\), and by (\(\lnot \)-PC\(_{\pi }\)) this is in turn equivalent in \(\textsf{DTEL}\) to \(\Circle _{\pi }\top \rightarrow \Circle _{\pi } \lnot \chi \). Hence

$$\begin{aligned} \vdash _{\textsf{DTEL}} (K\psi _{1} \wedge \cdots \wedge K\psi _{n}) \rightarrow K(\Circle _{\pi }\top \rightarrow \Circle _{\pi } \lnot (\chi _{1} \wedge \cdots \wedge \chi _{m})), \end{aligned}$$

so since \(K\psi _{1}, \ldots , K\psi _{n} \in x\), it follows that \(K(\Circle _{\pi }\top \rightarrow \Circle _{\pi } \lnot (\chi _{1} \wedge \cdots \wedge \chi _{m})) \in x\). Since \(f_{\pi }(x)\) is defined, also \(\Circle _{\pi }\top \in x\), so by (K-PC\(_{\pi }\)) we can deduce that \(\Circle _{\pi } K \lnot (\chi _{1} \wedge \cdots \wedge \chi _{m}) \in x\). We therefore have \(K \lnot (\chi _{1} \wedge \cdots \wedge \chi _{m}) \in f_{\pi }(x)\), so since \(z \sim f_{\pi }(x)\), \(\lnot (\chi _{1} \wedge \cdots \wedge \chi _{m}) \in z\), a contradiction. \(\square \)

Back to the Truth Lemma: suppose inductively the result holds for \(\varphi \) and let us show it holds for \(\Circle _{\pi } \varphi \). Observe that \(\Circle _{\pi } \varphi \in x\) iff \(f_{\pi }(x)\) is defined and \(\varphi \in f_{\pi }(x)\). By the inductive hypothesis, this is equivalent to \((f_{\pi }(x), [f_{\pi }(x)]) \models \varphi \); by Lemma 5, this is in turn equivalent to \((f_{\pi }(x), f_{\pi }[x]) \models \varphi \), which by definition holds iff \((x,[x]) \models \Circle _{\pi } \varphi \). \(\square \)

Completeness is an easy consequence: if \(\varphi \) is not a theorem of \(\textsf{DTEL}\), then \(\{\lnot \varphi \}\) is consistent and so can be extended by Lindenbaum’s lemma to some \(x \in X\); by Lemma 4, we have \((\mathcal {X},x,[x]) \not \models \varphi \). \(\square \)

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Bjorndahl, A. The Epistemology of Nondeterminism. J of Log Lang and Inf 31, 619–644 (2022). https://doi.org/10.1007/s10849-022-09389-4

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