Complexity of the Model Checking problem for inquisitive propositional and modal logic

TL;DR

We analyze the model checking problem for inquisitive propositional logic $\inqB$ and inquisitive modal logic $\inqM$, establishing that both are $\PSPACE$-complete. The approach combines an alternating-Turing-machine algorithm to place $\MC(\inqM)$ in $\PSPACE$ with a polynomial-space reduction from $\mathrm{TQBF}$ to $\MC(\inqB)$ to prove hardness, yielding the overall result. Central to the construction are switching models that encode Boolean valuations as information states, a collection of special formulas that capture valuation properties, and a translation of quantified Boolean formulas into inquisitive formulas whose truth is preserved under the information-state semantics. These results position inquisitive logics firmly within the PSPACE-complete landscape and suggest avenues for extending the techniques to related inquisitive frameworks.

Abstract

The aim of this paper is to study the complexity of the model checking problem MC for inquisitive propositional logic InqB and for inquisitive modal logic InqM, that is, the problem of deciding whether a given finite structure for the logic satisfies a given formula. In recent years, this problem has been thoroughly investigated for several variations of dependence and teams logics, systems closely related to inquisitive logic. Building upon some ideas presented by Yang, we prove that the model checking problems for InqB and InqM are both AP-complete.

Figures (6)

• Figure 1: A graphical representation of an information model for $\inqM$ over the set of atoms $\mathrm{AP} = \{p_0, p_1\}$ and with set of worlds $W = \{w_0,w_1,w_2\}$. The valuation function $V$ is represented by the colored rectangles in the first image: $V(p_0) = \{ w_0, w_2\}$ (the red rectangle) and $V(p_1) = \{w_0,w_1\}$ (the blue rectangle). The map $\Sigma$ is represented in the other images by depicting the maximal states of each $\Sigma(w)$ for $w\in W$. So for example $\Sigma(w_2) = \{\, \{w_0,w_1\},\, \{w_0,w_2\} \,\}$, as depicted in the last image. Notice that to obtain a graphical representation of an information model for $\inqB$, we require only the first image of the sequence, that is, a representation of the valuation function $V$.
• Figure 2: The encoding of the information model depicted in Figure \ref{['fig:inqModel']}. Each bit of the encoding correspond to the valuation of a propositional atom at a certain world. For example, the bit $\mathtt{1}$ at position $0$ indicates that $w_0 \in V(p_0)$, while the bit $\mathtt{0}$ at position $2$ indicates that $w_0 \notin V(p_2)$.
• Figure 3: The switching model $\mathcal{S}_l$. The circles represent the worlds of the model ($w_1^+,w_1^-,\dots$). The extension of the atomic propositions is depicted by the colored rectangles (in green the atoms $p_i$, in blue the atoms $q_i$). For example, the set $V_l(p_2) = \{ w_2^+ \}$ corresponds to the second (from the left) green rectangle.
• Figure 4: A $4$-switching $s$ (on the left) and a $2$-switching $t$ (on the right) over the model $\mathcal{S}_4$. $s$ corresponds to the Boolean valuation $\sigma: \mathrm{AP}_4 \to \{0,1\}$ defined as $\sigma(x_0) = 1$, $\sigma(x_1) = 0$, $\sigma(x_2) = 1$ and $\sigma(x_3) = 1$. $t$ corresponds to the Boolean valuation $\tau: \mathrm{AP}_2 \to \{0,1\}$ defined as $\tau(x_0) = 1$ and $\tau(x_1) = 0$.
• Figure 5: Semantics of $C_k^+$ and $C_k^-$.

Theorems & Definitions (22)

• Definition 1: Syntax of $\inqB$ and $\inqM$
• Definition 2: Information models
• Definition 3: Semantics of $\inqM$
• Proposition 4
• Proposition 5
• proof
• Theorem 6
• Definition 7: Switching model
• Definition 8
• Lemma 9