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Coursepack2

# Coursepack2 - Module 12 Computation and Configurations...

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1 Module 12 Computation and Configurations Formal Definition Examples

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2 Definitions Configuration Functional Definition Given the original program and the current configuration of a computation, someone should be able to complete the computation Contents of a configuration for a C++ program current instruction to be executed current value of all variables Computation Complete sequence of configurations
3 Computation 1 1 int main(int x,y) { 2 int r = x % y; 3 if (r== 0) goto 8; 4 x = y; 5 y = r; 6 r = x % y; 7 goto 3; 8 return y; } Input: 10 3 Line 1, x=?,y=?,r=? Line 2, x=10, y=3,r=? Line 3, x=10, y=3, r=1 Line 4, x=10, y=3, r=1 Line 5, x= 3, y=3, r=1 Line 6, x=3, y=1, r=1 Line 7, x=3, y=1, r=0 Line 3, x=3, y=1, r=0 Line 8, x=3, y=1, r=0 Output is 1

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4 Computation 2 int main(int x,y) { 2 int r = x % y; 3 if (r== 0) goto 8; 4 x = y; 5 y = r; 6 r = x % y; 7 goto 3; 8 return y; } Input: 53 10 Line 1, x=?,y=?,r=? Line 2, x=53, y=10, r=? Line 3, x= 53, y=10, r=3 Line 4, x=53, y=10, r=3 Line 5, x=10, y=10, r=3 Line 6, x=10, y=3, r=3 Line 7, x=10, y=3, r=1 Line 3, x=10, y=3, r=1 ...
5 Computations 1 and 2 Line 1, x=?,y=?,r=? Line 2, x=53, y=10, r=? Line 3, x= 53, y=10, r=3 Line 4, x=53, y=10, r=3 Line 5, x=10, y=10, r=3 Line 6, x=10, y=3, r=3 Line 7, x=10, y=3, r=1 Line 3, x=10, y=3, r=1 ... Line 1, x=?,y=?,r=? Line 2, x=10, y=3,r=? Line 3, x=10, y=3, r=1 Line 4, x=10, y=3, r=1 Line 5, x= 3, y=3, r=1 Line 6, x=3, y=1, r=1 Line 7, x=3, y=1, r=0 Line 3, x=3, y=1, r=0 Line 8, x=3, y=1, r=0 Output is 1

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6 Observation int main(int x,y) { 2 int r = x % y; 3 if (r== 0) goto 8; 4 x = y; 5 y = r; 6 r = x % y; 7 goto 3; 8 return y; } Line 3, x= 10, y=3, r=1 Program and current configuration Together, these two pieces of information are enough to complete the computation Are they enough to determine what the original input was? No! Both previous inputs, 10 3 as well as 53 10 eventually reached the same configuration (Line 3, x=10, y=3, r=1)
7 Module 13 Studying the internal structure of REC, the set of solvable problems Complexity theory overview Automata theory preview Motivating Problem string searching

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8 Studying REC Complexity Theory Automata Theory
9 Current picture of all languages Α ll Languages RE-REC Α ll languages - RE Half Solvable Not even half solvable Which language class should be studied further? REC Solvable

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10 Complexity Theory In complexity theory, we differentiate problems by how hard a problem is to solve Remember, all problems in REC are solvable Which problem is harder and why? Max: Input: list of n numbers Task: return largest of the n numbers Element Input: list of n numbers Task: return any of the n numbers REC RE - REC All languages - RE
11 Resource Usage * How do we formally measure the hardness of a problem?

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