Infinite State System Essay

Analysis of Infinite State Systems
Undergraduate Thesis
Submitted in partial fulfillment of the requirements of
BITS F421, Thesis by Somya Shubhangi
2014A7TS0124G
Under the supervision of
Dr. K Narayan Kumar
Chennai Mathematical Institute
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
K. K. BIRLA GOA CAMPUS
December 2017
Certificate
This is to certify that the thesis entitled, “Analysis of Infinite State Systems” and submitted by Somya Shubhangi ID No. 2014A7TS0124G in partial fulfillment of the requirements of BITS
F421 Thesis embodies the work done by him under my supervision.
Supervisor
Dr. K Narayan Kumar
Professor, Department of Computer Sc.,
Chennai Mathematical Institute
Date:
On Campus Supervisor
Dr. A Baskar
Associate Professor,

Suppose that for all k ≥ 0 there exist strings x, y, z such that xyz ∈ A and |y| ≥ k, and for all u, v, w with y = uvw and v 6= , there exists i ≥ 0 such that xuviwz /∈ A. Then A is not regular.
1.1.5 Myhill-Nerode Theorem
Let R ⊆ Σ
∗ be a regular set. A Myhill-Nerode relation for R is an equivalence relation ≡ satisfying the following three properties:
Chapter 1. Literature Review 3
(i) ≡ is a right congruence: for any x, y ∈ Σ* and a ∈ Σ, x ≡ y ⇒ xa ≡ ya;
(ii) ≡ refines R: for any x, y ∈ Σ*, x ≡ y ⇒ (x ∈ R ⇔ y ∈ R);
(iii) ≡ is of finite index, that is, ≡ has only finitely many equivalence classes. x ≡R y ⇔ ∀z ∈ Σ ∗ (xz ∈ R ⇔ yz ∈ R) (1.1)
Let R ⊆ Σ
∗
. The Myhill-Nerode Theorem states that the following statements are equivalent:
(a) R is regular;
(b) there exists a Myhill-Nerode relation for R;
(c) the relation ≡R is of finite index.
1.2 Pushdown Automata and Context-Free Languages
1.2.1 Context-Free Grammars and Languages
A context-free grammar (CFG) is a grammar is a quadruple G = (N, Σ, P, S), where
N is a finite set of nonterminal symbols,
Σ is a finite set of terminal symbols,
P is a finite subset of N × (N ∪ Σ)*, the productions,
S ∈ N (the start