Causal Agency and Responsibility: A Refinement of STIT Logic

Abstract

We propose a refinement of STIT logic to make it suitable to model causal agency and responsibility in basic multi-agent scenarios in which agents can interfere with one another. We do this by supplementing STIT semantics, first, with action types and, second, with a relation of opposing between action types. We exploit these novel elements to represent a test for potential causation, based on an intuitive notion of expected result of an action, and two tests for actual causation from the legal literature, i.e., the but-for and the NESS tests. We then introduce three new STIT operators modeling corresponding notions of causal responsibility, which we call potential, strong, and plain responsibility, and use them to provide a fine-grained analysis of a number of case studies involving both individual agents and groups.

Appendix: Soundness and Completeness of \( \mathcal {ALO}\)

The proof that the axiom system $\mathcal {ALO}$ is sound with respect to the class of all models for $\mathcal {L}_{\mathcal {ALO}}$ is routine, and is thus omitted. The proof of completeness consists of two main steps: (1) we identify a class of Kripke models with respect to which $\mathcal {ALO}$ is sound and complete; (2) we show that any formula satisfiable in a model of the identified class is also satisfiable in a model for $\mathcal {L}_{\mathcal {ALO}}$.

Step 1 Kripke semantics for $\mathcal {ALO}$

Definition 6

(Kripke model for $\mathcal {L}_{\mathcal {ALO}}$) A Kripke model for $\mathcal {L}_{\mathcal {ALO}}$ is a tuple $\mathcal {M}= \left\langle {W,Ag, \mathbf {T},R_{\square },R_{X},D,O, \pi }\right\rangle$ where $W\ne \varnothing$ is a set of possible worlds, $Ag$ and $\mathbf {T}$ are fixed finite sets of agents and action types, $R_{\square } \subseteq W\times W$ is an equivalence relation on W, $R_{X} \subseteq W \times W$ is a serial and functional relation on W, $D: \mathbf {IA}\rightarrow \wp (W)$ intuitively assigns to every individual action the set of worlds where it is performed, $O: \mathbf {Act}\rightarrow \wp (\mathbf {Act})$ intuitively assigns to every action the set of actions opposing it, and $\pi : Prop \rightarrow \wp (W)$ is a standard valuation. The functions D and $O$ are required to satisfy the following conditions, where, for any $I \subseteq Ag$ and $\alpha \in \mathbf {GA}$, $D(\alpha _I)= \bigcap _{i\in I}D(\alpha (i))$ (and so, for any $\alpha \in \mathbf {GA}$, $D(\alpha )= \bigcap _{i\in Ag}D(\alpha (i))$).

$$\begin{array}{ll} {\textit{\textbf{D}}} 1 &{} \text {For all } w \in W~\text {and}~i \in Ag, w\in \bigcup _{a_i \in \mathbf {A}_i} D(a_i) \\ {\textit{\textbf{D}}} 2 &{} \text {For all } a_i,b_i \in \mathbf {A}_i~\text {s.t.}~a_i \ne b_i , D(a_i)\cap D(b_i) = \varnothing \\ {\textit{\textbf{D}}} 3 &{} \text {For all } w_1,v_1,v_2 \in W,~if~w_1 \in D(\alpha )~\text {and}~w_1 R_{X} v_1 \text {and } v_1 R_{\square } v_2 , \text {then there is} \\ &{} w_2 \in W \text {s.t.} w_1 R_\square w_2, w_2 \in D(\alpha ) , \text {and}~w_2 R_{X} v_2 \\ {\textit{\textbf{D}}} 4 &{} \text {If there are}~w_1,\dots ,w_n,w \in W \text {s.t.} w_1 R_{\square } w_2 R_{\square } \dots w_n R_{\square } w~\text {and}~w_i \in D(\alpha _i) , \\ &{} \text {then there is } w' \in W~\text {s.t.}~w' R_{\square } w~\text {and}~w' \in D(\alpha )\\ { O1}&{} \text {For all } \alpha _I \in \mathbf {Act}, \alpha _I \notin O(\alpha _I)\\ { O2}&{} \text {For all } \alpha _I,\beta _J,\gamma _K\in \mathbf {Act},~\text {if}~\alpha _I\in O(\beta _J)~and~\beta _J \sqsubseteq \gamma _K ,~\text {then}~\alpha _I \in O(\gamma _K) \\ \end{array}$$

