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Theory Of Computation

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Theory Of Computation

Dr. Adam P. Anthony

Lectures 25 and 26

- Computer Science: do we need computers?
- Computation Theory
- Functions
- Turing Machines
- Universal Programming Languages
- The Halting problem

Computer science is no more about computers than astronomy is about telescopes.

- Edgser W. Dijkstra

- Insight: the computation is separate, in concept, from the computer
- A computer, then, is just some object that can carry out the computation
- Humans
- Brain, often supplemented by pencil, paper

- Charles Babbage
- Difference engine, analytical engine
- Controlled using clockwork-type components

- Difference engine, analytical engine
- ENIAC
- Controlled using vacuum tubes

- Intel 8080
- Controlled using micro-transistors

- Humans

- Can a simple calculator help you find your way around cleveland?
- How about a (dumb) phone?
- Aside from making calls

- How about a smart phone?
- How about a laptop?
- How about a desktop?
- Which of these count as computers?

- Some ‘computers’ are designed only to achieve a limited number of specific tasks, and to do that either at high speed or at a low cost:
- Digital Phones (Cell and otherwise)
- Encryption chips too!

- Various scientific measuring devices

- Digital Phones (Cell and otherwise)
- Others are considered General Purpose Computers
- Anything that can be computed, can be done on one of these machines

- The theory of computation aims to answer the following questions:
- What is a general purpose computer?
- What problems can I solve with a general purpose computer?
- Is this specific computer general purpose?
- Given a general purpose computer, how difficult will it be to solve a specific problem?

- Successful mathematician
- Cryptographer
- Helped build some early (classified) computation devices
- Many ideas predated the first computers
- Turing Machine
- Computability

- Helped define what is possible on a computer, and what is not

- A function is a mapping of inputs to outputs
- Sum(2+2) = 4
- Feet-Centimeter(500) = 15,240
- Sort([1,3,2,3,6,9,7,8,0]) = [0,1,2,3,3,6,7,8,9]
- Father(Bill Smith) = Edward Smith

- Some functions are computable
- Given the input, an algorithmic process can always be applied to get an exact answer for the output

- A general purpose computer can compute any computable function, and no others

- Control Unit: The actual machine
- Tape: infinitely long memory
- Read/Write Head: Used to read information from the tape, erase information, write new information
- Reads one character at a time
- Moves left/right one position at a time

- State: Description of current situation, based on tape values

State = START

- Each new Turing Machine has an alphabet of characters that it understands, and a set of states that help it make decisions
- Given the state the current character read by the the read/write head, and a program of execution, the Control Unitdecides to:
- Stop running (HALT state)
- Write over the current character
- Move one space left/right
- Change States

- Alphabet: {0,1,*}
- A single binary number is represented as *101010*

- States: ADD,RETURN,CARRY,OVERFLOW,HALT
- Program to increase a positive binary integer by 1:

- Multiple Turing machines are no more powerful (though possibly faster) than a single Turing Machine
- Any Turing Machine can ‘simulate’ another Turing Machine
- Result: We can use unambiguous complex commands in the control unit’s program!
- Command: “Move 5 Spaces to the left”
- Turing Machine reads: “Execute the Turing Machine routine that moves 5 spaces to the left”

- Command: “Move 5 Spaces to the left”
- Theoretically speaking, one should typically demonstrate the sub-program is computable first

Any function that can be computed using a Turing Machine is also computable using any other general purpose computer (i.e., the function is computable)

SO WHAT???

Who Cares?

- If Power = ‘number of functions I can compute,’ then a Turing machine is the most powerful computer imaginable
- Or, at least, it ties with any other computer
- It computes ALL of the computable functions!

- Control Unit = Processor
- Alphabet = {0,1}
- States = Op Codes
- Read/Write Head = BUS
- Programs = Software
- Tape = RAM
- Infinite?????
- No, but for most purposes it is long enough to solve the problem

- Tape = External Storage
- Only limited by the number of natural resources we can obtain from the entire universe (so, probably infinite!)

- Programming languages usually market their ‘features’
- Meant to make programming easier

- Bare-Bones Language:
- Only includes features that are 100% necessary to be equivalent to a Turing machine:
- Variable names: all variables are in binary
- clear statement: set a variable = 0 (clear X;)
- incrstatement: increase a variable by 1 ( incr X; )
- decrstatement: decrease a variable by 1 (decr X;)
- While/end: continue execution until a variable = 0
- while X not 0 do;
….

end;

- while X not 0 do;

- Only includes features that are 100% necessary to be equivalent to a Turing machine:

- Can you use Bare-Bones to:
- Set the variable Z = 4?
- Add X + Y = Z? (use one variable each for X,Y,Z)
- Copy the value of X into Y?

- Computer scientists have proven that any computer that can execute the Bare-Bones language is equivalent in power to a Turing Machine
- Heaven forbid!

- Useful conclusion:
- Any programming language does at least the same as Bare-Bones (hopefully more!) will also be Turing Equivalent
- The extra features are just for convenience

- Computers are just tools for completing computations
- Theory of computation: what is possible/impossible for all computers? What is computable?
- Turing Machine: imaginary ‘all powerful’ computer
- Church-Turing thesis states no computer can do better

- Modern computers are equivalent to Turing Machines
- Any algorithm we implement on a computer is computable

- It’d be nice to know, before we start if a problem is noncomputable
- Halting problem as an example

- Even if a problem is computable, it would be nice to know in advance if it is easy or hard to solve
- Even if we can solve a problem, it would be nice to know how long it will take to solve it
- Save effort in solving complex problems
- Take advantage of complexity

- Some problems can’t be solved.
- Consider: Given the source code for any computer program, can you analyze the code and decide if it will it ever stop running?

