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On the Semantics of Primary Cause in Hybrid Dynamic Domains

Shakil M. Khan, Asim Mehmood, Sandra Zilles

TL;DR

This paper proposes two definitions of primary cause in a hybrid action-theoretic framework, namely the hybrid temporal situation calculus, and proves that these two definitions are indeed equivalent and show that their definitions of causation have some intuitively justifiable properties.

Abstract

Reasoning about actual causes of observed effects is fundamental to the study of rationality. This important problem has been studied since the time of Aristotle, with formal mathematical accounts emerging recently. We live in a world where change due to actions can be both discrete and continuous, that is, hybrid. Yet, despite extensive research on actual causation, only few recent studies looked into causation with continuous change. Building on recent progress, in this paper we propose two definitions of primary cause in a hybrid action-theoretic framework, namely the hybrid temporal situation calculus. One of these is foundational in nature while the other formalizes causation through contributions, which can then be verified from a counterfactual perspective using a modified ``but-for'' test. We prove that these two definitions are indeed equivalent. We then show that our definitions of causation have some intuitively justifiable properties.

On the Semantics of Primary Cause in Hybrid Dynamic Domains

TL;DR

This paper proposes two definitions of primary cause in a hybrid action-theoretic framework, namely the hybrid temporal situation calculus, and proves that these two definitions are indeed equivalent and show that their definitions of causation have some intuitively justifiable properties.

Abstract

Reasoning about actual causes of observed effects is fundamental to the study of rationality. This important problem has been studied since the time of Aristotle, with formal mathematical accounts emerging recently. We live in a world where change due to actions can be both discrete and continuous, that is, hybrid. Yet, despite extensive research on actual causation, only few recent studies looked into causation with continuous change. Building on recent progress, in this paper we propose two definitions of primary cause in a hybrid action-theoretic framework, namely the hybrid temporal situation calculus. One of these is foundational in nature while the other formalizes causation through contributions, which can then be verified from a counterfactual perspective using a modified ``but-for'' test. We prove that these two definitions are indeed equivalent. We then show that our definitions of causation have some intuitively justifiable properties.
Paper Structure (19 sections, 26 theorems, 46 equations, 2 figures)

This paper contains 19 sections, 26 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Situation Calculus (SC)
  4. Hybrid Temporal Situation Calculus (HTSC)
  5. Example
  6. Actual Achievement Cause in the SC
  7. Example (cont'd).
  8. Primary Actual Cause in the HTSC
  9. Example (cont'd).
  10. Properties
  11. Primary Cause via Contribution
  12. A Modified But-For Test
  13. Example (cont'd).
  14. Discussion and Conclusion
  15. Dynamic Formulae.
  16. ...and 4 more sections

Key Result

proposition 1

Figures (2)

  • Figure 1: Primary Cause in Hybrid Domains: Primitive Case
  • Figure 2: Defused Situation $\sigma_2'$

Theorems & Definitions (43)

  • definition 1: Primary Cause KL21
  • definition 2: Hybrid Temporal Achievement Causal Setting
  • definition 3: Primary Temporal Achievement Cause
  • definition 4: Achievement Situation
  • definition 5: End Time of a Situation within a Context
  • proposition 1
  • proposition 2
  • Lemma 1: Uniqueness of Direct Cause
  • Lemma 2: Uniqueness of Achievement Situation
  • theorem 1: Uniqueness of Primary Cause of Temporal Effects
  • ...and 33 more