Definition 7

(Kripke semantics) Where $\varphi$ is a formula of $\mathcal {L}_{\mathcal {ALO}}$, $\mathcal {M}$ a Kripke model for $\mathcal {L}_{\mathcal {ALO}}$, and w a world in $\mathcal {M}$, the relation $\mathcal {M},w\models \varphi$ of truth of $\varphi$ at w is recursively defined by the the following conditions, where the constant $\perp$, the propositional variables, and the Boolean connectives have the usual interpretation:

$$\begin{array}{ll} \mathcal {M},w\models do(a_i) \Leftrightarrow w\in D(a_i) &{} \mathcal {M}, w \models \alpha _I \triangleright \beta _J \Leftrightarrow \alpha _I \in O(\beta _J)\\ \mathcal {M},w\models \square \varphi \Leftrightarrow \forall w'(wR_{\square } w'\Rightarrow M,w'\models \varphi )&{}\\ \mathcal {M},w\models X\varphi \Leftrightarrow \forall w'(wR_{X} w'\Rightarrow M,w'\models \varphi )&{}\\ \end{array}$$

Theorem 2

The axiom system $\mathcal {ALO}$ is sound and strongly complete with respect to the class of all Kripke models for $\mathcal {L}_{\mathcal {ALO}}$.

The proof of soundness is routine. The proof of completeness proceeds along standard lines and exploits the possibility of building a canonical model for any given $\mathcal {ALO}$-consistent set of formulas. Where w is any maximal $\mathcal {ALO}$-consistent set (henceforth: mcs), let:

• $w/\square = \{ \varphi \,|\, \square \varphi \in w\}$ and $w/X= \{ \varphi \,|\, X\varphi \in w\}$

• $pos_{\triangleright }(w) = \{ \alpha _I \triangleright \beta _J \,|\, \alpha _I \triangleright \beta _J \in w \}$ and $neg_{\triangleright }(w) = \{ \lnot ( \alpha _I \triangleright \beta _J) \,|\, \lnot ( \alpha _I \triangleright \beta _J) \in w \}$

Lemma 1

(Lindenbaum Lemma) Every consistent set can be extended to a mcs.

Proof

Standard.

Definition 8

(Canonical model) Let $w_0$ be a fixed mcs. The canonical model $\mathcal {M}$ around $w_0$ for $\mathcal {ALO}$ is a tuple $\mathcal {M}= \left\langle {W,Ag,\mathbf {T},R_{\square },R_{X},D, O, \pi }\right\rangle$, where $W = \{w \,|\, pos_{\triangleright }(w_0)\cup \, neg_{\triangleright }(w_0) \subseteq w \}$ for w ranging over mcs, $w R_{\square } v$ iff $w/\square \subseteq v$, $w R_{X} v$ iff $w/X\subseteq v$, $w\in D(a_i)$ iff $do(a_i) \in w$, $\alpha _I \in O(\beta _J)$ iff $\alpha _I \triangleright \beta _J \in w_0$, and $w\in \pi (p_i)$ iff $p_i \in w$.

The proofs of the following lemmas proceed as usual and are left to the reader.

Lemma 2

For all $w\in W$ and formula $\varphi$ of $\mathcal {L}_{\mathcal {ALO}}$, $\mathcal {M},w\models \varphi$ iff $\varphi \in w$.

Lemma 3

$\mathcal {M}$ is a Kripke model for $\mathcal {L}_{\mathcal {ALO}}$.

Since, by the Lindenbaum Lemma, any $\mathcal {ALO}$-consistent set of formulas can be extended to a mcs, Lemmas 2 and 3 imply that any $\mathcal {ALO}$-consistent set of formulas is satisfiable in a Kripke model for $\mathcal {L}_{\mathcal {ALO}}$, and hence that $\mathcal {ALO}$ is strongly complete with respect to the class of all Kripke models for $\mathcal {L}_{\mathcal {ALO}}$.

