CIS 511, 2008
Brief description:
The course provides an introduction to
the theory of computation. The treatment is mathematical, but the point
of view is that of Computer Science. Roughly speaking, the theory of
computation consists of three overlapping subareas: (1) formal languages
and automata; (2) Models of computation,
computability, and basics of recursive function theory; (3)
complexity theory.
The course will focus mostly on (1) and (2), but will
also cover some of (3): the classes P and NP, and NP-complete problems.
Applications
of (1) to programming (and natural) language specification and
parsing (top-down and bottom-up parsing) will be mentioned,
whenever appropriate.
Syllabus:
Topics will include: ((*) means: if time permits)
PART 1: Languages and Automata
- Basics of language theory: alphabets, strings, concatenation,
languages, operations on languages (including Kleene *)
- Deterministic finite automata (DFA's)
- The cross-product construction
- Nondeterministic finite automata (NFA's)
- From NFA's to DFA's, the subset algorithm (Rabin and Scott)
- An Application of NFA's: Text Search
- Regular languages and regular expressions
- From regular expressions to NFA's
- From NFA's to regular expressions (node elimination)
- Closure properties of the regular languages
- Applications of regular expressions: Lexical analysis, finding patterns
in text
- Right-invariant equivalence relations
- The Nerode/Myhill characterization theorem
- The pumping lemma for regular languages
- State equivalence, minimal DFA's
- Context-free grammars and context-free languages
- Leftmost derivations, rightmost derivations, parse trees
- The universality of leftmost derivations
- Cleaning-up context-free grammars (e-rules, chain rules)
- Chomsky Normal Form
- Right-linear grammars and regular languages
- Eliminating useless productions
- (*) Greibach Normal Form
- Tree domains, Gorn trees, and parse trees
- A strong pumping lemma for context-free languages: Ogden's lemma
- Pushdown Automata (PDA's), instantaneous descriptions, acceptance modes
- DPDA's (Deterministic PDA's)
- From context-free grammars to PDA's
- From PDA's to context-free grammars
PART II: Models of Computation and Basics of Recursive Function Theory
- Generalities on computability, Partial Functions
- RAM programs (Post machines)
- Turing Machines
- RAM computable functions are Turing computable
- Turing computable functions are RAM computable
- Primitive recursive functions
- Recursive and partial recursive functions
- The equivalence of Turing computable functions and partial recursive functions
- Recursively enumerable languages and recursive languages
- Pairing Functions
- Coding of RAM programs
- Unsolvability of the Halting Problem
- A universal RAM program
- (*) The Enumeration Theorem
- (*) The undecidability of Post's Correspondence Problem
- (*) Undecidable Properties of CFL's; Greibach's Theorem and applications
PART III: Computational Complexity
- The class P
- Examples of Problems
- Boolean Satisfiability
- The class NP
- Polynomial-time reductions
- NP-completeness
- Cook's Theorem
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