Coursepack2-2 - Module 12 Computation and Configurations...

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1 1 Module 12 • Computation and Configurations – Formal Definition –Examp les 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
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2 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 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 • ...
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3 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 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? •N o ! • Both previous inputs, 10 3 as well as 53 10 eventually reached the same configuration (Line 3, x=10, y=3, r=1)
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4 7 Module 13 • Studying the internal structure of REC, the set of solvable problems – Complexity theory overview – Automata theory preview • Motivating Problem – string searching 8 Studying REC Complexity Theory Automata Theory
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5 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 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 –E lem en t • Input: list of n numbers • Task: return any of the n numbers REC RE - REC All languages -RE
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6 11 Resource Usage * • How do we formally measure the hardness of a problem?
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Coursepack2-2 - Module 12 Computation and Configurations...

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