- Motivation and Objectives
This course is about mathematical logic with a focus on automated
reasoning techniques that are useful in reasoning about programs.
In the first part of the course we cover Propositional and
First-Order logic and some of the classical results like sound and
complete proof systems, compactness, and decidability of the
satisfiability/validity problems. In the second part we focus on
decision procedures for various theories that arise while reasoning
about assertions in programs.
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Course Outline
-
Zeroth-Order/Propositional Logic:
- Proofs in arithmetic
- Propositional logic, proof
- Decision procedure, completeness and compactness
-
First-Order Logic:
- Proof systems
- Undecidability
- Completeness and compactness
-
Theories and Decision Procedures:
- Equality and Uninterpreted Functions (EUF)
- Linear Arithmetic
- Nelson-Oppen combination
- Array logics
- Constrained Horn Clauses
- Meeting Schedule
3:30pm Mon/Wed
First meeting will be at 3:30pm on Wed 24th Feb
2021.
Meetings are online on Google Meet.
Google Meet
Meeting Link
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Lectures
- (24-Feb-2021) Course Overview: Mathematical Logic, Decision Procedures + Course Logistics.
Recording
Part 1,
Recording
Part 2
- (01-Mar-2021) Zeroeth-Order Logic:
Syntax. [Slides]
[Recording Part 1]
[Recording Part 2].
- (03-Mar-2021) Zoroeth-Order Logic:
Models. [Slides] [Recording]
- (08-Mar-2021) Zoroeth-Order Logic:
Theories to Models. [Slides] [Recording]
- (10-Mar-2021) Intro to Propositional
Logic. [Slides]
[Recording]
- (15-Mar-2021)
Consistency. [Slides]
[Recording]
- (17-Mar-2021)
Completeness. [Slides]
[Recording]
- (22-Mar-2021)
Decidability / Resolution. [Slides]
[Recording]
- (24-Mar-2021 and 29-Mar-2021) DPLL Satisfiability
Procedure. [Slides]
[Whiteboard notes
[Recording
24 Mar]
[Recording
29 Mar]
- (31-Mar-2021)
First Order Logic. [Slides]
[Recording]
- (05-Apr-2021)
First Order Logic (Proof System). [Slides]
[Recording]
- (07-Apr-2021)
First Order Logic (Universal Fragment). [Slides]
[Recording]
- (12-Apr-2021)
First Order Logic (Existential
Fragment). [Slides]
[Recording]
- (19-Apr-2021)
Theories to Models. [Slides]
[Recording]
- (26,28-Apr-2021 and 03-May-2021)
Equality and Uninterpreted Functions (EUF). [Slides]
[Recording
26-Apr-2021]
[Recording
28-Apr-2021]
[Recording
03-May-2021]
- (05, 10, 12-May-2021)
Linear Arithmetic. [Slides]
[Recording
05-May-2021, Part 1]
[Recording
05-May-2021, Part 2]
[Recording
10-May-2021]
[Recording 12-May-2021]
- (17, 19-May-2021)
Array Logic. [Slides]
[Recording 19-May-2021]
- (24-May-2021)
Constrained Horn Clauses (Sumanth Prabhu). [Slides]
[Recording]
- (26-May-2021)
Model-Based Projection (Stanly Samuel). [Slides]
[Recording]
- (31-May-2021)
Wrap-up lecture [Slides]
[Recording]
- Books and other material
- First-order Logic and automated theorem proving, Melvin Fitting,
Springer-Verlag, 1990.
- Logic for Computer Science -- Foundations for Automatic Theorem
Proving, Jean H. Gallier.
- Decision Procedures, Kroening and Strichman, Springer, 2008.
- The Calculus of Computation, Bradley and Manna, Springer, 2007.
- Computability and logic, George Boolos, John Burgess and Richard Jeffrey,
Cambridge U Press, 2007.
- An Introduction to Logic, Madhavan Mukund and S P Suresh, Lecture
Notes, Chennai Mathematical Institute (2011).
- Assignments
- Seminar Topics
- Decision Procedure for Real Arithmetic (Seidenberg and
Tarski) (Mohith and Upamanyu, 10am Tue 11th May)
Slides.
- Bit Vector Arithmetic (Kroening-Stichman (2e)
Ch~6). Nabarun and Prathamesh (10am Tue 18th May). Slides.
- Maximal Specification Synthesis (Albarghouthi, Dillig,
Gurfinkel, POPL 2016) (Logical Abduction). Alvin (10am, Tue
25th May).
- Undecidability of First-Order Logic (Reference: Boolos,
Burgess, and Jeffrey) K. Sreehari (10am, Tue 1st June)
- A Linear-Time algorithm for Horn SAT (Dowling and Gallier,
J. Logic Programming 1984
Link to
paper). Sai Sanjeev and Shreepranav (3:30pm, Wed 2nd
June). Slides.
- Uninterpreted Programs (Mathur, Madhusudan, Vishwanathan, PoPL
2019). Ajinkya (10am, Thu 3rd June)
- Decision Procedure for Presburger Logic (Reference: Boolos,
Burgess, and Jeffrey). Gaurish and Prashik (3:30pm, Wed 9th June).
- E-Matching for Quantified Formulas (Greg Nelson PhD thesis (1980),
De Moura and Bjorner CADE 2007)
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Weightage for evaluation:
-
Mid-semester Exam: 20%
-
Assignments + Quizes + Seminar: 40%
-
Final Exam: 40%
- Exam Schedule:
-
Mid-semester Exam:
3:30pm Wed 21st April 2021.
-
Final Exam: 3:30pm Fri, 04 June 2021.