EECS 376 Midterm
Decider
1. A TM M is a decider if for some language L, M accepts all input x in L and rejects all input x not in L. 2. No looping can occur. 3. Can only decide 1 language
Recognizer
1. A TM M is a recognizer if for some language A, M accepts all input x in A and rejects/loops on all input x not in A. 2. Every machine is a recognizer for its own language. 3. A recognizer can recognize only 1 language
Turing Machine Model Assumptions
1. Assume paper is divided into discrete cells 2. Assume a finite number of symbols can be written in each cell 3. Assume paper is infinite 4. Assume we can only look at one cell at a time 5. Assume we can only remember a fixed amount of information. There will be a finite set of states.
Properties of a Minimal Spanning Tree
1. Connected 2. Has the least total edge cost 3. Forms a tree
Types of proofs
1. Direct 2. Proof by contradiction 3. Proof by contraposition 4. Proof by induction
Church-Turing Thesis
1. Every function that is computable by any mechanical device is computable by a Turing Machine. 2. Every function that is not computable by a Turing Machine cannot be computed by any computer.
Sigma*
1. The set of all finite length strings over Sigma. 2. Is infinite 3. A language is a subset of this
Kolmogorov Complexity
1. The size in bits of the shortest U program that outputs some value w. 2. For every U, Ku(w) is incomputable 3. No program can compute Ku(w) correctly on all binary strings.
The class of recognizable languages is closed under...
1. Union 2. Intersection
Alphabet
A finite set of symbols
Computable
A function is computable if there exists a program M s.t. given w as input, M halts and outputs f(w).
Decidable
A language A is decidable if there exists an M s.t. M decides A.
Turing Complete
A model is Turing complete if it can implement all algorithms that a TM can.
String
A sequence of symbols from an alphabet
Language
A set of all "yes" instances for some input. Encodes a "ye/no" problem.
Principle of Optimality (Optimal Substructure)
A substructure of an optimal substructure is itself optimal. Example: A subpath of any shortest path is itself a shortest path.
If a language A is undecidable, then what does this say about A or A-complement
At least one of A or A-complement must be unrecognizable.
Combination theorem from HW
C(n, k) = C(n-1, k-1) + C(n-1, k)
If A and A-complement are both recognizable, then A is...
Decidable
If A is Turing-Reducible to B and B is decidable, then A is
Decidable
If A is Turing-Reducible to B and A is decidable, then B is
Decidable/Undecidable
If A is Turing-Reducible to B and B is undecidable, then A is
Decidable/Undecidable
Every language is recognizable? (T/F)
False
Kruskal's Algorithm
For each edge in the set of edges E sorted by weight, add the next edge if it does not induce a cycle.
Euclid's Algorithm
GCD(a, b) = GCD(b, a mod b), finds GCD, run-time = O(log(a+b))
Exchange Property of MSTs
If T is an MST, and we construct a new tree S, which exchanges an edge from T for an edge not in T. Then w(T) <= W(S).
D&C Multiplication
Karatsuba Algorithm N1 x N2 = M2*10^n + (M1-M2-M3)10^(n/2)+M3 M1 = (a+b) x (c+d) M2 = a x c M3 = b x d Time complexity: 3 T(n/2) recursive calls, 2 O(n) subtractions. Applying Master's Theorem: T(n) = O(n^(log_{2}^{3})) = O(n^1.585)
Negative Result
No algorithm can solve a given problem
Closest Point Problem
Procedure: 1. Sort the list of points 2. Go through the list and manually check the distance of points within some fixed amount. This will be done in a constant number of steps. 3. Return closest distance we find. Runtime: O(nlogn) to sort points. O(1) to check closest distance to other desks for each desk. Total time complexity is O(nlogn) + O(n) = O(nlogn).
lossless data compression
Represent data w/o losing info.
Potential Function
S Maps an algorithm to the set of real numbers s.t. 1. S strictly decreases 2. S has a lower bound
Lossy data compression
Some loss of data is OK
Proving Big O:
There are positive real-numbers C and K s.t. 0 <= f(x) <= C*g(x) for every x >= k.
Positive Result
There exists an algorithm that solves a given problem
Intrinsic Complexity
Time complexity of the most efficient algorithm that solves a problem.
If A is Turing-Reducible to B and A is undecidable, then B is
Undecidable
An incomputable problem is one that takes
an uncountable number of steps
Little-o notation
f(n) < M(g(n)), where M is some constant. Describes an upper bound that cannot be tight.
Big-o notation
f(n) <= M(g(n)), where M is some constant. Describes an upper bound that can be tight or loose.
