Meetings

Click on the theme item for the meeting plan for that theme.
Click on the meeting item for references, exercises, and additional reading related to it.

Theme 1 : Describing Sets : Mathematical Logic & Proofs - 27 meetings

Meeting 01 : Mon, Jul 29, 09:00 am-09:50 am

Acad : Introduction to the course. The story and the thread the course will follow. Reference textbooks.
Admin : Evaluation plans, TA and Instructor contact hours, course homepage, moodle, mailing list.

Meeting 02 : Tue, Jul 30, 08:00 am-08:50 am

Propositional Logic. Syntax. Semantics, Truth tables. Implication.

Meeting 03 : Wed, Jul 31, 12:00pm-12:50pm

Tautology, Contradiction, Contingents, Satisfiability. Algorithmics. Implication. Contrapostive, Converse, Inverse, Examples.

Meeting 04 : Fri, Aug 02, 11:00 am-11:50 am

Logical Equivalences. Checking if a formula is a tautology. Algorithmic question.

Meeting 05 : Mon, Aug 05, 09:00 am-09:50 am

Arguments and argument forms, validity of argument forms. Rules of inferences. Why are we not adding more rules of inferences? Discussion. Axiomatizations.

Meeting 06 : Tue, Aug 06, 08:00 am-08:50 am

Example deductions/derivations using rules of inferences. Four examples.

Meeting 07 : Wed, Aug 07, 12:00pm-12:50pm

Resolution refutation. Examples. Axiomatization of Proposition logic. Limitations of Propositional Logic. Need for a more powerful language.

Meeting 08 : Fri, Aug 09, 11:00 am-11:50 am

Predicate Logic. An informal introduction to quantifiers and their informal semantics.

Meeting 09 : Tue, Aug 13, 08:00 am-08:50 am

Translating English sentences to predicate logic. Use of function symbols and examples.

Meeting 10 : Wed, Aug 14, 12:00pm-12:50pm

Syntax for predicate logic. Well-formed formulas. The parse tree. Free variables, Scope of the quantifier, bound variables.

Meeting 11 : Fri, Aug 16, 11:00 am-11:50 am

Semantics of Predicate Logic. Models. Environment (variable assignments). Truth of a formula under a model. Examples of model which satisfy a formula and model which do not.

Meeting 12 : Mon, Aug 19, 09:00 am-09:50 am

More examples of models. Validity checking problem (undecidabile). Satisfiability problem. Arguments in predicate logic. Valid argument forms. Rules of inferences - UG, UI, EG, EI.

Meeting 13 : Tue, Aug 20, 08:00 am-08:50 am

Examples of valid arguments in predicate logic. Using them to establish logical equivalences. Axiomatization of predicate logic. Axioms, rules of inferences, proofs, theorems, completeness, soundness.

Meeting 14 : Wed, Aug 21, 12:00pm-12:50pm

Tutorial

Meeting 15 : Fri, Aug 23, 11:00 am-11:50 am

How to develop a theory based on predicate logic. Attempts on Theory of natural numbers, Theory of graphs. Peano's axioms.

Meeting 16 : Tue, Aug 27, 08:00 am-08:50 am

Mathematical Proofs. Proof techniques: Direct proof. The structure of a proof.Indirect proof or Proving the contrapositive. Example(s). Proof by contradiction. Examples. Proof by cases. Examples.

Meeting 17 : Wed, Aug 28, 12:00pm-12:50pm

A conjecture and confirming the conjecture by a proof. Existence proofs. Non-constructive proofs. Three examples.

Meeting 18 : Fri, Aug 30, 11:00 am-11:50 am

Principle of mathematical induction. Three examples.

Meeting 19 : Wed, Sep 04, 12:00pm-12:50pm

Strong Induction. Structural Induction. Examples. Well-ordering vs Mathematical Induction.

Meeting 20 : Fri, Sep 06, 11:00 am-11:50 am

More example proofs. Infiniteness of prime numbers. How do we define infiniteness? Finiteness? Notion of cardinality of sets. "Equal cardinality" as a relation. Schroeder-Bernstein theorem. Finite and infinite sets. Examples.

Meeting 21 : Mon, Sep 09, 09:00 am-09:50 am

Meeting 22 : Fri, Sep 13, 08:00 am-08:50 am

Meeting 23 : Mon, Sep 16, 09:00 am-09:50 am

Cardinality. Finite and Infinte subsets. Any infinite subset of N has the same cardinality as N. Countable and countably infinite sets. Uncountability. Cantor's diagonalization. More examples of uncountable sets. Countability of NxN. Union, Cartesian product of countable sets. Diagonalization in set form - there cannot be an surjection from a set to its power set. Infinitely many types of infinities.

Meeting 24 : Tue, Sep 17, 08:00 am-08:50 am

An unexpected use of the technique. Are there computational tasks which are not effectively programmable? Computational Tasks vs Languages. Programs vs Strings. A non-constructive proof.

