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Le Nozze di Giustizia. Interactions between Artificial Intelligence, Law, Logic, Language and Computation with some case studies in Traffic Regulations and Health Care

Joost J. Joosten, Manuela Montoya García

TL;DR

This paper examines the potential and limits of AI in public administrations by focusing on rule-based AI and formal methods. It argues that logical, computational, and mathematical considerations—exemplified through traffic regulations and health-care governance—constrain what automated systems can safely and fairly do. By exposing underspecification, inconsistencies, and hidden dynamics in law, the authors advocate prudent, case-by-case regulation and the use of formal methods to maintain transparency and accountability. The overarching message is that while AI can accelerate administration, safeguarding core societal values requires careful governance, monitoring, and a pragmatic acceptance of the 'efficiency of inefficiency.'

Abstract

An important aim of this paper is to convey some basics of mathematical logic to the legal community working with Artificial Intelligence. After analysing what AI is, we decide to delimit ourselves to rule-based AI leaving Neural Networks and Machine Learning aside. Rule based AI allows for Formal methods which are described in a rudimentary form. We will then see how mathematical logic interacts with legal rule-based AI practice. We shall see how mathematical logic imposes limitations and complications to AI applications. We classify the limitations and interactions between mathematical logic and legal AI in three categories: logical, computational and mathematical. The examples to showcase the interactions will largely come from European traffic regulations. The paper closes off with some reflections on how and where AI could be used and on basic mechanisms that shape society.

Le Nozze di Giustizia. Interactions between Artificial Intelligence, Law, Logic, Language and Computation with some case studies in Traffic Regulations and Health Care

TL;DR

This paper examines the potential and limits of AI in public administrations by focusing on rule-based AI and formal methods. It argues that logical, computational, and mathematical considerations—exemplified through traffic regulations and health-care governance—constrain what automated systems can safely and fairly do. By exposing underspecification, inconsistencies, and hidden dynamics in law, the authors advocate prudent, case-by-case regulation and the use of formal methods to maintain transparency and accountability. The overarching message is that while AI can accelerate administration, safeguarding core societal values requires careful governance, monitoring, and a pragmatic acceptance of the 'efficiency of inefficiency.'

Abstract

An important aim of this paper is to convey some basics of mathematical logic to the legal community working with Artificial Intelligence. After analysing what AI is, we decide to delimit ourselves to rule-based AI leaving Neural Networks and Machine Learning aside. Rule based AI allows for Formal methods which are described in a rudimentary form. We will then see how mathematical logic interacts with legal rule-based AI practice. We shall see how mathematical logic imposes limitations and complications to AI applications. We classify the limitations and interactions between mathematical logic and legal AI in three categories: logical, computational and mathematical. The examples to showcase the interactions will largely come from European traffic regulations. The paper closes off with some reflections on how and where AI could be used and on basic mechanisms that shape society.
Paper Structure (26 sections, 1 theorem, 7 equations, 5 figures)

This paper contains 26 sections, 1 theorem, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. What is Artificial Intelligence?
  3. What is intelligence?
  4. Then, what is artificial intelligence?
  5. Should we use Artificial Intelligence?
  6. AI in public administrations
  7. Rule-based AI and Formal Methods
  8. Formal ontologies and limitations
  9. Logical Lingerings
  10. Halting programs and looping programs
  11. The Undecidability of the Halting problem
  12. Computational Complications
  13. Mathematical Mischiefs
  14. Fundamental AI problems rarely show up
  15. Mathematical mischiefs
  16. ...and 11 more sections

Key Result

Theorem 1

There cannot exist a computer program $H$ that solves the halting problem telling whether an arbitrary computer program $\Pi$ will eventually end its computation when it starts on input $x$.

Figures (5)

  • Figure 1: Formal ontologies should be chosen carefully. Mathematical rigour can only be obtained for the relation between expressions in a Formal Language with respect to the Formal Semantics of this formal language. This relation is the only one depicted with a double arrow. All other relations are meaningful on the basis of trust and common sense and formal methods in full for them are impossible.
  • Figure 2: The Mandelbrot set
  • Figure 3: Compensating reduced weekly rest periods
  • Figure 4: In the middle line we have the tachograph recording second by second. The upper line is a minute-to-minute interpretation in case UNIX time would be used. The lower line is a minute-to-minute interpretation in case UTC time would be used.
  • Figure 5: The Allegory of Good and Bad Governance (Ambrogio Lorenzetti)

Theorems & Definitions (1)

  • Theorem : Unsolvability of the Halting problem