Lecture 8: Context-Free Languages and Grammars - Prof. David Evans, Study notes of Discrete Mathematics

Download Lecture 8: Context-Free Languages and Grammars - Prof. David Evans and more Study notes Discrete Mathematics in PDF only on Docsity! 1 David Evans http://www.cs.virginia.edu/evans cs302: Theory of Computation University of Virginia Computer Science Lecture 8: Context-Free Languages 2Lecture 8: Language Menu • Review Machine Models of Computing • Linguistic Model of Computing • Challenge Problem Near-Solution - Liuyi (Eric) Zhang • Context-Free Grammars 3Lecture 8: Language Deterministic Machine Models 010110101101010101 “Yes” or “No” Memory Machine Languages Finite States DFA Regular + Stack DPDA [this week] + Tape TM [later] 4Lecture 8: Language 010110101101010101 “Yes” or “No” Nondeterministic Machine Models Memory Machine Languages Finite States NFA Regular + Stack NDPDA [this week] + Tape NDTM [later] 5Lecture 8: Language Where did these models come from? 6Lecture 8: Language Modeling Human Intellect • Turing Machine (Alan Turing, 1936) – Modeling Human Computers • DFAs – McCulloch and Pitts, “A logical calculus of the ideas immanent in nervous activity”, 1943 – S. C. Kleene, Representation of Events in Nerve Nets and Finite Automata, 1956 – Claude Shannon and John McCarthy, Automata Studies, 1956 2 7Lecture 8: Language Out theoretical objective is not dependent on the assumptions fitting exactly. It is a familiar strategem of science, when faced with a body of data too complex to be mastered as a whole, to select some limited domain of experiences, some simple situations, and to undertake to construct a model to fit these at least approximately. Having set up such a model, the next step is to seek a thorough understanding of the model itself. S. C. Kleene, Representation of Events in Nerve Nets and Finite Automata, 1956 8Lecture 8: Language Language-Based Models of Computation 9Lecture 8: Language Noam Chomsky (born 1928), MIT Linguistics Professor and Leftist Political Activist 10Lecture 8: Language Hugo Chávez at the United Nations (20 Sept 2006) 11Lecture 8: Language I don’t know anybody who’s ever read a Chomsky book, He does not write page turners, he writes page stoppers. There are a lot of bent pages in Noam Chomsky’s books, and they are usually at about Page 16. Alan Dershowitz 12Lecture 8: Language “I must admit to taking a copy of Noam Chomsky’s Syntactic Structures along with me on my honeymoon in 1961. During odd moments, while crossing the Atlantic in an ocean liner and while camping in Europe, I read that book rather thoroughly and tried to answer some basic theoretical questions. Here was a marvelous thing: a mathematical theory of language in which I could use a computer programmer’s intuition! The mathematical, linguistic, and algorithmic parts of my life had previously been totally separate. During the ensuing years those three aspects became steadily more intertwined; and by the end of the 1960s I found myself a Professor of Computer Science at Stanford University, primarily because of work that I had done with respect to languages for computer programming.” Donald Knuth 5 25Lecture 8: Language Continue • Sort till the last bit • Bottom-up 26Lecture 8: Language Memory 27Lecture 8: Language Compare • Heap Sort –Requires ~ n log2 n comparisons –4 bytes to store integer • Bit Sort: –Requires ~ 32n comparisons –Memory to store (up to 33) memory addresses that separate the 1s and 0s 28Lecture 8: Language Challenge Recap 29Lecture 8: Language Language Classes Finite Described by a finite set, DFA with no cycles, Regular grammar with no recursion Regular Described by a DFA, NFA, Regular grammar Context-Free Described by a NDPDA, Context-Free Grammar 30Lecture 8: Language Context-Free Grammar A → αγβ One variable Any sequence of variables and terminals Why is it called “Context-Free”? 6 31Lecture 8: Language Example 1 S → 0S0 S → 1S1 S → ε 32Lecture 8: Language Example 2 Define a CFG that generates the language: { w | w ∈ {0, 1}* and w has an equal number of 0s and 1s } 33Lecture 8: Language Review Questions • How do you prove a grammar is context-free? • How do you prove a grammar is not context-free? • How do you prove a language is context-free? • How do you prove a language is not context-free? 34Lecture 8: Language Charge • Return PS2 and PS2 Comments now • PS3 is posted on the course site (Due in 1 week, Feb 19) • Thursday: –Equivalence of NDPDAs and CFGs (sketch) –Non-Context-Free Languages