A Decidable Multi-agent Logic for Reasoning About Actions, Instruments, and Norms

Abstract

We formally introduce a novel, yet ubiquitous, category of norms: norms of instrumentality. Norms of this category describe which actions are obligatory, or prohibited, as instruments for certain purposes. We propose the Logic of Agency and Norms ($\mathsf {LAN}$) that enables reasoning about actions, instrumentality, and normative principles in a multi-agent setting. Leveraging $\mathsf {LAN}$, we formalize norms of instrumentality and compare them to two prevalent norm categories: norms to be and norms to do. Last, we pose principles relating the three categories and evaluate their validity vis-à-vis notions of deliberative acting. On a technical note, the logic will be shown decidable via the finite model property.

A Finite Model Property and Decidability

In this appendix, we provide the main technical contribution of this paper: we show that $\mathsf {LAN}$ is decidable (Corollary 1), via proving the finite model property (FMP) for the logic (Theorem 2). Our strategy is, accordingly: first, we show that every satisfiable formula is satisfiable on a treelike model (Lemma 1). Second, we show that the depth of the treelike model can be bounded (Lemma 2). Last, we prove that the breadth of the model can be bounded (Lemma 3).

Lemma 1

Every formula $\phi \in \mathcal {L}_{\mathsf {LAN}}$ satisfiable on a $\mathsf {LAN}$-model, is satisfiable at the root of a treelike $\mathsf {LAN}$-model.

Proof

Let $M = (W, \{W_{\mathfrak {d}^{\alpha _{i}}_{j}}:\mathfrak {d}^{\alpha _{i}}_{j} \in \mathcal {L}_{\mathsf {LAN}}\}, \{W_{\mathfrak {v}^{\alpha _i}}:\mathfrak {v}^{\alpha _i}\in \mathcal {L}_{\mathsf {LAN}}\}, R, R_{\mathsf {N}},V)$ be a $\mathsf {LAN}$-model with $w \in W$ and assume $M, w \models \phi$ (i.e. $\phi$ is satisfiable). To show that $\phi$ is satisfiable at the root of a treelike model we evoke an unraveling procedure similar to the one in [5, Ch. 2.1]. We define the treelike model $M^t$ as follows:

$$\begin{aligned} M^t = (W^t, \{W_{\mathfrak {d}^{\alpha _{i}}_{j}}^t:\mathfrak {d}^{\alpha _{i}}_{j} \in \mathcal {L}_{\mathsf {LAN}}\}, \{W_{\mathfrak {v}^{\alpha _i}}^t:\mathfrak {v}^{\alpha _i}\in \mathcal {L}_{\mathsf {LAN}}\}, R^t, R_{\mathsf {N}}^t,V^t),\text { where} \end{aligned}$$

• $W^t \subseteq \bigcup _{n \in \mathbb {N}} W^{n}$ is the set of all finite sequences $(w,w_{1},{...},w_{n})$ s.t. $wRw_{1}$, $w_{1}Rw_{2}$, ..., $w_{n-1}Rw_{n}$;

• For each $\alpha _{i} \in Agt$ and each $\mathfrak {d}^{\alpha _{i}}_{j} \in Wit^{\alpha _{i}}$, $W_{\mathfrak {d}^{\alpha _{i}}_{j}}^t \subseteq W^t$ is the set of all finite sequences $(w,w_{1},{...},w_{n})$ s.t. $w_{n} \in W_{\mathfrak {d}^{\alpha _{i}}_{j}}$;

• For each $\alpha _{i} \in Agt$, $W_{\mathfrak {v}^{\alpha _i}}^t \subseteq W^t$ is the set of all finite sequences $(w,w_{1},{...},w_{n})$ s.t. $w_{n} \in W_{\mathfrak {v}^{\alpha _i}}$;

• For all $\varvec{w}, \varvec{u} \in W^t$, $\varvec{w}R^t\varvec{u}$ iff $\varvec{w} = (w,w_{1},{...},w_{n})$, $\varvec{u} = (w,w_{1}{...},w_{n},w_{n+1})$, and $w_{n}Rw_{n+1}$;

• For all $\varvec{w}, \varvec{u} \in W^t$, $\varvec{w}R_{\mathsf {N}}^t\varvec{u}$ iff $\varvec{w} = (w,w_{1},{...},w_{n})$, $\varvec{u} = (w,w_{1}{...},w_{n},w_{n+1})$, and $w_{n}R_{\mathsf {N}}w_{n+1}$;

• For all $\varvec{w} \in W^t$, $\varvec{w} = (w,w_{1},{...},w_{n}) \in V^t(p)$ iff $w_{n} \in V(p)$.

