1. What is actual causality and its formalization?
Actual causality refers to the causality of a specific event that has actually happened, such as 'John died because Alice shot him'. It is different from general causes like 'smoking causes cancer'. Formal approaches for modeling actual causality have been developed, with one of the most prominent being the model by Halpern and Pearl. This model uses structural equations to describe dependencies between endogenous and exogenous variables. Halpern and Pearl have provided three definitions of actual causality using counterfactual reasoning: original, updated, and modified definitions. The formal language developed in this model is used to define notions like normality, blame, accountability, and responsibility. The Halpern and Pearl formalism has been extended to other frameworks, but a general Kripke model for actual causality based on their framework has not been studied yet. In this work, a causal Kripke model is developed, introducing a modal language for causal reasoning with uncertainty, temporality, possibility, and epistemic knowledge. This model formalizes notions like sufficient causality, blame, responsibility, normality, and explanations. It provides a more natural definition of sufficient causality by considering nearby contexts, which Halpern's causal model does not support. The developed causal Kripke models offer a straightforward way to describe nearby contexts and define sufficient causality as intended by Halpern. The paper includes preliminaries on causal models and logic of causality, examples to motivate the development of causal Kripke semantics, definitions of causal Kripke models and a modal logic of actual causality, generalizations of Halpern-Pearl definitions, modeling examples, and future research directions.
read more
2. What is a causal model?
A causal model describes the world in terms of variables with given ranges, exogenous and endogenous variables, and structural equations. It is a recursive model without cyclic dependencies, where context determines the values of endogenous variables. Exogenous variables are given and not subject to interventions. The model's signature consists of a set of exogenous and endogenous variables, and their ranges. The model is defined by a pair (S, F), where S is the signature and F is a set of structural equations. A direct cause or parent relationship is established between variables based on specific conditions. Causal models are used to understand the relationships and dependencies between variables in a given context.
read more
3. What is the basic language L C for describing causality?
The basic language L C for describing causality is an extension of propositional logic. It includes primitive events of the form X = x, where X V is an endogenous variable and x R(X ). The language is defined by a recursion formula and uses intervention formulas to define satisfaction. It is used by Halpern and Pearl to provide different definitions of causality, including original, updated, and modified definitions. The language is essential for understanding causal relationships and analyzing causal settings.
read more
4. How do causal models incorporate notions like possibility, knowledge, and belief?
Causal models incorporate notions like possibility, knowledge, and belief by extending the basic language of propositional logic. These models allow for reasoning in scenarios involving temporality, uncertainty, and accessibility. For example, in the Umbrella scenario, the possibility of rain in London in the future is considered a cause for Alice taking her umbrella. Similarly, in the Chess example, the possibility of the king being in check due to the knight's movement is a cause for the player to move the knight. These examples demonstrate that causal reasoning naturally involves considering possibilities, knowledge, and beliefs.
read more