Review of 136, Preview of 186 Peter Cappello Department of Computer - - PDF document

Review of 136, Preview of 186

Peter Cappello Department of Computer Science University of California, Santa Barbara Santa Barbara, CA 93106 cappello@cs.ucsb.edu

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Review of 136

The course develops:

• a precise description
• proven assertions

surrounding 2 language classes:

• Regular languages
• Context-free languages

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For each language class C, we are concerned primarily with:

• a meta-grammar for specifying a grammar for a language in C
• a machine model that is capable of recognizing any language in C
• Proving that the class of languages that can be specified grammat-

ically equals the class of languages that can be recognized by the machine model.

• its closure properties
• tools for proving that a language is not in C
• which properties of C are decidable.

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Regular Languages

Meta-grammar: Regular expressions. Machine model: Finite automata, transition graphs, nondeterministic finite automata. Grammatical specification = machine recognition: Kleene’s the-

• rem: Regular expressions = FA = TG.

Closure properties:

• Union
• Concatenation
• Kleene star
• Complement

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• Intersection
• Quotient with a language

Proving that a language is not in C:

• Pumping lemma
• Myhill-Nerode theorem

Decidable properties:

• Emptiness and Equivalence (specified as RE or as FA)
• Finiteness

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Context-Free Languages

Meta-grammar: Context-free grammars. Machine model: Pushdown automata. Grammatical specification = machine recognition: CFG = PDA. Closure properties:

• Union
• Concatenation
• Kleene star
• Not Complement
• Not Intersection

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• Intersection with a Regular language

Proving that a language is not in C: Pumping lemma Decidable properties:

• Emptiness
• Uselessness
• Finiteness
• Membership - The CYK algorithm

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Preview of 186

• Turing Machines
• Undecidable Problems
• Intractable Problems

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Turing Machines

• What is a Turing machine?
• Extending the basic model (e.g., multi-head, multidimensional tape,

nondeterminism)

• Restricting the model (e.g., semi-infinite tape, multi-stack machines,

counter machines)

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Undecidable Problems

• A language that is not recursively enumerable (diagonalization)
• Reducing an undecidable problem to another problem
• Undecidable problems about Turing machines (e.g., emptiness)

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Intractable Problems

• The class P: Problems solvable in polynomial time (e.g., MST)
• The class NP: Problems solvable in nondeterministic polynomial

time (e.g., TSP)

• NP-complete problem (e.g., SAT)
• Polynomial time reductions

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