The model $M^t$ is clearly treelike. Further, Prop. 2.14 and 2.15 of [5] imply:

• (1) For any formula $\psi \in \mathcal {L}_{\mathsf {LAN}}$, each $u \in W$, and each $\varvec{u} \in W^t$ of the form

  $$\begin{aligned} (w, w_{1}, {...}, u), \text { we have that } M,u \models \psi \text { iff } M^t,\varvec{u} \models \psi . \end{aligned}$$

This result, together with the assumption $M, w \models \phi$, implies $M^t, (w) \models \phi$, where (w) is the root of the treelike model $M^t$. To complete the proof, we must argue that $M^t$ is a $\mathsf {LAN}$-model, i.e., it satisfies conditions (R3)–(R6) of Definition 5:

(R3) Let $\varvec{w}, \varvec{u}, \varvec{v} \in W^t$ and suppose $\varvec{w}R_{\mathsf {N}}^t\varvec{u}$ and $\varvec{w}R_{\mathsf {N}}^t\varvec{v}$. By definition of $R_{\mathsf {N}}^t$ we get (i) $\varvec{w}$ is a sequence of the form $(w,w_{1},{...},w_{n})$, (ii) $\varvec{u}$ is a sequence $(w,w_{1},{...},w_{n},w_{n+1})$, (iii) $\varvec{v}$ is a sequence $(w,w_{1},{...},w_{n},w_{n+1}')$, (iv) $w_{n}R_{\mathsf {N}}w_{n+1}$, and (v) $w_{n}R_{\mathsf {N}}w_{n+1}'$. Since the original model M satisfies (R3), it follows from (iv) and (v) that $w_{n+1} = w_{n+1}'$, which, together with (ii) and (iii), implies $\varvec{u} = \varvec{v}$.

(R4) Let $\varvec{w}, \varvec{u} \in W^t$ and assume $\varvec{w}R_{\mathsf {N}}^t\varvec{u}$. By definition of $R_{\mathsf {N}}^t$ we get (i) $\varvec{w}$ is a sequence of the form $(w,w_{1},{...},w_{n})$, (ii) $\varvec{u}$ is a sequence $(w,w_{1},{...},w_{n},w_{n+1})$, and (iii) $w_{n}R_{\mathsf {N}}w_{n+1}$. Since the original model M satisfies (R4), it follows from (iii) that $w_{n}Rw_{n+1}$, which, together with (i) and (ii), implies $\varvec{w}R^t\varvec{u}$.

(R5) Let $\varvec{w} \in W^t$ and $Agt = \{\alpha _{1}, ..., \alpha _{n}\}$. Suppose there are (not necessarily distinct) action-types $\varDelta _{1}, {...}, \varDelta _{n} \in Act_{\mathsf {LAN}}$ s.t. for $1 \le i \le n$ there exist $\varvec{u}_{i} \in W^t$ s.t. $\varvec{w}R^t\varvec{u}_{i}$ and $\varvec{u}_{i} \in W_{t(\varDelta _{i}^{\alpha _{i}})}^{t}$. It follows that $\varvec{w}$ is of the form $(w,w_{1},{...},w_{n})$ and each $\varvec{u}_{i}$ is of the form $(w,w_{1},{...},w_{n},u_{i})$ with $w_{n}Ru_{i}$. The model M satisfies condition (R5), and hence there exists a world $v \in W$ s.t. $w_{n}Rv$ and $v \in W_{t(\varDelta _{1}^{\alpha _{1}})} \cap \cdots \cap W_{t(\varDelta _{n}^{\alpha _{n}})}$. By definition of $M^t$, we have $\varvec{v} = (w,w_{1},{...},w_{n},v) \in W^t$, implying that $\varvec{w}R^{t}\varvec{v}$ and $\varvec{v} \in W_{t(\varDelta _{1}^{\alpha _{1}})}^{t} \cap \cdots \cap W_{t(\varDelta _{n}^{\alpha _{n}})}^{t}$.

