💻 computer science

Inquisitive Action Logic

This paper introduces Inquisitive Action Logic (InqAL), a multi-agent modal framework that extends traditional action logics by modeling agentive determination through questions, and establishes its theoretical foundations via an axiomatization, completeness, decidability, and a representation theorem linking neighborhood frames to concurrent game structures.

Original authors: Ivano Ciardelli (University of Padua)

Published 2026-07-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ivano Ciardelli (University of Padua)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Not Just What Happens, But Who Decides

Imagine you are watching a magic show. A magician pulls a rabbit out of a hat.

Traditional Logic asks: "Did the magician force the rabbit to appear?" (Yes/No).
This Paper's Logic (InqAL) asks a deeper question: "Did the magician decide exactly what kind of rabbit it would be, or was that just luck?"

The author, Ivano Ciardelli, introduces a new way of thinking about actions. He argues that when we do something, we don't just force a specific result; we often determine the answer to a question.

The Clay Animal Analogy

To explain this, the paper uses a story about two friends, Alice and Bob, making clay animals.

Alice shapes the clay. She decides if it's a cat, a dog, or a cow.
Bob paints the animal. He decides if it's red, blue, or green.
They work at the same time, without looking at what the other is doing.

The Traditional View:
If they make a blue cat, traditional logic might say, "Alice forced the cat shape" and "Bob forced the blue color."

The New View (InqAL):
The paper says we should look at it as determining answers to questions.

Alice determines the answer to the question: "What shape is the animal?" (She makes sure the answer is settled, even if she doesn't know the color yet).
Bob determines the answer to the question: "What color is the animal?"
Crucially, Alice does not determine the color. Even if she makes a cat, the color could still be red, blue, or green depending on Bob. So, she hasn't "settled" the color question.

The paper creates a mathematical language to say exactly this: "Alice determines the shape, but not the color."

How the Logic Works (The "Neighborhood" Metaphor)

To make this precise, the author uses a concept called a "Neighborhood."

Imagine every possible outcome (a red cat, a blue dog, etc.) is a house in a neighborhood.

Alice's Actions: When Alice chooses to make a "cat," she narrows the world down to a specific neighborhood containing only cat houses (Red Cat, Blue Cat, Green Cat).
Bob's Actions: When Bob chooses "red," he narrows the world to a neighborhood of only red houses (Red Cat, Red Dog, Red Cow).

The logic checks two things to see if someone "determined" a question:

Was the answer uncertain before? (Was the neighborhood mixed with different shapes?)
Is the answer settled after the action? (Is the new neighborhood Alice created only cats?)

If the answer to both is "Yes," then Alice has determined the shape.

The "Game" Behind the Scenes

The paper also solves a puzzle about how these "neighborhoods" are created. It proves that if you have a group of people playing a game where everyone chooses actions at the same time (like Alice and Bob), their "power" to create these neighborhoods follows three strict rules:

You must have choices: You can't have zero actions.
Everyone sees the same total possibilities: The total pool of outcomes available to Alice is the same size as the pool available to Bob.
Your choices don't block each other: If Alice picks a "cat" neighborhood and Bob picks a "red" neighborhood, there must be at least one outcome that is both a cat and red. They can't accidentally create a situation where no outcome is possible.

The paper proves that only systems that follow these three rules can be modeled as a real game where people act together.

Why This Matters (Without Getting Too Technical)

The paper shows that this new logic (InqAL) is just as powerful as other existing logics used to study cooperation and power, but it is much more efficient.

The Translation Problem: If you try to translate a sentence from this new logic into the old logic, the sentence might get exponentially longer (like turning a short poem into a novel just to say the same thing).
The Solution: InqAL keeps the sentences short and sweet by using "questions" directly in the math, rather than forcing everything into "statements."

Summary

The paper builds a new tool for thinking about agency (the power to act). It moves beyond asking "Can I force this result?" to ask "Do I control the answer to this question?"

Old Logic: "I can make the animal a cat."
New Logic (InqAL): "I can decide what shape the animal is, even if I don't decide the color."

The author proves that this new way of thinking is mathematically sound, can be used to solve problems, and offers a clearer, more compact way to describe how agents influence the world around them.

Technical Summary: Inquisitive Action Logic

Problem and Motivation
Traditional modal logics of agency, such as Coalition Logic (CL), focus primarily on what facts an agent (or coalition) can force through their actions. However, a distinct and equally important aspect of agency is what an agent determines or influences. For instance, in a scenario where Alice molds a clay animal's shape and Bob paints its color, Alice determines the shape but not the color. Standard modal logic struggles to formalize "determination" because the object of determination is often a question (e.g., "what is the shape?") rather than a specific proposition.

