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Three Traditions in the Logic of Action: Bringing them Together

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Krister Segerberg on Logic of Actions

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 1))

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We propose a Dynamic Logic of Propositional Control (DL-PC) that is equipped with two dynamic modal operators: one of ability and one of action. We integrate into DL-PC the concept of ‘seeing to it that’ (abbreviated by stit) as studied by Belnap, Horty and others. We prove decidability of DL-PC satisfiability and establish the relation with the logic of the Chellas stit opertor.

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  1. 1.

    Our syntax is actually a bit more restrictive: instead of \(p \!\! \leftarrow \!\! \varphi \) it only allows for assignments to either true or false, written \(+ p\) and \(- p\). The more general assignment \(p \!\! \leftarrow \!\! \varphi \) can however be simulated by the dynamic logic program \( ( \varphi ? ; + p) \cup ( \lnot \varphi ? ; - p) \), where ‘\(?\)’ is test, ‘\(;\)’ is sequential composition, and ‘\(;\cup \)’ is nondeterministic composition.

  2. 2.

    To see this, suppose that \({\langle \!\langle }{i, +p}{\rangle \!\rangle } \top \) is the only SFA of \(\varphi \) such that \(\mathcal {I}( \nu _{ {\langle \!\langle }{i, +p}{\rangle \!\rangle }\top } ) = 1 \). Then \(\mathcal {S}_\mathcal {I}( {\mathrm {nil}}) = \{ ( i, +p) \} \). Now suppose \(\mathcal {I}\) is such that \(\mathcal {I}( \nu _{ {\mathrm {X}}{\langle \!\langle }{i, +p}{\rangle \!\rangle }\top } ) = 1 \) and \(\mathcal {I}( \nu _{ \langle i, +p \rangle {\langle \!\langle }{i, +p}{\rangle \!\rangle }\top } ) = 0 \): then \(\mathcal {S}_\mathcal {I}\) would be ill-defined if we hadn’t we introduced the fresh action \(( i_0, +\nu )\).

  3. 3.

    The original definition is equivalent to ours in the case of discrete BT structures: it stipulates that there is some \(m'\) such that \(m< m'\) and \(m' \) belongs to both \(h_1\) and \(h_2\).

  4. 4.

    The original definition is equivalent: \(\mathcal {C}\) is function from \(\mathsf {Ag}\times {\mathsf {Mom}}\) to \(2^{2^\mathcal {H}}\) mapping each agent and each moment into a partition of \(\mathcal {H}_m\).

  5. 5.

    The original definition is: for every moment \(m\), if \(H_{i}\) is some set in \(\mathcal {C}(i, m)\) for every \(i\in \mathsf {Ag}\) then \(\bigcap _{i\in \mathsf {Ag}} H_{i} \ne \emptyset \).


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The first and third author acknowledge the support of the EU coordinated action SINTELNET. The fourth author acknowledges the support of the program Marie Curie People Action Trentino (project LASTS).

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Correspondence to Andreas Herzig .

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Herzig, A., Lima, T., Lorini, E., Troquard, N. (2014). Three Traditions in the Logic of Action: Bringing them Together. In: Trypuz, R. (eds) Krister Segerberg on Logic of Actions. Outstanding Contributions to Logic, vol 1. Springer, Dordrecht.

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