(R6) Let $\varvec{w} \in W^t$ and $\alpha _{i} \in Agt$. Assume there is a $\varvec{v} \in W^t$ s.t. $\varvec{w}R^t\varvec{v}$ and $\varvec{v} \in W_{\mathfrak {v}^{\alpha _i}}^{t}$. This implies $\varvec{w} = (w,w_{1},{...},w_{n})$ and $\varvec{v} = (w,w_{1},{...},w_{n},v)$ with $w_{n}Rv$. Since M satisfies (R6), there is a u s.t. $w_{n}Ru$ and $u \in W - W_{\mathfrak {v}^{\alpha _i}}$. By definition of $M^t$, there is a $\varvec{u} = (w,w_{1},{...},w_{n},u) \in W^t$ s.t. $\varvec{w}R^t\varvec{u}$ and $\varvec{u} \in W^{t} - W_{\mathfrak {v}^{\alpha _i}}^{t}$.

For the second transformation we define the following auxiliary concepts:

Definition 7

(Degree $deg(\cdot )$). The modal degree is recursively defined as:

• $deg(p) = deg(\mathfrak {d}^{\alpha _{i}}_{j}) = deg(\mathfrak {v}^{\alpha _i}) = 0$;

• $deg(\lnot \phi ) = \deg (\phi )$;

• $deg(\phi \rightarrow \psi ) = max\{\deg (\phi ),deg(\psi )\}$;

• .

Definition 8

(Height $height(\cdot )$ and Depth). Let M be a treelike model. We define the height of a node w in M recursively as follows:

• $height(w) = 0$, if w is the root of M;

• $height(w) = height(u) + 1$, if uRw in M.

The depth of M is the maximum height among all the worlds in M.

Lemma 2

Every formula $\phi$ satisfiable at the root of a treelike $\mathsf {LAN}$-model, is satisfiable at the root of a treelike $\mathsf {LAN}$-model with finite depth (specifically, with a depth equal to $deg(\phi )$).

Proof

Let $M = (W, \{W_{\mathfrak {d}^{\alpha _{i}}_{j}}:\mathfrak {d}^{\alpha _{i}}_{j} \in \mathcal {L}_{\mathsf {LAN}}\}, \{W_{\mathfrak {v}^{\alpha _i}}:\mathfrak {v}^{\alpha _i}\in \mathcal {L}_{\mathsf {LAN}}\}, R, R_{\mathsf {N}},V)$ be a treelike $\mathsf {LAN}$-model with root $w\in W$ and assume $M, w \models \phi$. We first construct a treelike model $M^d$ of finite depth by restricting the depth of $M^d$ to $deg(\phi )$ and argue that $\phi$ is satisfiable at the root w of $M^{d}$. We define $M^d$ as follows:

$$\begin{aligned} M^{d} = (W^{d}, \{W_{\mathfrak {d}^{\alpha _{i}}_{j}}^{d}:\mathfrak {d}^{\alpha _{i}}_{j} \in \mathcal {L}_{\mathsf {LAN}}\}, \{W_{\mathfrak {v}^{\alpha _i}}^{d}:\mathfrak {v}^{\alpha _i}\in \mathcal {L}_{\mathsf {LAN}}\}, R^{d}, R_{\mathsf {N}}^{d},V^{d}), \text { where} \end{aligned}$$

• For all $w \in W$, $w \in W^{d}$ iff $height(w) \le deg(\phi )$;

• For all $\mathfrak {d}^{\alpha _{i}}_{j} \in \mathcal {L}_{\mathsf {LAN}}$, $W_{\mathfrak {d}^{\alpha _{i}}_{j}}^{d} = W_{\mathfrak {d}^{\alpha _{i}}_{j}} \cap W^{d}$;

• For all $\mathfrak {v}^{\alpha _i}\in \mathcal {L}_{\mathsf {LAN}}$, $W_{\mathfrak {v}^{\alpha _i}}^{d} = W_{\mathfrak {v}^{\alpha _i}} \cap W^{d}$;

• $R^{d} = R \cap (W^{d} \times W^{d})$;

• $R_{\mathsf {N}}^{d} = R_{\mathsf {N}} \cap (W^d \times W^{d})$;

• For all $p \in Var$, $V^{d}(p) = V(p) \cap W^{d}$.

