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assign3 - CS 220 Winter 2011 Assignment 3 Due February 17 1...

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CS 220 Winter 2011, Assignment 3, Due: February 17 1. Show that the function f ( x, y ) = x * y cannot be computed by any L 1 -program. 2. Consider a programming language Q whose only non-I/O instructions are of the forms: x 0, x 1, x x + y , x x - y (monus), do x ... end . Denote by Q i the set of all programs with do -loop depth i . Show that a function is computable by a G -program with running time bounded by an elementary function if and only if it is computable by a Q 1 -program. 3. Show that for i 1, a function is computable by a Q i -program if and only if it is computable by an L i +1 -program. 4. From the definition of partial recursive functions, show that every partial recursive function can be computed by a G program. 5. Consider a DFA (deterministic finite automaton) M operating on the upper-right quadrant of the plane, filled with λ ’s, with the boundaries delimited by $’s. M starts in its intital state on the $ in the cell at the origin (location (0 , 0)). M has a finite number of states and can only read and not write. Note that M has now four directions of moves. Show that the halting problem for such DFAs is undecidable.
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