Meeting 25 : Wed, Sep 18, 12:00pm-12:50pm

A constructive proof that there are problems for which no effective programs exist using diagonalization. In particular, the problem of given a program P and an input x, the task of testing if it halts on x, undecidable.

Meeting 26 : Fri, Sep 20, 11:00 am-11:50 am

Is it because we do not know the x in advance? The variant of the problem where only the program is the input to us and x in which we are interested in checking P is halting or not is fixed apriori to be the string "LCCS". A proof that even this is not effectively programmable.

Meeting 27 : Sat, Sep 28, 09:00 am-12:00 pm

Completeness of Propositional Logic. Proof of Completeness Theorem. Eliminating the premises. Completeness theorem with empty premises. Reintroducing the premises. Completing the first and the last step.
Proofs for the formula from the lines of the truth table. Combining proofs. Deduction theorem (statement).
Deduction Theorem. Completing the proof of completeness theorem

Theme 2 : Sizes of Sets : Counting & Combinatorics - 14 meetings

Meeting 28 : Mon, Sep 23, 09:00 am-09:50 am

Back to logic : Applying the proof techniques that we learned. Soundness Theorem for Propositional Logic. Proof by induction. Completeness theorem (statement).

Meeting 29 : Tue, Sep 24, 08:00 am-08:50 am

Soundness and Completeness Theorems for First-Order Logic. Godel's Incompleteness Theorem.

Meeting 30 : Wed, Sep 25, 12:00pm-12:50pm

Expressibility of First Order Logic. Theory for graphs. 2-colorability cannot be expressed by a formula. 2-colorabilty cannot be axiomatized finitely but can be expressed by an infinite number of formulas.

Meeting 31 : Fri, Sep 27, 11:00 am-11:50 am

Connectedness of Graphs is not even infinitely axiomatizable.

Meeting 32 : Tue, Oct 01, 08:00 am-08:50 am

Birds eye-view : Second Order Logic, Existential Second Order, Universal Second Order, Monadic Second order Logics. Modes of Truth. Modal Logic: syntax and semantics. Truth in a world. Example formulas.

Meeting 33 : Fri, Oct 04, 11:00 am-11:50 am

Modelling mutual exclusion. Safety and Liveness. Truth in an execution path - a sequence of worlds evolving over time. Need of Linear-time Temporal Logic (LTL).

Meeting 34 : Mon, Oct 07, 09:00 am-09:50 am

Syntax and Semantics. An overview of how it is used in the design and correction of systems. Modelling systems using LTL. Mutual exclusion: an example.

Meeting 35 : Wed, Oct 09, 12:00pm-12:50pm

Meeting 36 : Fri, Oct 11, 11:00 am-11:50 am

Meeting 37 : Mon, Oct 14, 09:00 am-09:50 am

Meeting 38 : Wed, Oct 16, 12:00pm-12:50pm

To Be Announced

Meeting 39 : Fri, Oct 18, 11:00 am-11:50 am

To Be Announced

Meeting 40 : Mon, Oct 21, 09:00 am-09:50 am

To Be Announced

Meeting 41 : Tue, Oct 22, 08:00 am-08:50 am

To Be Announced

Theme 3 : Structured Sets : Relational & Algebraic Structures
- 11 meetings
Meeting 42 : Wed, Oct 23, 12:00pm-12:50pm
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 43 : Fri, Oct 25, 11:00 am-11:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 44 : Mon, Oct 28, 09:00 am-09:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 45 : Tue, Oct 29, 08:00 am-08:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 46 : Wed, Oct 30, 12:00pm-12:50pm
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 47 : Fri, Nov 01, 11:00 am-11:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 48 : Mon, Nov 04, 09:00 am-09:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 49 : Tue, Nov 05, 08:00 am-08:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 50 : Wed, Nov 06, 12:00pm-12:50pm
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 51 : Mon, Nov 11, 09:00 am-09:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 52 : Wed, Nov 13, 12:00pm-12:50pm
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Evaluation Meetings
- 6 meetings
Meeting 53 : Mon, Aug 26, 09:00 am-09:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 54 : Tue, Sep 03, 08:00 am-08:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 55 : Mon, Sep 30, 09:00 am-09:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 56 : Tue, Oct 15, 08:00 am-08:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 57 : Fri, Nov 08, 11:00 am-11:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 58 : Mon, Nov 18, 09:00 am-12:00 pm
Endsemester Examination
References
Exercises
Reading
Endsemester Examination
References : None

CS6030 - Logic and Combinatorics for Computer Science

Today : Sat, Dec 21, 2024

Announcements

Meetings

References	[KR] Book Section 1.5. For discussion on axiomatization refer class notes.
Exercises
Reading

References	[KR] Book Section 1.3. For some examples, refer class notes.
Exercises
Reading

References	[HR] (relevant section uploaded to moodle) Section 2.2
Exercises
Reading