The model $M^d$ is treelike with finite depth. Further, Lem. 2.33 in [5] gives us:

From (2) we conclude that $M^{d},w \models \phi$. Last, we show that $M^{d}$ is a $\mathsf {LAN}$-model:

(R3) Let $w,u,v \in W^{d}$ and assume $wR_{\mathsf {N}}^{d}u$ and $wR_{\mathsf {N}}^{d}v$. By definition of $M^{d}$, we know that $w,u,v \in W$ and that $wR_{\mathsf {N}}u$ and $wR_{\mathsf {N}}v$. Since the original model M satisfies property (R3), we have that $u = v$.

(R4) Let $w,u \in W^{d}$ and assume $wR_{\mathsf {N}}^{d}u$. By definition of $M^{d}$, we get $w,u \in W$ and $wR_{\mathsf {N}}u$. Since M satisfies property (R4), it follows that wRu. By the fact that $w,u \in W^{d}$ and the definition of $M^{d}$, we obtain $wR^{d}u$.

(R5) Let $w \in W^{d}$ and $Agt = \{\alpha _{1}, ..., \alpha _{n}\}$. Suppose there are (not necessarily distinct) complex action-types $\varDelta _{1}, {...}, \varDelta _{n} \in Act_{\mathsf {LAN}}$ s.t. for $1 \le i \le n$ there exist $u_{i} \in W^{d}$ s.t. $wR^{d}u_{i}$ and $u_{i} \in W_{t(\varDelta _{i}^{\alpha _{i}})}^{d}$. By definition of $M^{d}$, it follows that $wRu_{i}$ holds for each $i \in \{1, {...}, n\}$ with $height(u_{i}) \le deg(\phi )$. Since M satisfies (R5), we know there exists a $v \in W$ s.t. wRv and $v \in W_{t(\varDelta _{1}^{\alpha _{1}})} \cap \cdots \cap W_{t(\varDelta _{n}^{\alpha _{n}})}$. We know $v \in W^{d}$ since $height(v) = height(u_{i}) \le deg(\phi )$, which implies $wR^{d}v$ and $v \in W_{t(\varDelta _{1}^{\alpha _{1}})}^{d} \cap \cdots \cap W_{t(\varDelta _{n}^{\alpha _{n}})}^{d}$ by definition of $M^{d}$.

(R6) Let $w \in W^{d}$ and $\alpha _{i} \in Agt$. Assume there exists a $v \in W^{d}$ s.t. $wR^{d}v$ and $v \in W_{\mathfrak {v}^{\alpha _i}}^{d}$. By definition of $M^{d}$, we know that wRv holds with $height(v) \le deg(\phi )$. Since M satisfies (R6), we know there exists a $u \in W$ s.t. wRu and $u \in W - W_{\mathfrak {v}^{\alpha _i}}$. Since $height(u) = height(v) \le deg(\phi )$, it follows that $u \in W^{d}$, $wR^{d}u$, and $u \in W^{d} - W_{\mathfrak {v}^{\alpha _i}}^{d}$.

Lemma 3

Every formula $\phi$ satisfiable at the root of a treelike $\mathsf {LAN}$-model with finite depth equal to $deg(\phi )$, is satisfiable at the root of a treelike $\mathsf {LAN}$-model with finite depth and finite branching (i.e., $\phi$ is satisfiable on a finite model).

