Rices Theorem
Recall that a language is a set of strings (over a given alphabet).
Denition: A property of languages is a set of languages (over a given
alphabet).
Examples:
The set of all nite languag
THE PUMPING LEMMA FOR
REGULAR LANGUAGES
The Pumping Lemma is a useful result for proving that languages are
not regular.
(Note that I cant think up a DFA for the language isnt a proof that its
not reg
The Bounds of Computability
1. The Church-Turing Thesis
All the models of computation discussed in this textbook
Turing Machines
variants: 1-way & 2-way tape, multitape, [non]deterministic
Par tial
Chapter 4: Markov Algorithms
A. A. Markov (1950s): Russian mathematician
Transduction paradigm: Computation as string rewriting
How a Markov Algorithm works:
Markov algorithm schema: a set of rules fo
Recursive and Recursively Enumerable Sets
Partial recursive functions can be used to classify sets of natural
numbers.
Turing machines can be used to classify sets of natural numbers and sets
of strin
Chapter 3: An Introduction to Recursion Theory
Recursive Function Theory uses
Simple initial functions
Forms for building complicated functions out of simpler functions
Primitive recursive functions
Chapter 2: Varieties of Turing Machine
The TM with
a 2-way innite tape,
taking its input from the tape,
writing its output on the tape,
moving and writing in separate steps
is not the only TM model th
Chapter 1: Turing Machines
COMPUTATIONAL PARADIGMS
A computation, or an algorithm, is an intuitive notion.
Well formalize that intuition in several ways in this class.
Some elements an algorithm shoul
Chapter 0: Mathematical Preliminaries
SET NOTATION
Some important sets
N
Z
Q
R
the set of natural numbers: cfw_0, 1, 2, .
the set of integers: cfw_., -3, -2, -1, 0, 1, 2, 3, .
the set of rational numb
The Bounds of Practical Computing: NP-Completeness
Polynomial-time complexity
The boundary between "computable in a reasonable time" and "not computable in
a reasonable time" can be estimated by "[not
The Clique Problem is NP-Complete
Proof: Reduce CNF-Satisability to Clique.
Given a CNF expression with k clauses, we construct a graph which has a k-clique
iff the expression is satisable.
Each verte
The Satisability Problem
Input:
A propositional logic expression using only the operators and, or, and not
Decision:
Is there an assignment of truth values to the propositional variables that
makes th
THE CHOMSKY HIERARCHY
(Reading: Section 12.7)
The Chomsky hierarchy, named for linguist Noam Chomsky who studied
formal grammars, describes a hierarchy of languages, ranging from the
very simple regul