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CPSC 121: Models of Computation
2010 Winter Term 2
Introduction to Induction
Steve Wolfman
1
Outline
Prereqs, Learning Goals, and Quiz Notes
Problems and Discussion
Single-Elimination Tournaments
Binary Trees
A First Pattern for Induction
Lecture 7:
Decidable and semi-decidable languages, the Halting problem
Valentine Kabanets
September 24, 2013
1
N P and P
Recall that an NTM M is said to halt on an input x i all of its computation branches halt. In
other words, the height of the computati
Lecture 3:
Nondeterministic Finite Automata
Valentine Kabanets
September 11, 2013
1
Nondeterministic Finite Automaton (NFA)
Informally, an NFA has more than one option for the next transition. For example, in state q upon
reading 0, it may enter state p1
Lecture 14:
P , N P , and search to decision reductions
Valentine Kabanets
October 22, 2013
1
Complexity Theory
We take a closer look at the class of decidable problems, and want to classify these problems
according to the amount of time/space they requir
Lecture 12:
Im not provable
Valentine Kabanets
October 10, 2013
1
Second Proof of Gdels First Incompleteness Theorem
o
Here we give another proof of Gdels First Incompleteness Theorem. In this proof, we exhibit an
o
explicit true but not provable sentence
Lecture 16:
NP-complete versions of SAT
Valentine Kabanets
October 29, 2013
1
NP-complete versions of SAT
We showed last time that SAT is NP-complete, where SAT is to decide if a given propositional
formula (x1 , . . . , xn ) is satisable.
We will show th
Lecture 13:
Gdels Second Incompleteness Theorem, and Tarskis Theorem
o
Valentine Kabanets
October 15, 2013
1
1.1
Gdels Second Incompleteness Theorem
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Consistency
We say that a proof system P is consistent if P does not prove both A and A for some sentenc
Lecture 20:
Randomized complexity
Valentine Kabanets
November 12, 2013
1
Randomized complexity classes
A language L RP if there is a deterministic polytime TM M (x, r) such that
1. for all x L, Prr [M (x, r) accepts] 1/2, and
2. for all x L, Prr [M (x, r)
Lecture 19:
NL=coNL
Valentine Kabanets
November 7, 2013
1
N L-completeness
To talk about N L-completeness, we need to rene our notion of a reduction. Well talk about
logspace-computable reductions (recall our denition of a space-bounded TM that also has a
Lecture 10:
Applications of the Recursion Theorem
Valentine Kabanets
October 3, 2013
1
Care with mapping-reductions
In class, we argued that the language ET M = cfw_ M | L(M ) = is undecidable. We proved it by
assuming that there is a decider for ET M ,
Lecture 6:
Universal TM and Nondeterministic TMs
Valentine Kabanets
September 19, 2013
1
Universal Turing machine
Turing (1936) showed that there exists a machine U that, when given as input (appropriate encoding
of) M, w of a TM M and an input w to M , w
Lecture 8:
Rices Theorem
Valentine Kabanets
September 26, 2013
1
Another example of undecidability
Theorem 1. The language
EQT M = cfw_ M1 , M2 | L(M1 ) = L(M2 )
is undecidable.
Proof. Suppose it is decidable by some decider R. We reduce ET M to EQT M .
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CPSC 121: Models of Computation
2011 Winter Term 1
Number Representation
Steve Wolfman, based on notes by
Patrice Belleville and others
1
Outline
Prereqs, Learning Goals, and Quiz Notes
Prelude: Additive Inverse
Problems and Discussion
Clo
snick
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CPSC 121: Models of Computation
2011 Winter Term 1
Number Representation
Steve Wolfman, based on notes by
Patrice Belleville and others
1
Outline
Prereqs, Learning Goals, and Quiz Notes
Prelude: Additive Inverse
Problems and Discussion
Clo
snick
snack
CPSC 121: Models of Computation
2009 Winter Term 1
Sequential Circuits (Mealy Machines)
Steve Wolfman, based on notes by
Patrice Belleville and others
1
Outline
Prereqs, Learning Goals, and !Quiz Notes
Problems and Discussion
A Pushbutton L
Mei Qin Chen
citadel-math231
WeBWorK assignment number Homework16 is due : 10/28/2011 at 10:00am EDT.
The
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Lecture 15:
NP-completeness
Valentine Kabanets
October 24, 2013
1
Polytime mapping-reductions
We say that A is polytime reducible to B if there is a polytime computable function f (a reduction)
such that, for every string x, x A i f (x) B. We use the nota
Lecture 18:
PSPACE
Valentine Kabanets
November 5, 2013
1
Determinism vs. Nondeterminism
We have looked at the P vs. NP question, which is basically a question whether nondetemrinism
can be eciently removed, without a huge (exponential) increase in the run
Lecture 5:
Pumping Lemma, variants of Turing machines
Valentine Kabanets
September 17, 2013
1
Non-regular languages
Consider the language L = cfw_0n 1n | n 0. Intuitively, L cannot be accepted by any DFA since
the DFA cannot count (as they have only a con
Lecture 9:
Mapping reductions, and the Recursion Theorem
Valentine Kabanets
October 1, 2013
1
Reductions
We will consider a special kind of reductions: mapping reductions.
Denition 1. Language A is m-reducible to B (denoted A < B) if there is a computable
Lecture 4:
Regular expressions versus Finite Automata
Valentine Kabanets
September 12, 2013
1
Regular expressions
We have the following inductive denition of regular expressions.
The following are regular expressions: , , and a for every a .
If R1 and R
Lecture 11:
Applications of Computability Theory: Randomness and Logic
Valentine Kabanets
October 14, 2013
1
Kolmogorov Complexity
Usually, when we say that a string x is random, we have some probability distribution in mind
and we mean that x is chosen a