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COT5310hw6key

# COT5310hw6key - COT 5310 Homework 6 Key Fall 2007 1 Using...

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COT 5310 Homework 6 Key Fall 2007 1. Using reduction from the complement of the Halting Problem, show the undecidability of the problem to determine if an arbitrary partial recursive function, f , has a summation upper bound. This means that there is an M , such that the sum of all values in the range of f (repeats are added in and divergence just adds 0) is M . The set HALT = {( f, x ) : f ( x ) ↓} , therefore CoHALT = {( f, x ) : f ( x ) ↑} . The set of partial func- tions f with a summation upper bound can be described as UB = { f : x y [ y > x ( f ( y ) = 0 or f ( y ) )] } . or in other words, only finitely many of the outputs can be non-zero. To show that CoHALT m UB we define g ( ( f, x ) ) = g f,x as g f,x ( y ) = μ t [STP( f, x, t )] + 1 . Then if ( f, x ) ∈ CoHALT, g f,x ( y ) for all y N so dom( g f,x ) = g f,x UB. If ( f, x ) / CoHALT, g f,x ( y ) 1 for all y N . This means that no summation upper bound exists and g f,x / UB. 2. Use one of the versions of Rice’s Theorem to show the undecidability of the problem to determine if an arbitrary partial recursive function, f , has a summation upper bound. This means that there is an M , such that the sum of all values in the range of f

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COT5310hw6key - COT 5310 Homework 6 Key Fall 2007 1 Using...

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