Semantics and Verification

M1 IF course, ENS de Lyon.

Overview

The goal of formal verification is to check automatically that some programs or systems
are correct with respect to their requirements.

In this course we present mathematical models of programs and systems and we
describe classes of properties which can be automatically checked on these models.

Content of the course:

• Modelling with Labelled Transition Systems
• Linear-Time Properties
  • Definition and Examples
  • Characterizations via Order Theory and Topology
  • ω-Regular Properties and Büchi Automata
  • Linear Temporal Logic
• Bisimulation and Modal Logic
  • Bisimulation and Trace Equivalence
  • Hennessy-Milner Logic
  • Bisimulation and Logical Equivalence

Prerequisites from the L3 year of the Computer Science Department of ENS de Lyon:

• Fondements de l'informatique.
• Logique.

Bibliography:

• Course notes [pdf] (regularly updated).
• The course is mostly based on the book:
  • Baier, C. and Katoen, J.-P., Principles of Model Checking, MIT Press, 2008.

Homework

• The homework in [pdf].
• Part I to be completed for the 7th of March 2023 (see instructions inside).
• Part II to be completed for the 21st of March 28th of March 2023 (see instructions inside).

Final Exam

• Allowed documents: handwritten or typeset course notes. No article. No book.

Courses

• Course 1 (Jan. 11th)
  • General overview of the course.
    • See Chap. 1 of the book for an informal introduction to verification (up to 2008).
  • Modelling Concurrent Systems (Chap. 2 of the book).

• Course 2 (Jan. 18th)
  
  • Linear-Time Properties:
    • General definitions and Examples.
    • Safety properties.

• Course 3 (Jan. 25th)
  
  • Linear-Time Properties:
    • Liveness properties.
    • The Decomposition Theorem.
  • Topological Approach:
    • Generalities.
    • The Topological Decomposition Theorem.
    • Spaces of ω-Words.

• Course 4 (Feb. 1st)
  
  • Partial Orders, Complete Lattices, Closure Operators and Galois Connections.

• Course 5 (Feb. 8th)
  
  • Observable Properties.
  • The Logic LML (introduction).

• Course 6 (Feb. 22nd)
  
  • The Logic LML.
  • Introduction to the Homework.
  • Extending LML with Fixpoints.

• Course 7 (March 1st)
  • The Logic LTL.
  • ω-Regular Properties (§4.3 of the book):
    • ω-Regular Expressions (§4.3.1).
    • Non-Deterministic Büchi Automata (NBAs) (§4.3.2).

• Course 8 (March 8th)
  • NBAs and ω-Regular Expressions (§4.3.2).
  • Deterministic Büchi Automata (DBAs) (§4.3.3).
  • Generalized Büchi automata (GNBAs) and closure under intersection of ω-regular languages (§4.3.4).
  • Translation of LTL to (G)NBA's (§5.2 of the book).

• Course 9 (March 22nd).
  • Bisimulation.
  • Modal Logics of Transition Systems (beginning).

• Course 10 (March 29th).
  • Modal Logics of Transition Systems:
    • The Hennessy-Milner Property.
    • Modal Saturation.
    • Boolean Algebras with Operators.
    • Ultrafilter Extensions of Kripke Models.

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