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. Slides 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

Slides for the lecture on 27/10/2021 Exercises Sheet for the session on 27/10/2021

Exam (10/11/2021)

The second part of the module (six lectures from 17/11/21) is dedicated to decision procedures for SAT and SMT (Satisfiability Modulo Theory) problems. Here is a brief description (more details):

• SAT problem: DPLL and CDCL algorithms
• SMT problem: CDCL(T) algorithm and Z3 API
• Satisfiability problem in theories for: Equality with uninterpreted functions, Linear rational arithmetic, Linear integer arithmetic, bit vectors, arrays and pointers.
• Combination of decidable theories: Nelson-Oppen algorithm

Practice: modeling in Z3 API.

The evaluation for this part at the first session is composed of 60% of written exam, 40% of partial evaluation (during the period). For the second session, the evaluation is a written exam.