Chapter 1. Computation theory


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1 Chapter 1. Computation theory In this chapter we will describe computation logic for the machines. This topic is a wide interdisciplinary field, so that the students can work in an interdisciplinary context. The field of computational logic include all kind of computational logic for machines or computers that is use of logic in computer science. Outline Introduction Turing Machine Binary System Boolean Algebra and Logic Gates Circuit Model of Computation Algorithm Models of computation P and NP Problems Motivations and ideas underlying quantum computation Introduction Let us briefly summarize developments of the computational logic. At 17th century Leibniz introduced an idea about the computational logic to determine the truthvalue by using an algebraic logic and building a special language of alphabets George Boole proposed a (boolean) logic. This logic based on logical calculus of truth values. This algebra is the algebra of two values (0,1).... The search for algorithm to determine the truthvalue of sentences was appeared mid 1920's. The algorithm answer the problem as Yes or No. This algorithm is known as decision algorithm. A complete decision algorithm gives answer of the problem in a finite time. An incomplete decision algorithm never finishes the computation if the answer is No, or may or may not finishes the computation if the answer is Yes. For significantly large problems the incomplete algorithm is important. In 1935 Alan Turing proposed a theoretical model of a device that can execute and possible algorithm. This device is known as Turing Machine. John von Neumann and his collaborators was designed first multipurpose programmable computer based on principles of the Turing Machine. Computation theory is very important because computers were first proposed and devised theoretically. Turing Machine Turing machine is a theoretical model of a device that can execute any possible algorithm. Turing machine has the following abilities: It can read and write on a tape It can move right or left on a tape
2 It halt (stop running) when it reach special accept state or special reject state, otherwise the machine never halt and continue loops. It has infinity memory! Example: Determine for the following statement Turing machine HALT or continue to LOOPS. for i=1 to infinity if 2*i is not some of two primes then HALT (STOP) It is obvious that this is an infinite LOOPS statement. Unfortunately, such an infinite LOOPS detector cannot exist. (There is no algorithm that detect an infinite loop). This problem is known as HALTING problem. Turing machine has the following components: {Q,,,, q 0, q accept, q reject, t} Q is finite set of state (s 0, s 1, s 2,...) is input alphabet ( ) is tape alphapet q 0 is the start state q accept is the accept state q reject is the reject state is transition function from one stateto another by a given condition. # is blank symbol(not in, #2 ) For each transition, the tape begins in a state s, then read a symbol a, then change to a state q(p q) and write a symbol b, moves left or right (L or R). If we are dealing Turing Machine as computers then our input alphabet will be ={0,1} (The input will be binary). a 0 a 1 a 2 a 3... a n # # # CONTROL Turing machine can be used as; function evaluator( Read w then write f(w)), language accepters (Reads w then write 1 if w is accepted otherwise write 0 or continue to read new value), problem deciders (read a decision problem coded in binary and it halts by writing 1 on tape if the answer YES otherwise if the answer NO it writes 0) etc. Example. Design a Turing Machine, erase input and write blank. Solution: ERASE Turing machine
3 Example: Design a Turing machine that will decide whether the binary input represent an EVEN number. Solution: EVEN Turing machine Basic principles: Move right to the first blank then move left one cell. If the current cell is 0, the number is even. If the current cell is 1, the number is odd. Erase the tape and write 0 or 1, accordingly. Binary System and Boolean Algebra Classical computers use binary digits. Information is represented electronically in a computer using voltages of different levels. For instance, a 0 could be represented by a 0 voltage and a 1 could be represented by a voltage of 3 volts. Human have 10 fingers which has led to adaptation of decimal (base 10) number system. Decimal number system consists of 10 digits (09). Binary number system (base 2) the digit takes values 0 or 1. Converting binary to decimal: Using power of 2 we can convert any binary number to the decimal number. For example: Converting decimal to binary Dividing given decimal number by 2 and writing the remainder in each division one can convert the number from decimal to binary. As an example convert the number 83 to the binary. Divided Divisior Remainder 83 83/ / / / / / Remove 1 last cell then the number can be written as:(from bottom to up):
4 Divided Divisior Remainder Then the number / / / / / / / Remove 1 last cell Conversion of Decimal Fractions to Binary There are various methods to convert a decimal fraction to the binary. One of them is based on multiplication of decimal fraction by 2. As an example consider conversion of 0.125: Number 2xNumber Integer Part Result Sometimes decimal fractions produce infinite binary fraction. Example: Convert decimal fraction 0.1 to binary: 2xNumber 0.1x2 0.2x2 0.4x2 0.8x2 0.6x2 0.2x2 0.4x2 0.8x2 0.6x2 Integer part Then 0.1= Conversion of Binary Fractions to the Decimal A binary fraction of a number can be converted to a decimal number using powers of (2). For example Boolean Algebra
