Shanxia Wang

Shanxia Wang

We introduce Value Coalition Logic, a typed assignment-based reconstruction of classical coalition logic. The strategic semantics is unchanged: coalitional ability is still interpreted by the standard one-step game-form clause. The change is at the atomic level. Instead of flat propositional valuations, states carry total assignments of values to finitely typed variables. As a result, exhaustivity and mutual exclusion of alternative values are built into the semantics, rather than imposed as external coherence constraints. We prove that, over each fixed finite typed signature, Value Coalition Logic is truth-equivalent to propositional coalition logic over coherent valuations. This correspondence yields a sound and complete Hilbert-style axiomatisation obtained by adding finite-domain value-coherence axioms to the standard axioms of coalition logic. The main contribution is structural. Projecting ordinary coalitional ability onto a single value domain yields quotient game forms, projected effectivity families, and strategic value-range hypergraphs. These structures support set-valued strategic exclusion, transversal polarity for disjoint coalitions, exact boundary duality between the empty and grand coalitions, and a measure of residual value indeterminacy. Thus the logic is conservative in its strategic modality, but exposes value-level invariants that are hidden in flat propositional encodings.

Shanxia Wang

Coalition Logic studies what coalitions can enforce. Recent work treats inability as simple non-ability: $\neg\Eff{C}φ$. This conflates two distinct configurations -- a coalition unable to force $φ$ may still force $\negφ$, retaining adversarial control rather than genuine inability. We introduce \textbf{Full Inability} ($\FI$): the symmetric condition in which a coalition can enforce neither a proposition nor its negation. Combining coalitional effectivity with propositional negation yields a four-fold spectrum: \textbf{Full Control} ($\FC$), \textbf{Positive Determination} ($\PD$), \textbf{Adverse Determination} ($\AD$), and \textbf{Full Inability} ($\FI$). These categories partition a coalition's strategic status exhaustively and exclusively. We establish their algebraic and order-theoretic structure. Under $α$-duality, propositional negation and coalition complementation generate a Klein four-group symmetry. In playable models, the four power regions are order-convex in the powerset lattice, yielding interval-stable verification of inability. We axiomatize $\CLFI$, a definitional extension treating Full Inability as a primitive modality. Via elimination translation, we prove soundness, completeness, and conservativity over Coalition Logic. The extension preserves expressive power and complexity ($\PSPACE$-complete), but provides direct proof-theoretic access to symmetric inability, strategic dependence, propositional dummyhood, and containment verification.

Shanxia Wang

Coalition Logic is primarily concerned with what coalitions can achieve, whereas what coalitions cannot achieve -- their \emph{inability} -- has received comparatively little explicit attention. This asymmetry matters in artificial intelligence and safety-critical multi-agent systems, where one often needs to specify not merely what agents are instructed or disposed not to do, but what they are \emph{unable} to bring about. We develop a conservative extension of Coalition Logic with an explicit inability operator, interpreted as the negation of coalition ability. This operator is introduced as a conservative and formally tractable starting point for studying inability as a modal concept in its own right. We prove soundness, completeness, and conservativity over standard Coalition Logic, and analyse the resulting modal behaviour: anti-monotonicity with respect to coalition inclusion, contravariance with respect to goal strength, asymmetric interaction with conjunction and disjunction, failure of superadditivity, non-equivalence with opponent ability, and the connection between grand-coalition inability and systemic impossibility. Making this definable operator explicit reveals a systematic modal structure governing the limits of agency, and supports reasoning about constraints, negative capabilities, and impossibility in multi-agent systems.

Shanxia Wang

We develop a logic of secrecy on simplicial models for multi-agent systems. Standard simplicial models provide a geometric semantics for knowledge by representing global states as facets of a chromatic simplicial complex and agents' local states as coloured vertices. However, secrecy cannot in general be captured as a genuinely new modality by relying on the ordinary simplicial knowledge structure alone. This motivates the introduction of an additional secrecy layer. To this end, we define \emph{simplicial secrecy models}, which enrich standard simplicial epistemic models with agent-relative secrecy neighborhood functions attached to local states. On this basis, we introduce a primitive secrecy operator $S_aφ$. Semantically, $S_aφ$ holds when agent $a$ knows $φ$ in the ordinary simplicial sense and, moreover, the truth set of $φ$ belongs to one of the designated secrecy neighborhoods associated with $a$'s current local state. The clause for secrecy thus combines an ordinary knowledge requirement with an additional local-state-based neighborhood requirement, while the frame condition ensures that designated secrecy events remain non-trivial from the perspective of every other agent. We formulate a system $\mathsf{SSL}$ for the resulting language and show that it is sound with respect to the class of simplicial secrecy models. For the genuinely multi-agent case $|A|\ge 2$, we prove completeness by first constructing an auxiliary-colour canonical model and then representing it inside the original class of pure $A$-chromatic simplicial secrecy models. The resulting framework yields a primitive, local-state-based, and geometrically grounded account of secrecy on simplicial models, together with a sound axiomatization and, in the genuinely multi-agent case, a complete one.

Shanxia Wang

We introduce a dual-threshold probabilistic knowing value logic for uncertain multi-agent settings. The framework captures within a single formalism both probabilistic-threshold attitudes toward propositions and high-confidence attitudes toward term values, thereby connecting probabilistic epistemic logic with classical knowing value logic. It is especially motivated by privacy-sensitive scenarios in which an attacker assigns high posterior probability to a candidate sensitive value without guaranteeing that it is the true one. The main idea is to separate the threshold domains of propositional and value-oriented operators. While $K_i^θ$ ranges over the full rational threshold interval, the knowing-value operator $Kv_i^η(t)$ is restricted to $(\frac{1}{2},1]$. This high-threshold restriction has a structural effect: once $η>\frac{1}{2}$, two distinct values cannot both satisfy the threshold, so uniqueness becomes automatic. Over probabilistic models with countably additive measures, $Kv_i^η(t)$ is interpreted as non-factive high-confidence value locking. We establish sound axiomatic systems for the framework and develop a two-layer construction based on type-space distributions and assignment-configuration mappings. This resolves the joint realization problem arising from probabilistic mass allocation and value-sensitive constraints, and yields a structured weak-completeness theorem for the high-threshold fragment.