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My University Life - PowerPoint PPT Presentation

My University Life

Given two positive integers x and y, design a tm that. computes x y. ... design a tm that multiplies two integers in unary notation. ... – powerpoint ppt presentation.

• Turing machines are not only interesting as language accepters, they provide us with a simple abstract model for digital computers in general.
• The input for a computation is all the non-blank symbols on the tape at the initial time. The output is whatever on the tape after the computation.
• A function f with domain D is computable if there exists some TM
• We will see shortly, all the common mathematical functions, no matter how complicated, are computable.
• Given two positive integers x and y, design a TM that
• computes xy.
• For simplicity, we use unary notation in which any positive integer x is represented by w(x) in 1, such that w(x) x. We assume that w(x) and w(y) are on the tape in unary notation, separated by a single 0, with the head on the leftmost symbol of w(x). After the computation, w(xy) will be on the tape followed by a single 0, with the head at the left end of the result.
• We therefore want to design a TM for performing the computation
• All we need to do is to move the separating 0 to the right end of w(y), so that the addition amounts to nothing more than the coalescing of the two strings. To achieve this, we construct with
• Unary notation is very convenient for programming TMs. The resulting programs are much simpler than binary or decimal representations.
• Other basic operations are copying strings and simple comparisons. These can also be done easily on a TM.
• Design a TM that copies strings of 1s.
• More precisely, we want a TM to perform
• To solve the problem, we implement the following intuitive process
• 1. Replace every 1 by x.
• 2. Find the rightmost x and replace it with 1.
• 3. Travel to the right end and create a 1 there.
• 4. Repeat steps 2 and 3 until there are no more xs.
• A Turing machine implementing the process is
• where is the only final state. You should try to trace the program with some input strings.
• Design a TM as a comparer. More precisely, we want a TM to perform
• The idea is to match each 1 on the left of dividing 0 with the 1 on the right. At the end of the matching, we will have on the tape, one of the following
• Depending on whether or In the 1st case, we encounter the blank at the right of the working place when we attempt to match another 1. In the 2nd case, we still find a 1 on the right when all 1s on the left have been replaced. We use these signs to enter the two different states. The complete program is left as exercise.
• Study how to combine together the TMs which perform basic operations into a single TM for complicated tasks.
• We start with a high-level description (block diagram or pseudo-code), then refine it successively until the program is in the actual language with which we are working.
• Design a TM that compute the function
• where x and y are positive integers in unary representation.
• The computation of f(x,y) can be described by the diagram below.
• Design a TM that multiplies two integers in unary notation.
• Assume the initial and final tape contents are as follows 00y0x and 0xy0y0x.
• The multiplication can be seen as a repeated copying of the multiplicand y for each 1 in the multiplier x. The following pseudo-code shows the main steps of the process.
• 1. Repeat the following steps until x contains no more 1s
• Find a 1 in x and replace it with another symbol a.
• Replace the leftmost 0 by 0y.
• 2. Replace all as with 1s.
• The Turings thesis states that any computation that can be carried out by mechanical means (digital computer) can be performed by some Turing machine.
• No one has been able to suggest a problem, solvable by an algorithm, for which a TM program cannot be written.
• Construct TMs to compute the following functions, where x and y are positive integers represented in unary.
• Write the pseudo-code for the TM program that compute the following functions (you may use adder, copier or multiplier in your program.)

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