5 Logic circuits are basis of the classical computer system. To appreciate operation of the computer system you have to understand digital logic and Boolean algebra. Boolean Algebra is both a formalization of the algebraic aspects of logic. A Boolean variable can take two values 1(or T or ON) or 0 (F or OFF). Basic Boolean operations are AND, OR and NOT. We can make more complicated functions from the basic Boolean operations. AND operation is function of two variable (A, B). Boolean function representation F(A,B)=A.B. Table shows result of the AND operation. Truth Table AND Gate A B F OR operation is function of two variable (A,B). Boolean function representation F(A,B)=A+B Truth Table OR Gate A B F NOT operation replace a Boolean function with its compliment F(A)=A'. Complement 1=1'=0, Complement 0=0'=1. A' is read as NOT A ot Compliment A. Truth Table NOT Gate or inverter A F Basic theorems: Boolean operations have the following properties:
6 Algebra X+0=X X+1=1 X+X=X (X')'=X X+X'=1 Dual of the operation X*0=0 X*1=1 X*X=X (X')'=X X*X'=0 Dual of a Boolean expression can be obtained by replacing ANDs with ORs, ORs with ANDs, 1s with 0s, 0s with 1s. Theorems of the Boolean operations: Commutative X+Y=Y+X Associative (X+Y)+Z=X+(Y+Z) Distributive A(B+C)=AB+AC DeMorgan's Law (X+Y)'=X'Y' Dual Laws X*Y=Y*X (X*Y)*Z=X*(Y*Z) A+BC=(A+B)(A+C) (XY)'=X'+Y' Dual of Distributive law only valid for Boolean Algebra! Circuit Model of Computation As we mentioned before we can construct more complicated Boolean functions from the basic Boolean functions. The followings include some examples. Three input AND function Example: Factorize the Boolean expression A+B'CD, then implement the circuit. A+B'CD=(A+B')(A+CD) =(A+B')(A+C)(A+D) Its logic gate is:
7 We can construct the NAND and NOR gates from the combinations of the basic gates. A NAD gate can be thought as AND gate followed by and inverter and NOR gate can be thought as OR gate followed by an inverter. The last gate type is XOR gate. Its function is: Algorithm An algorithm is defined as a specific set of instructions for solving a problem. The word "algorithm" is a distortion of alkhwārizmī, a Persian mathematician who wrote an influential treatise about algebraic methods. The process of applying an algorithm to an input to obtain an output is called a computation. Some examples Everyday algorithm An algorithm is a step by step representation of a program. An algorithm can be presented for solving a mathematical problem by using a computer as well as to plan our daily life. In order to make an algorithm our plan include INPUT PROCESS ASSUMPTIONS (if any) OUTPUT Example: Calling a friend on phone
8 Pick up the phone Code friend s number (INPUT) Press Call buton (PROCESS) If busy hang up phone then call again after 5 min. (ASSUMPTION) If no answering, then hang up and aster 2 hour call again (ASSUMPTION) Talk to friend (OUTPUT) Example: Find the greatest common divisor (GCD) of two integers, m and n. Euclid's Algorithm: while m is greater than zero: If n is greater than m, swap m and n. Subtract n from m. n is the GCD Let us apply this algorithm for the numbers m=6 and n=21 m>0. n>m then m=21, n=6 216= =9 96=3 Then m=6; n=3 63=3 33=0 therefore n=3 is GCD HW. Write an algorithm to Determine prime numbers. How many prime numbers between 1 and For more algorithm visit: For factoring algorithms read the notes: We will compare factoring algorithms have been presented quantum computers and classical computers. Models of computation In this section we treat concept of computation and algorithm. Mechanical machines compute some output from some input. Everyday we use our computer to solve some typical problems. For example, given a list of numbers {8,12,98,1,52}, sort them in a order from smallest number to bigger... The output becomes {1,8,12,52,98}. Below some models of the computation are briefly summarized. At the beginning of this chapter we have discussed Turing Machine. Note that all computations that could ever be carried out on any other mathematical machinemodels can also be carried out by Turing machines We also discussed Boolean (logical) circuits provide a model of computation that is not equivalent with the Turing machine. We have introduce some models of the computation which
9 a) Random Access Machines b) Straight Line Programs and Circuits (Boolean logical circuits) c) Turing Machines d) Decision Trees e) Machines That Make Random Choices P and NP Problems (Polynomial and NonDeterministic Polynomial problems) When you run a program one of the important question is: How long does it take to execute the program. Suppose that you have a set of number. In order to sort the numbers you can write an algorithm. The executing time probably will be proportional to N 2 or Nlog(N) or similar expressions. If the executing time proportional to N, N 2 or any power of N then the problem is called Polynomial (P) problem. For a given problem if the executing time proportional to the Nth power of a number i.e e N, or x N then the problem is known as NP (NonDeterministic Polynomial problems). Note that A Major application is in the cryptography area, Sipser says, where the security of cryptographic codes is often ensured by the complexity of a computational task. Example. Factorization of a number as product of the prime numbers is the problem between P and NP. Solution: You can write a simple division algorithm. Example. A traveling salesman has to travel a bunch of cities, in a such way that using the shrtest distance. This problem is between NP and NP hard. Solution: We have to calculate all alternative paths to find shortest distance. Table shows number of operations. The average time to determine shortest optimum path, with a computer operating per second can be calculated as: #path/ #cities Alternatives #path Timing 2 (12) 1(21)! 1 microsec 3 (123)(132) 2 (31)! 0.5 microsec 4 (1234) (1342) (1423) (1324) (1243) (1432) 6 (41)! 6 microsec (51)! 24 microsec (61)! 0.12 msec (101)! 0.36 sec (151)! 1 year Motivations and ideas underlying quantum computation Please read the articles given in the pages
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