Proof

Let $M = (W, \{W_{\mathfrak {d}^{\alpha _{i}}_{j}}:\mathfrak {d}^{\alpha _{i}}_{j} \in \mathcal {L}_{\mathsf {LAN}}\}, \{W_{\mathfrak {v}^{\alpha _i}}:\mathfrak {v}^{\alpha _i}\in \mathcal {L}_{\mathsf {LAN}}\}, R, R_{\mathsf {N}},V)$ be a treelike $\mathsf {LAN}$-model with depth equal to $deg(\phi )$ with root $w\in W$ and assume $M, w \models \phi$. Let $Var(\phi )$ be the set of propositional variables occurring in $\phi$. We define the set Atoms as $Var(\phi ) \cup Wit \cup \{\mathfrak {v}^{\alpha _i}: \alpha _{i} \in Agt\}$. By Prop. 2.29 in [5], we know there are only a finite number of modal formulae (up to logical equivalence) built from the finite set Atoms with degree less than or equal to $deg(\phi )$. We use $\varTheta$ to denote this collection of (equivalence classes of) formulae.

Using $\varTheta$, we first provide a selection procedure, similar to Thm. 2.34 of [5], to construct a finite model $M^f$ and show that the root of this model satisfies $\phi$. Last, we show that $M^f$ is indeed a $\mathsf {LAN}$-model. We construct $M^f$ as follows:

$$M^{f} = (W^{f}, \{W_{\mathfrak {d}^{\alpha _{i}}_{j}}^{f}:\mathfrak {d}^{\alpha _{i}}_{j} \in \mathcal {L}_{\mathsf {LAN}}\}, \{W_{\mathfrak {v}^{\alpha _i}}^{f}:\mathfrak {v}^{\alpha _i}\in \mathcal {L}_{\mathsf {LAN}}\}, R^{f}, R_{\mathsf {N}}^{f},V^{f}), \text { where} $$

• $W^{f}$ is the set obtained from the selection procedure (below);

• For all $\mathfrak {d}^{\alpha _{i}}_{j} \in \mathcal {L}_{\mathsf {LAN}}$, $W_{\mathfrak {d}^{\alpha _{i}}_{j}}^{f} = W_{\mathfrak {d}^{\alpha _{i}}_{j}} \cap W^{f}$;

• For all $\mathfrak {v}^{\alpha _i}\in \mathcal {L}_{\mathsf {LAN}}$, $W_{\mathfrak {v}^{\alpha _i}}^{f} = W_{\mathfrak {v}^{\alpha _i}} \cap W^{f}$;

• $R^{f} = R \cap (W^{f} \times W^{f})$;

• $R_{\mathsf {N}}^{f} = R_{\mathsf {N}} \cap (W^{f} \times W^{f})$;

• For all $p \in Var$, $V^{f}(p) = V(p) \cap W^{f}$.

Selection Procedure. We build our domain $W^{f}$ by selecting a sequence of states $S_{0}, S_{1}, ..., S_{deg(\phi )}$ up to a height of $deg(\phi )$, where $S_{0} = \{w\}$. Each subscript i of $S_{i}$ represents that the states contained in the associated set are at a height of i in the original model M. Suppose that the sets $S_{0}$, $S_{1}$, ..., $S_{i}$ have already been chosen; we now explain how to select the set $S_{i+1}$ with $i+1 \le deg(\phi )$. For each formula $\psi \in \varTheta$ equivalent to a formula of the form or $\langle \mathsf {N}\rangle \chi$ with $deg(\psi ) \le deg(\phi ) - i$ s.t. $M,u \models \psi$ for some $u \in S_{i} \subseteq W$, we choose a $v \in W$ s.t. uRv (or, $uR_{\mathsf {N}}v$, depending on the modality in $\psi$) and $M,v \models \chi$. We define the domain $W^{f} = S_{0} \cup S_{1} \cup {...} \cup S_{deg(\phi )}$.

The next statement is a consequence of this selection procedure [5, pp. 76–77]:

• (3) For any formula $\psi \in \varTheta$ s.t. $deg(\psi ) \le deg(\phi )$ and any world $u \in W^{f}$ s.t.

  $$\begin{aligned} height(u) \le deg(\phi ) - \deg (\psi ), M,u \models \psi \text { iff }M^{f},u \models \psi . \end{aligned}$$

From (3), together with $M,w\models \phi$, $\phi \in \varTheta$, $deg(\phi ) \le deg(\phi )$, $w \in W^{f}\!$, and $height(w) \le deg(\phi )$, we infer $M^{f},w \models \phi$. We show that $M^{f}$ is an $\mathsf {LAN}$-model:

(R3) Let $w, u, v \in W^{f}$ and assume $wR_{\mathsf {N}}^{f}u$ and $wR_{\mathsf {N}}^{f}v$. By definition of $M^{f}$, $wR_{\mathsf {N}}u$ and $wR_{\mathsf {N}}v$ hold. Since the model M satisfies (R3), we obtain $u=v$.