Step 2 From Kripke models for $\mathcal {L}_{\mathcal {ALO}}$ to models for $\mathcal {L}_{\mathcal {ALO}}$

Assume that $\varphi _0$ is $\mathcal {ALO}$-consistent. Then, by Theorem 2, there are a Kripke model $\mathcal {M}$ and a world $w_0\in \mathcal {M}$ such that $\mathcal {M},w_0\models \varphi _0$. We show that $\mathcal {M}$ can be transformed into a model for $\mathcal {L}_{\mathcal {ALO}}$ in which $\varphi _0$ is satisfiable. By adapting the techniques used in [21], we proceed in steps by unraveling $\mathcal {M}$.

Step 2.1 Initial unraveling

Definition 9

(Unraveling of $\mathcal {M}$) Let $\mathcal {M}'=\left\langle { W',Ag, \mathbf {T}, R'_{\square },R'_{X},D',O, \pi ' }\right\rangle$ be s.t.

• $W'$ is the set of all sequences $\overrightarrow{w_{n}}= (w_1, w_2, \dots , w_n)$ s.t. $w_1 R_{\square } w_0$ and $w_i R_{X} w_{i+1}$, for all $1 \le i < n$

• $\overrightarrow{w_{n}} R'_{\square } \overrightarrow{v_{m}}$ iff (i) $n = m$, (ii) $w_{i} R_{\square } v_{i}$ for all $1\le i \le n$, and (iii) $w_{i} \in D(a_i)$ iff $v_{i} \in D(a_i)$ for all $1\le i < n$ and $a_i \in \mathbf {IA}$

• $\overrightarrow{w_{n}} R'_{X} \overrightarrow{v_{m}}$ iff $\overrightarrow{v_{m}} = (\overrightarrow{w_{n}}, v_m)$ where $w_n R_{X} v_m$

• $\overrightarrow{w_{n}} \in D'(a_i)$ iff $w_n \in D(a_i)$

• $\overrightarrow{w_{n}} \in \pi '(p)$ iff $w_n \in \pi (p)$

Proposition 3

If $\overrightarrow{w_n} \in W'$ and $w_n R_{\square } v$, then there is $\overrightarrow{v_n}\in W'$ s.t. $v_n = v$ and $\overrightarrow{w_n} R_{\square } \overrightarrow{v_n}$.

Proof

Immediate from Definition 9, since $\mathcal {M}$ satisfies condition D3.

The following proposition follows straightforwardly from Definition 9.

Proposition 4

$\mathcal {M}'$ is a Kripke model for $\mathcal {L}_{\mathcal {ALO}}$.

Theorem 3

Let $f: W' \rightarrow W$ be the mapping defined by setting: for all $\overrightarrow{w_n} \in W'$, $f(\overrightarrow{w_n}) = w_n$. Then, for all $\overrightarrow{w_n}\in W'$, $\mathcal {M}', \overrightarrow{w_n} \models \varphi$ iff $\mathcal {M}, f( \overrightarrow{w_n}) \models \varphi$.

Proof

It suffices to show that the map $f: W' \rightarrow W$ such that $f(\overrightarrow{w_{n}}) = w_n$, for all $\overrightarrow{w_n} \in W'$ defines a bounded morphism from $\mathcal {M}'$ to $\mathcal {M}$ (see [4]). Specifically, letting $R_{\otimes }$ be either $R_{\square }$ or $R_{X}$, we need to check that

• (1) if $\overrightarrow{w_n} R'_{\otimes } \overrightarrow{v_m}$, then $f(\overrightarrow{w_n} ) R_{\otimes } f( \overrightarrow{v_m})$

• (2) if $f(\overrightarrow{w_n} ) R_{\otimes } v_m$, then there is $\overrightarrow{u_k} \in W'$ s.t. $f(\overrightarrow{u_k}) = v_m$ and $\overrightarrow{w_n} R'_{\otimes } \overrightarrow{u_k}$

• (3) $\overrightarrow{w_n} \in D'(a_i)$ iff $f(\overrightarrow{w_n}) \in D(a_i)$

• (4) $\overrightarrow{w_n} \in \pi '(p)$ iff $f(\overrightarrow{w_n}) \in \pi (p)$

The proof follows immediately from Definition 9 in all cases, except in case (2) when $\otimes$ is $\square$. But this case coincides with Proposition 3.

