- Meeting 01 : Tue, Jan 03, 10:00 pm-10:50 pm
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Introduction. Outline of the course. Administrative details. The science vs engineering perspectives towards computation. The theory to aim for.
- Meeting 02 : Wed, Jan 04, 09:00 am-09:50 am
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Engineering side : Need of system modeling and verification. Intel FDIV bug. Push-button system as a simple system to study. The notion of a state.
References | : | Section 1.1 of HMU textbook.
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Exercises | : | Correctness Proof strategy for Push Button Switch
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Reading | : | Pentium FDIV flaw |
- Meeting 03 : Thu, Jan 05, 01:00 pm-01:50 pm
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Science side : Notion of computation. Compass-straight edge construction from Ancient greeks. Computation as problem solving. View of the push button switch as a problem solver. Coffee-Tea machine as an additional example.
- Meeting 04 : Tue, Jan 10, 10:00 pm-10:50 pm
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What language does Push-button switch know? How do we express languages formally? The language for languages : Alphabets, Strings, Sigma*, Languages. Languages vs Problems.
References | : | Section 1.4 and 2.2 of HMU textbook.
Kozen Book Lecture 3.
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- Meeting 05 : Wed, Jan 11, 09:00 am-09:50 am
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Modeling the push-button switch, Finite states, Finite alphabet, start state, transitions. Need of final states for the theory. Language accepted by a machine (informally).
References | : | Section 1.5 of HMU textbook.
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- Meeting 06 : Thu, Jan 12, 01:00 pm-01:50 pm
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Extended transition functions, Language accepted by a finite state automaton. A formal proof that the push-button switch is indeed correct by design. The decimal-mod-3 machine.
References | : | HMU Book Section 2.2.4,
Kozen Book Lecture 3 and 4.
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- Meeting 07 : Mon, Jan 16, 11:00 am-11:50 am
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Proof of the correctness of decimal-mod3 machine. Regular Languages.
Operations on languages. Closure of Regular Languages under complementation.
References | : | Lecture 4 in Kozen's Book.
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Exercises | : | Attempt on the formal proof that Regular languages are closed under complementation.
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- Meeting 08 : Tue, Jan 17, 10:00 am-10:50 am
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Product construction and the formal proof of correctness. Regular Languages are closed under complementation, intersection. DeMorgan's Law for union case.
References | : | Lecture 4 in Kozen's Book.
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Exercises | : | Formally prove that regular languages are closed under union using the product construction (without using the DeMorgan's law).
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- Meeting 09 : Mon, Jan 23, 11:00 am-11:50 am
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Are all languages regular? Question 5 in the Tutorial -1. Developing a proof strategy. A complete proof that the language is not regular.
References | : | Lecture 11 in Kozen, Section 4.1 in HMU.
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- Meeting 10 : Tue, Jan 24, 10:00 am-10:50 am
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Another complete proof : a^nb^n is not regular. Another one, d^ma^nb^n is not regular. Observations from the proofs. Abstraction of the proof strategies as a Lemma. Pumping Lemma for Regular Languages. Proof as a game with the demon.
References | : | Lecture 11 in Kozen's Book.
Chapter 4 in HMU Book.
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Exercises | : | We wrote pumping lemma as implied by the proofs that we did in the class. Write a complete proof of the pumping lemma combining the proofs of the claims that we have done in the class.
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Reading | : | A natural question about which you could read and think : Is the converse of pumping Lemma true?. If the RHS is true for a language A, does it imply that the language A is regular?
If yes, how will we construct the automaton just from the fact that it satisfies the RHS?
If no, we have to show an example of a language A satisfying the RHS of the pumping lemma . Aha, but then we need a method of showing that some language is not regular without using the pumping lemma !. |
- Meeting 11 : Wed, Jan 25, 01:00 pm-01:50 pm
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Myhill-Nerode Relations. (
Notes ).
References | : | Chapter on Myhill-Nerode relations in Kozen's Book.
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Exercises | : | We showed in class a right congruence property for the Myhill-Nerode relations defined from an automata. How about a left-congruence relation? Try to define this notion, and argue.
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- Meeting 12 : Mon, Jan 30, 11:00 am-11:50 am
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Myhill-Nerode theorem. Forward direction. Showing non-regularity of languages using Myhill-Nerode theorem.
References | : | The notes sent to the course mailing list
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Exercises | : | Try showing that { a^p : p is a prime } is not regular using MN theorem.
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- Meeting 13 : Tue, Jan 31, 10:00 am-10:50 am
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Proof of reverse direction in Myhill-Nerode Theorem. Showing lower bounds on the number of states of a finite automata.
References | : | The notes sent to the course mailing list
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- Meeting 14 : Wed, Feb 01, 09:00 am-09:50 am
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Myhill-Nerode relations, refinement, the coarsest Myhill-Nerode relation.
References | : | The notes sent to the course mailing list
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- Meeting 15 : Thu, Feb 02, 01:00 pm-02:00 pm
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Coarsest MN relation, Connections to minimization, Examples of Minimization Process. Need of an algorithm.
References | : | Notes sent to the course mailing list.
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- Meeting 16 : Mon, Feb 06, 11:00 am-11:50 am
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A criterion for merging of states. Marking Algorithm, Examples
References | : | Kozen Book - Lecture 13, last section (for the criterion). Lecture 14, for the algorithm and the examples.
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- Meeting 17 : Tue, Feb 07, 10:00 am-10:50 am
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Proof of correctness of Marking Algorithm for Minimization of Automaton. An introduction to non-determinism.
References | : | Induction Proof done in class.
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