Teachers: Stéphane Demri (LMF, ENS Paris-Saclay) & Mihaela Sighireanu (LMF, ENS Paris-Saclay)

This module presents the logical aspects of AI that are known to be essential to reason formally and symbolically on logical theories and for knowledge representation. Along the years, the logical aspects of AI had their own pace of development, in parallel to what has been done in formal logic. New scientific questions have been raised and this topic is currently at the crossroads of AI and logic, with its own research agenda. A lot of efforts have been devoted to the formalisation of concepts, some of them coming from the philosophical aspects of AI. The necessity to build bridges between several areas such as formal verification or automated deduction, has also been a driving force to develop this subject. This module proposes a large overview of logical formalisms related to AI issues.

Sessions will take place in room 1Z53 (ENS Paris-Saclay, Gif-sur-Yvette) on Wednesdays from 14:00 to 18:30, starting on September 22nd, 2021. Some sessions will use the lab room 1S53.

The two parts are independent. The validation of each part gives 4 ECTS.

The first part of the module (six lectures from 22/09/21) is dedicated to an introduction of description logics (4 lectures) and on temporal logics to reason about multi-agent systems (2 lectures). Here is a brief description:

• Introduction to description logics: knowledge representation, logical formalism, introduction to the basic description logic ALC, extensions
• Tableaux methods: tableaux-style proof system for ALC decision problems, extensions
• Computational complexity: complexity of concept satisfiability for ALC, knowledge base consistency for ALC, undecidable problems for ALC extensions
• Introduction to temporal logics for multi-agent systems: concurrent game structures, logics ATL, ATL* and variants, model-checking, extension with resources and variant with incomplete information (if time permits).

Each session of this part is divided into a lecture (2/3) and an exercises session (1/3).

The evaluation for this part is a written exam, planned to take place on November 10th from 2pm to 5pm, room 1Z53. Lecture notes and exercises sheets with correction authorised.

Material

Slides for the lecture on 22/09/2021 Exercises Sheet for the session on 22/09/2021

Slides for the lecture on 29/09/2021 Exercises Sheet for the session on 29/09/2021

Slides for the lecture on 06/10/2021 Exercises Sheet for the session on 06/10/2021

Slides for the lecture on 13/10/2021 Exercises Sheet for the session on 13/10/2021

Slides for the lecture on 20/10/2021 Exercises Sheet for the session on 20/10/2021

The second part of the module (six lectures from ??/11/21) is dedicated to the problems SAT and SMT (Satisfiability Modulo Theory). Here is a brief description (more details on the lecture notes):

• Problem SAT, algorithm DPLL
• Learning, back jump, algorithm CDCL
• Logic programming with Prolog
• Problem SMT, naive algorithm
• Examples of theories and related decision procedures : Difference logic (Floyd algorithm), Linear arithmetic on rational numbers (simplex algorithm, Fourier-Motzkin elimination), Equality theory (Union-Find structure, congruence closure)
• Combination of decision procedures: Nelson-Oppen algorithm, Shostak method
• Algorithm CDCL(T)
• Modern SMT techniques: splitting on demand, delayed theory combination, MCSAT

Practice: modeling in Prolog and implementation of a mini SMT-solver.

The evaluation for this part at the first session is composed of 50% of written exam and 50% of project. The project mark is computed as 50% from the first version (deadline on October 25th) and 50% from the final version (deadline on December 20th). For the second session, the evaluation is a written exam.