Corollary 1

$\mathcal {M}', \overrightarrow{w_0} \models \varphi _0$

Step 2.2   Divide histories in $\mathcal {M}'$ that remain undivided

Definition 10

Let $ND(\varphi _0)$ be the maximum degree of nested $X$ operators in $\varphi _0$ and $\mathcal {M}''= \left\langle {W',Ag, \mathbf {T}, R''_{\square },R'_{X},D',O, \pi ' }\right\rangle$, where

$\overrightarrow{w_n} R''_{\square } \overrightarrow{v_m} \text { iff } {\left\{ \begin{array}{ll} \overrightarrow{w_n} R'_{\square } \overrightarrow{v_m} &{} \text {if } n \le ND(\varphi _0) \\ \overrightarrow{w_n} = \overrightarrow{v_m} &{} \text {if } ND(\varphi _0) < n \\ \end{array}\right. }$

It is immediate to check that $\mathcal {M}''$ is still a Kripke model for $\mathcal {L}_{\mathcal {ALO}}$. The following proposition follows straightforwardly from Theorem 3 and from the fact that sequences equivalent in $\mathcal {M}'$ are separated only at an $R'_{X}$-depth that is higher than the maximum degree of nesting of $X$ operators in $\varphi _0$.

Proposition 5

$\mathcal {M}'',\overrightarrow{w_0}\models \varphi _0$.

For any sequence $\sigma$, let $last(\sigma )$ be the last element of $\sigma$.

Definition 11

(History in $\mathcal {M}''$) A history in $\mathcal {M}''$ is a function $h: \mathbb {N} \rightarrow W'$ such that (i) $h(1) R''_{\square } \overrightarrow{w_0}$ and (ii) $h(n+1) = (h(n), v)$ where $last(h(n))R_{X} v$. The set of all histories in $\mathcal {M}''$ is denoted with Hist.

The following proposition follows from Definitions 10 and 11.

Proposition 6

For every $\overrightarrow{w_n} \in W''$ there is a unique history h s.t. $h(n) = \overrightarrow{w_n}$.

We will denote with $h_{\overrightarrow{w_n}}$ the unique history associated with $\overrightarrow{w_n}$.

Step 2.3   Transform $\mathcal {M}''$ in a model for $\mathcal {L}_{\mathcal {ALO}}$

We start by defining a DBT structure based on $\mathcal {M}''$. For any $\overrightarrow{w_n} \in W'$, let $[ \overrightarrow{w_n}]$ be the equivalence class of $\overrightarrow{w_n}$ under $R''_{\square }$.

Definition 12

(DBT structure based on $\mathcal {M}''$) Define $\mathcal {T}= \left\langle Mom,<\right\rangle$ so that

• $Mom = \{ [\overrightarrow{w_n}] \,|\, n\in \mathbb {N} \text { and } \overrightarrow{w_n} \in W' \}$

• $[\overrightarrow{w_n}] < [\overrightarrow{v_m}]$ iff $n < m$ and all prefixes of length n of sequences in $[\overrightarrow{v_m}]$ are in $[\overrightarrow{w_n}]$

It is not difficult to check that $\mathcal {T}$ is a DBT structure. For any $\overrightarrow{w_n}\in W'$, let $\mathbf {X}(\overrightarrow{w_{n}})$ be the unique successor of $\overrightarrow{w_n}$ along $R'_{X}$.

Definition 13

(History in $\left\langle Mom,<\right\rangle$) A history $h^*$ in $\mathcal {T}$ is a function $h^*: \mathbb {N} \rightarrow Mom$ s.t. (i) $h^*(1) = [\overrightarrow{w_0}]$ and (ii) $h^*(n+1) = [\mathbf {X}(\overrightarrow{w_{n}}) ]$ for some $\overrightarrow{w_n} \in h^*(n)$. The set of all histories in $\left\langle Mom,<\right\rangle$ is denoted with $Hist^*$.

