By Eitan Gurari

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Sum1 := 0 sum2 := 0 do if eof then accept do /* Guess where the next input value belongs. */ read x sum1 := sum1 + x or read x write x sum2 := sum2 + x until sum1 = sum2 /* Check for the correctness of the guesses, with respect to the portion of the input consumed so far. 7 A nondeterministic program for partitioning the input into two subsets of equal sums of elements. For instance, on input "2, 1, 3, 4, 2" the possible outputs are "2, 1, 3", "1, 3, 2", "2, 4", and "4, 2". On the other hand, no output is defined for input "2, 3".

Finally B gives the output S' of A to Tg, and assumes the same output S as the one obtained from Tg. 1 a. Find all the alphabets that use only symbols from the set {a, b, c}. Which of the alphabets is unary? Which is binary? b. Let S be a set of t symbols. How many unary alphabets can be constructed from the symbols of S? How many binary alphabets? 2 For each of the following conditions find all the strings over the alphabet {a, b} that satisfy the condition. a. No symbol is repeated in . b. The length of is 3, that is, | | = 3.

Two LOOP programs are said to be equivalent if on identical input values they produce the same output values. 7 The following problems are known to be undecidable. Can you show that they are partially decidable? a. Hilbert's tenth problem b. 1 Let be the set { m | m = 2i for some integer i }. Show that the problem of multiplication of numbers from is reducible to the problem of addition of integer numbers. 2 Show that the nonemptiness problem for programs is reducible to the acceptance problem for programs.