#### Amdahl's Law

(a) Illustrate Amdahl's law in terms of speedup vs. sequential portion of program by showing the speedup for N = 8 processors when the sequential portion of the program grows from 1% to 25%. (b) ( Amdahl's law) With sequential execution occurring 15% of the time: (i) What is the maximum speed...

#### Logic: Turing Machines

Please see the attached file for the fully formatted problems. (a) We wish to design a Turing machine which, using monadic notation, inputs a pair (in, n) of positive integers in standard starting position (on an otherwise blank tape), and which halts scanning the rightmost of a string of in is ...

#### Turing-recognizable language

Let A be a turing-recognizable language consisting of descriptions of Turing machines, {&#61665;M1&#61681;, &#61665;M2&#61681;,...}, where every Mi is a decider. Prove that some decidable language D is not decided by any decider Mi whose description appears in A. (Hint: You may find it helpful to co...

#### Turing Machine Probabilities

Let B be a probabilistic polynomial time Turing machine and let C be a language where, for some fixed 0 < &#61646;1 < &#61646;2 < 1, a. w &#61647; C implies Pr [B accepts w] &#61603; &#61646;1, and b. w &#61646; C implies Pr [B accepts w] &#61619; &#61646;2. Show that C &#61646; BPP. HINT: ...

#### Automata

Let &#61542; be a 3cnf-formula. An &#61625; assignment to the variables of &#61542; is one where each clause contains two literals with unequal truth values. In other words an &#61625; -assignment satisfies &#61542; without assigning three true literals in any clause. a. Show that the negation ...

#### Automata and Computability

Consider the problem of testing whether a Turing machine M on an input w ever attempts to move its head left when its head is on the left-most tape cell. Formulate this problem as a language and show that it is undecidable.

#### Automata and Computability

A Turing machine with doubly infinite tape is similar to an ordinary Turing machine except that its tape is infinite to the left as well as to the right. The tape is initially filled with blanks except for the portion that contains the input. Computation is defined as usual except that the head ne...

#### Turing machine with input in unary notation

Devise a Turing machine with input given in unary notation (i.e., a string of n 1's denotes the integer n, and numbers are delimited by 0's) such that the machine produces the following output: 0 if x is divisible by 4 1 if x is congruent to 1 modulo 4 2 if x is congruent to 2 modulo 4 3...