It is immediate to see that the following pair of functions define a one-to-one correspondence between the set of histories in $\mathcal {M}''$ and the set of histories in $\mathcal {T}$:

• $\gamma : Hist \rightarrow Hist^*$ is s.t. $\gamma _h(n)= [h(n)]$ for all $n \in \mathbb {N}$

• $\beta : Hist^* \rightarrow Hist$ is s.t. $\beta _{h^*}(n) \in h^*(n)$ for all $n \in \mathbb {N}$

Given a moment $[\overrightarrow{w_n}] \in Mom$ and a history $h^* \in Hist^*$, $h^*$ passes through m iff there is $n\in \mathbb {N}$ s.t. $h^*(n)=[ \overrightarrow{w_n}]$. The set $S([\overrightarrow{w_n}])$ of successors of $[\overrightarrow{w_n}]$ is the set of equivalence classes $[\overrightarrow{v_{n+1}}] \in Mom$ s.t. $[\overrightarrow{w_n}]$ contains the prefixes of length n of all sequences in $[\overrightarrow{v_{n+1}}]$. A transition is any pair $\tau = ([\overrightarrow{w_n}], [\overrightarrow{v_{n+1}}])$ s.t. $[\overrightarrow{v_{n+1}}] \in S([\overrightarrow{w_n}])$.

Definition 14

(Model for $\mathcal {L}_{\mathcal {ALO}}$ based on $\mathcal {M}''$) Let $M= \left\langle {\mathcal {T}, Ag,\mathbf{T} , Act, \mathbf {O},V}\right\rangle$ be s.t. $\mathcal {T}$ is as in Definition 12, $Ag$ and $\mathbf{T}$ are as in $\mathcal {M}''$, $\mathbf {O}= O$, and the functions $Act : Tr \rightarrow \mathbf {GA}$ and $V: Prop \rightarrow \wp (Ind)$ are defined as follows:

• $Act(([\overrightarrow{w_n}], [\overrightarrow{v_{n+1}}])) = \alpha$ iff $\overrightarrow{v_{n}} \in D''(\alpha )$

• $[\overrightarrow{w_n}]/h^*\in V(p_i)$ iff $\beta _{h^*}(n) \in \pi ''(p_i)$

Observe that the functions Act is well defined. Specifically:

• 1. If $[\overrightarrow{v_{n+1}}] = [\overrightarrow{u_{n+1}}]$, then $Act([\overrightarrow{w_n}], [\overrightarrow{v_{n+1}}])= Act([\overrightarrow{w_n}], [\overrightarrow{u_{n+1}}])$. In fact, if $\overrightarrow{v_{n+1}}R_{\square }'' \overrightarrow{u_{n+1}}$, then, by the definition of $R_{\square }''$, $\overrightarrow{v_n}\in D''(\alpha )$ iff $\overrightarrow{u_n}\in D''(\alpha )$.

• 2. For any transition $\tau = ([\overrightarrow{w_n}], [\overrightarrow{v_{n+1}}])$, there is $\alpha \in \mathbf {GA}$ s.t. $v_n \in D''(\alpha )$. This is an immediate consequence of condition D1.

• 3. For any transition $\tau = ([\overrightarrow{w_n}], [\overrightarrow{v_{n+1}}])$, there is a unique $\alpha \in \mathbf {GA}$ s.t. $v_n \in D''(\alpha )$. This is an immediate consequence of condition D2.

The proof of the following proposition is routine.

Proposition 7

$M$ is a model for $\mathcal {L}_{\mathcal {ALO}}$.

Given Proposition 5, the following proposition ensures that $\varphi _0$ is satisfiable in $M$.

