Closure properties
- 1. Prof.Neeraj Bhargava Abhishek Kumar Department of Computer Science School of Engineering & System Sciences, MDS, University Ajmer, Rajasthan, India
- 2. Have a Grammar G with production rules of the accepted form. Define a FSA with one state for each auxiliary symbol, plus one more state. Make the initial state the one corresponding to the initial symbol. Add one more state, F and let it be the final state. If P=>aQ, where P and Q are auxiliaries and a is a symbol in the final alphabet, make an edge from P to Q, with label a. If P=>a, make an edge going from P to F with label a.
- 3. Given a FSA, define a grammar with auxiliary symbols corresponding to the states. The initial symbol is the one corresponding to the initial state. Add production rules P=>aQ for all edges from P to Q. Change all production rules P=>aF to P=>a.
- 4. Regular languages are closed with respect to various operations, including concatenation. If M is a regular language and N is a regular language, then we can define MN is the language made up of all the strings of the form sm sn where sn where sm is in M and sn is in N. Then MN also is regular.
- 5. If M and N are regular languages then so is the union of M and N. This is defined as the set of strings that are in M or are in N (or both). This also is a regular language.
- 6. Yet another equivalent definition! A regular expression over an alphabet is a pattern defined using letters from the alphabet combined using notation for union, concatenation, wild card, counts. The regular expession defines a set of strings, that is, a language
- 8. Invented what came to be called Turing machines. Proved various things about them, including Halting problem Worked in Bletchley Park during WWII: built a machine to help decode German codes (codes changed). Given medal (in secret) Worked in general area, including spending time at Princeton with Van Neumann, others. Wrote papers on computer chess, notion of Turing test Arrested & convicted for homosexual acts. Made to take hormone drugs. Appeared to be okay… Committed suicide
- 9. Pick 1: Please think about it first instead of or before trying to look it up. Informal proofs are okay. ◦ CFG and PDA equivalence If you believe this, then there does exist a PDA for the homework problem. ◦ Regular expression and (either) FSA or Regular grammar equivalence Possibilities for presentation topics in Turing work, history Keep up postings Shai videos