This paper introduces Inquisitive Action Logic (InqAL) to address this gap. The core problem is to provide a logical framework that captures agentive determination as a modal relation between an agent and a question. The authors argue that determining a question means the answer is not settled a priori but becomes settled once the agent's action is fixed. This requires a logic capable of handling both declarative statements and interrogative formulas (questions) within a multi-agent action setting.

Methodology
The methodology proceeds in three main stages:

Semantic Foundation (Concurrent Game Structures): The logic is grounded in Concurrent Game Structures (CGS), where state transitions result from the simultaneous actions of multiple agents. The authors define actual effectivity functions, which map an agent's available actions to the sets of possible outcomes (neighborhoods) those actions produce.
Characterization of Neighborhood Frames: A significant technical step involves characterizing which multi-agent neighborhood frames arise from CGSs. The authors prove that a neighborhood frame corresponds to a CGS if and only if it satisfies three conditions:
- Existence of actions: Every agent has at least one available action at every world.
- Uniform range: The union of all neighborhoods for any agent at a world equals the set of all possible outcomes at that world.
- Independence: The intersection of any selection of neighborhoods (one for each agent) is non-empty.
Logical Framework (InqNLA and InqAL):
- The authors first define Inquisitive Multi-Agent Neighborhood Logic (InqNLA), a multi-agent extension of Inquisitive Neighborhood Logic (InqNL). This logic uses a binary modality $\Rightarrow_a$ acting as a strict conditional over neighborhoods, alongside inquisitive disjunction ( $\vee$ ) to form questions.
- InqAL is then defined as the fragment of InqNLA interpreted specifically over CGSs (via their induced actual effectivity functions).
- Key modalities are defined to capture agency:
  - $\Diamond_a \alpha$ : Agent $a$ can force $\alpha$ (standard ability).
  - $\Box_a \phi$ : The question $\phi$ is settled a priori (regardless of actions).
  - $\boxplus_a \phi$ : The question $\phi$ is settled once agent $a$ 's action is fixed.
  - $\boxtimes_a \phi := \neg \Box_a \phi \land \boxplus_a \phi$ : Agent $a$ determines the question $\phi$ .

Key Contributions and Results

Representation Theorem: The paper establishes a representation theorem for actual effectivity functions of single agents (Theorem 2.2). It provides exact conditions under which a multi-agent neighborhood frame arises from a concurrent game structure. This resolves an open problem for the single-agent case and characterizes the class of models for InqAL.
Axiomatization: The authors present a sound and complete axiomatization for InqAL. The system extends the axioms for InqNLA with three specific schemata corresponding to the characterization conditions:
- $\Diamond_a \top$ (Existence of actions).
- $\boxplus_a \alpha \leftrightarrow \boxplus_b \alpha$ for declaratives $\alpha$ (Uniform range).
- A complex independence axiom ensuring that agents' choices do not preclude each other's outcomes.
Completeness and Decidability: Using a finite canonical model construction, the authors prove that InqAL is strongly complete with respect to CGSs. Consequently, the logic possesses the finite model property, which establishes its decidability.
Expressive Power Comparison: The paper compares InqAL with Socially Friendly Coalition Logic (SFCL) [20]. It demonstrates that InqAL is expressively equivalent to the individual-agent fragment of SFCL regarding statements (declaratives). However, the translation from InqAL to SFCL can lead to an exponential increase in formula size due to the need to resolve inquisitive formulas into their declarative components. Conversely, InqAL offers a more succinct way to express properties related to enabling and determining outcomes.

Significance and Claims
The paper claims that InqAL provides a natural formal analysis of "agentive determination" by treating determination as a modal relation to questions. By integrating inquisitive logic with action logic, the framework allows for the precise articulation of claims like "Alice determines the shape but not the color," which are difficult to express in standard logics of ability.

The authors note that the representation theorem for actual effectivity functions is of independent interest, as it clarifies the structural relationship between game-theoretic models and neighborhood semantics. Furthermore, the paper highlights that while SFCL captures similar properties (enabling others while forcing one's own goals), InqAL achieves this with a different logical architecture that handles questions natively, offering potential advantages in succinctness and conceptual clarity regarding the nature of determination.

The paper concludes modestly, acknowledging that extending these results to coalitions (groups of agents) remains an open challenge due to the complex interplay of overlapping effectivity functions, and that future work could involve integrating temporal operators to create an inquisitive version of strategic logics like ATL.