Proposition 8

For all formulas $\varphi$ of $\mathcal {L}_{\mathcal {ALO}}$ and world $\overrightarrow{w_n}$ in $\mathcal {M}''$, $\mathcal {M}'', \overrightarrow{w_n} \models \varphi$ iff $M, [\overrightarrow{w_n}]\ \gamma _{h_{\overrightarrow{w_n}}} \models \varphi$.

Proof

The proof is by induction on the complexity of $\varphi$. We only prove the cases (1) $\varphi = X\psi$ and (2) $\varphi = do(a_i)$. The case in which $\varphi =\alpha _I \triangleright \beta _J$ is straightforward because the semantics of $\alpha _I \triangleright \beta _J$ does not depend on the index at which the formula is evaluated and because $\mathbf {O}= O''$.

$$\begin{array}{llll} (1) &{} \mathcal {M}'', \overrightarrow{w_n} \models X\psi \Leftrightarrow \mathcal {M}'', \mathbf {X}(\overrightarrow{w_{n}}) \models \psi &{}\text {by the Kripke semantics of } X\psi \\ &{} \Leftrightarrow M, [\mathbf {X}(\overrightarrow{w_{n}})]/ \gamma _{h_{\mathbf {X}(\overrightarrow{w_{n}})}} \models \psi &{} \text {by induction hypothesis}\\ &{} \Leftrightarrow M, [\mathbf {X}(\overrightarrow{w_{n}})]/ \gamma _{h_{\overrightarrow{w_{n}}}} \models \psi &{} \text {as}\, h_{\mathbf {X}(\overrightarrow{w_n})}(n) = h_{\overrightarrow{w_n}}(n) ,\text { by prop}.\,6\\ &{} \Leftrightarrow M, \gamma _{h_{\overrightarrow{w_{n}}}}(n+1)/ \gamma _{h_{\overrightarrow{w_{n}}}} \models \psi &{} \text {as}\, [\mathbf {X}(\overrightarrow{w_{n}})]= \gamma _{h_{\overrightarrow{w_{n}}}}(n+1)\,\text {by def}.\, \gamma \,\text {and prop}.\,6 \\ &{} \Leftrightarrow M, [\overrightarrow{w_{n}}]/ \gamma _{h_{\overrightarrow{w_{n}}}} \models X\psi &{} \text {by the semantics of}\, X\psi \\ \end{array}$$

$$\begin{array}{llll} (2) &{} \mathcal {M}'', \overrightarrow{w_n} \models do(a_i) \Leftrightarrow \overrightarrow{w_n} \in D''(a_i)&{} \text {by the Kripke semantics of}\, {do}(a_i) \\ &{} \Leftrightarrow Act(([\overrightarrow{w_n}], [\mathbf {X}(\overrightarrow{w_{n}})]))(i) = a_i &{}\text {by def}.\, Act \\ &{} \Leftrightarrow Act(([\overrightarrow{w_n}], \gamma _{h_{\overrightarrow{w_{n}}}}(n+1)))(i) = a_i &{} \text {as}\, [\mathbf {X}(\overrightarrow{w_{n}})]= \gamma _{h_{\overrightarrow{w_{n}}}}(n+1)\, \text {by def}.\, \gamma \text {and prop}.\, 6\\ &{} \Leftrightarrow M, [\overrightarrow{w_n}]/ \gamma _{h_{\overrightarrow{w_n}}} \models do(a_i)&{} \text {by the semantics of}\, do(a_i) \\ \end{array}$$

Since $\mathcal {M}'', \overrightarrow{w_0} \models \varphi _0$, Proposition 8 implies that $M, [\overrightarrow{w_0}]/ \gamma _{h_{\overrightarrow{w_0}}} \models \varphi _0$. Hence, any proposition satisfiable in a Kripke model for $\mathcal {L}_{\mathcal {ALO}}$ is satisfiable in a model for $\mathcal {L}_{\mathcal {ALO}}$. Given Theorem 2, this ensures that any $\mathcal {ALO}$-consistent sentence is satisfiable in a model for $\mathcal {L}_{\mathcal {ALO}}$. Thus, $\mathcal {ALO}$ is weakly complete with respect to the class of all models for $\mathcal {L}_{\mathcal {ALO}}$.