Does this program halt?

int X = 3

while( X > 0)

x = x -1

Does this program halt?

int X = 3

repeat

x = x +1;

until x = 0

How about this program?

virtual void estimate_sigmas(){

sigmas = std::vector< std::vector<Matrix> >(num_clusters);

for(inti = 0; i<num_clusters; i++){

sigmas[i] = std::vector<Matrix>(num_clusters);

}

for(inti = 0; i<num_clusters; i++){

for(int j = 0; j<num_clusters; j++){

sigmas[i][j] = zero_matrix<double>(proper_size,proper_size);

}

}

Vector temp_v(proper_size);

edge_iteratorebg,end;

intc_i,c_j;

for(tie(ebg,end) = edges(data); ebg!=end; ++ebg){

if(data[*ebg].type == edge_type && data[*ebg].exists){

c_i = data[source(*ebg,data)].clustering();

c_j = data[target(*ebg,data)].clustering();

temp_v = get_edge_vector(*ebg) - ic_means[c_i][c_j]

sigmas[c_i][c_j] = sigmas[c_i][c_j] + outer_prod(temp_v,trans(temp_v))/observed_edge_prob(c_i,c_j);

}

}

}

- Sometimes, we can work out answers for simple, example inputs of hard problems, but:
- What algorithm did you use to decide for the first two programs?
- Can you generalize it to the third?

- To prove something is not computable, we’ll use the following strategy:
- Assume that there is an algorithm that can solve the problem all the time
- Show that, regardless of how the algorithm works, that there is at least one case where the algorithm will fail
- Contradicts part 1, which claimed it ‘always works’

- Assume, for the sake of contradiction, that there exists a computer program that, given any other computer program as input, can tell us if it stops:
- STOPS(Program)

- Make a new program:
- Opposite(X):
- If STOPS(X), then run forever.
- Otherwise, stop!

- Opposite(X):
- Opposite is a program, which itself accepts program code as input. What happens when we try to run
- Opposite(Oppisite)?
- If STOPS(Opposite), then Opposite will run forever
- Otherwise, stop!

- Opposite(Oppisite)?

- Existence proof: Since there’s one program that exists which we can’t compute if it halts, then there may be (probably are) others
- If there’s one problem that seems computable, but is not, then there are others
- Look up Wang Tiles for an interesting example!

- Program Analysis: “Does my program compute X?”
- Any place in the code where X is computed, add a HALT command
- Changes to “Does my program ever halt?”

- Any place in the code where X is computed, add a HALT command

- Algorithmic Complexity refers to how many resources (time and memory) a computer will need to solve a problem
- How long will it take to process all the data?
- How much space (Memory) will we need?
- If we use more space, will it take less time?
- Are some problems harder to solve than others?
- Can we figure that out before we try to solve them?

- How can we take advantage of complexity?

- It’s one thing to take a specific algorithm and say it’s complex (or not):
PROCEDURE Add(X,Y):

sum = X + Y

RETURN sum

- Because solving the problem, and doing so efficiently, are two different things:
PROCEDURE BadAdd(X,Y):

Z = 1000000000

sum = 0

REPEAT

Z = Z - 1

UNTIL Z = 0

sum = X + Y

RETURN sum

- To say that a PROBLEM is difficult, you need to prove that there are no easy ways to solve it

- Looked at briefly in chapter 5
- Principal method for analyzing algorithm complexity:
- How many steps does it take to complete the entire algorithm?

- Steps are often based on the size of the input:
PROCEDURE add-all(L):

sum = 0

count = 0

WHILE count < length(L):

sum = sum + L[count]

RETURN sum

- How many steps to add a list with 10 numbers? 1000 numbers?

- Many algorithms only need enough space to hold the input data
- Procedure add-all (L) Only needs enough space to store L

- Others, because the problem is more difficult, use supplementary data
- EX: Binary Search Trees

- Still others use extra memory to be faster
- EX: Dynamic Programming Fibonacci sequence

- The most reoccurring algorithmic runtimes are (in order) constant, log(N), N, N*log(N), N2, N3, and an
- A polynomial problem is any problem for which the best known algorithm for solving it has time complexity that is no worse than a polynomial function f(N) = Nd where d can be any number and N is the size of the input.
- All problems that we can solve with an exact solution is a reasonable amount of time are in the polynomial class
- Problems outside this class are referred to as intractible
- For short, we refer to the entire set of all polynomial problems in the world a the set P

- A Nondeterministic machine: is a theoretical machine that just knows how to solve a problem, no matter how hard it may be
- A Nondeterministic Polynomial Problem is a problem for which the best known algorithm for solving has a polynomial runtime, but its execution would require a nondeterministic machine
- Another intuition: these are problems for which finding the solution is hard, but checking the solution for correctness is easy
- We refer to the set of problems in this domain as NP

NP DOES NOT MEAN

“NOT POLYNOMIAL”

- In General, the class of problems in NP consists of problems that are difficult, but useful
- Traveling Salesman example—best solution is exponential

- Within a single class, some problems are harder than others
- In P, it is harder to sort a list than it is to add two numbers

- In the class NP, we identify a set of problems that are the most difficult to solve in the entire set
- Called NP-Complete problems
- The speed at which we can solve these problems determines how fast we can solve the lesser problems

- All problems that are in the set P must also be in the set NP
- Why?

- Big, unknown question in Computer Science:
- DOES P = NP???

- What does it mean if P = NP?
- One approach: Find a polynomial solution for an NP-Complete problem
- Thousands have tried, all have failed

- Most people believe, but can’t prove P NP