(R4) Let $w, u \in W^{f}$ and assume $wR_{\mathsf {N}}^{f}u$. By definition of $M^{f}$, $wR_{\mathsf {N}}u$ must hold. Since the original model M satisfies (R4), we have wRu, and because $R^{f}$ is the set R restricted to $W^{f}$, which contains w and u, we infer $wR^{f}u$.

(R5) Let $w \in W^{f}$ and let $Agt = \{\alpha _{1}, {...}, \alpha _{n}\}$. Suppose there are (not necessarily distinct) complex action-types $\varDelta _{1}, ..., \varDelta _{n}\in Act_{\mathsf {LAN}}$ s.t. for all $1 \le i \le n$ there exists a $u_{i} \in W^{f}$ s.t. $wR^{f}u_{i}$ and $u_{i} \in W_{t(\varDelta _{i}^{\alpha _{i}})}^{f}$. By definition of $M^{f}$, this implies $wRu_{i}$, $u_{i} \in W_{t(\varDelta _{i}^{\alpha _{i}})}$, and $height(u_{i}) \le deg(\phi )$ for each $i \in \{1, {...}, n\}$. Since M satisfies (R5), we know that there exists a v such that wRv and $v \in W_{t(\varDelta _{1}^{\alpha _{1}})} \cap \cdots \cap W_{t(\varDelta _{n}^{\alpha _{n}})}$, i.e., . Observe that because $height(w) + 1 = height(u_{i}) \le deg(\phi )$ that $1 \le deg(\phi )$, implying that , because $deg(\bigwedge _{1 \le i \le n}t(\varDelta _{i}^{\alpha _{i}})) = 0$. Consequently, by the selection procedure a $v' \in W$ such that $wRv'$ and $M, v' \models \bigwedge _{1 \le i \le n}t(\varDelta _{i}^{\alpha _{i}})$ must have been selected and placed in $S_{height(v')}$. Hence, there exists a $v' \in W^{f}$ s.t. $wR^{f}v'$ and $v' \in W_{t(\varDelta _{1}^{\alpha _{1}})}^{f} \cap \cdots \cap W_{t(\varDelta _{n}^{\alpha _{n}})}^{f}$.

(R6) Let $w \in W^{f}$, $\alpha _{i}\in Agt$, and assume there is a $v \in W^{f}$ s.t. $wR^{f}v$ and $v \in W_{\mathfrak {v}^{\alpha _i}}^{f}$. By definition of $M^{f}$ we infer wRv and $v \in W_{\mathfrak {v}^{\alpha _i}}$ with $height(v) \le deg(\phi )$; hence, there exists a $u \in W$ s.t. wRu and $u \in W - W_{\mathfrak {v}^{\alpha _i}}$ with $height(u) \le deg(\phi )$. It follows that . Since $height(w) = height(v) + 1 \le deg(\phi )$, we know that $1 \le deg(\phi )$, and so, . By the selection procedure, a $u' \in W$ s.t. $wRu'$ and $u' \in W - W_{\mathfrak {v}^{\alpha _i}}$ must have been chosen and placed in $S_{height(u)}$; hence, $u' \in W^{f}$, $wR^{f}u'$, and $u' \in W^{f} - W_{\mathfrak {v}^{\alpha _i}}^{f}$.

Theorem 2. $\mathsf {LAN}$ has the finite (tree) model property, i.e., every satsifiable formula is satisfiable on a finite, treelike model.

Proof

Follows from Lemmas 1, 2, and 3.

Corollary 1. The satisfiability problem of $\mathsf {LAN}$ is decidable.

Proof

By [5, Thm. 6.15], we know that if a normal modal logic is finitely axiomatizable and has the FMP, then it is decidable, which is the case for $\mathsf {LAN}$.