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INHERENT LIMITATIONS OF COMPUTER PROGRAMS CSci 4011

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GENERALIZED GEOGRAPHY b a e c d f g i h

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We show that FG P GG GG IS PSPACE-HARD We convert a formula into 〈 G,b 〉 such that: Player E has winning strategy in if and only if Player I has winning strategy in 〈 G,b 〉 For simplicity we assume is of the form: = x 1 x 2 x 3 … x k [ ] where is in cnf

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b c TRUEFALSE x1x1 x2x2 xkxk c1c1 c2c2 cncn x1x1 (x 1 x 1 x 2 ) ( x 1 x 2 x 2 ) … x2x2 x2x2 x1x1 x1x1 x2x2

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b TRUEFALSE x1x1 c1c1 x1x1 x1x1 x1x1 c x 1 [ (x 1 x 1 x 1 ) ]

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n x n GO, chess and checkers can be shown to be PSPACE-hard Question: Is Chess PSPACE complete? No, because determining whether Player I has a winning strategy takes constant time and space

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Office hours next week: Tuesday 2-4pm Wednesday 10am-noon Thursday 10am-noon Friday 3-5pm FINAL EXAM SATURDAY, MAY 18 8:00AM – 10:00AM. Reminder: one page (8.5×11”) “cheat sheet”

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1*(0 ∪ 01) 0 q0q0 q2q2 1 q1q1 0 1 {0 k 1 k | k ≥ 0} R → LRL | b L → a | b S → R | ε { ww | w ∈ {0,1}* } MODELS

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0 → 0, R readwritemove → , R q accept 0 → 0, R → , R 0 → 0, R → , L TURING MACHINES UNBOUNDED TAPE 0 q0q0 q0q0 q1q1 q2q2 q1q1 q2q2 0

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A TM = { M,w | M is a TM that accepts string w } A TM is undecidable:(proof by contradiction) Assume there is a program accepts to decide A TM. accepts( M ,w) = true if M(w) accepts false if M(w) does not. Construct a new TM LLPF that on input M , runs accepts( M , M ) and “does the opposite”: LLPF(PROG ) =if (accepts( PROG, PROG )) then reject; else accept. LLPF

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REDUCTIONS A m B if there is a computable ƒ so that w A ƒ(w) B ƒ is called a reduction from A to B A P B if there is a poly-time computable ƒ so that w A ƒ(w) B

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COMPLEXITY THEORY P = Problems where it is easy to find the answer. NP = Problems where it’s easy to check the answer. If P = NP then generation is as easy as recognition. Is there a fast program for this problem? PSPACE = Problems that can be solved in polynomial space. If P = PSPACE then TIME is as powerful as SPACE.

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COMPLETE PROBLEMS NP:3SAT,SUBSET-SUM,HAMPATH, VERTEX COVER, 3-COLOR,CLIQUE,… PSPACE:TQBF,FG,GEOGRAPHY,… If C is a class of languages and B is a language, then B is C-Complete if: 1. B ∈ C. 2. ∀ A ∈ C, A ≤ P C (i.e. B is C-Hard)

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SPACE COMPLEXITY 100 200 300 P VS NP 100 200 300 POTPOURRI 100 200 300 FINAL JEOPARDY 400

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A tight asymptotic bound on the function ƒ(n) = n 3/2 + n log(n 2 )

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A proof that SET-PARTITION ∈ NP, where SET-PARTITION = { T = {x 1,x 2,…,x n } | ∃ S ⊆ T so that Σ x ∈ S x = Σ y S y }.

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A proof that 3-COLOR ≤ P 4-COLOR, where 4-COLOR = { G | G is a graph that is 4-colorable. }

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A proof of SUBSET-SUM ≤ P SET-PARTITION, where SET-PARTITION = { T = {x 1,x 2,…,x n } | ∃ S ⊆ T so that Σ x ∈ S x = Σ y S y }.

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A winning strategy for Player I in this GG instance: a b c d

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A proof that PATH ∈ PSPACE, where PATH = { G,s,t | G is a directed graph with a path from s to t}

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A proof that MIN-FORMULA ∈ PSPACE, where MIN-FORMULA = { | is a formula and ∀ formulae . | | < | | ⇒ ∃ x. (x) (x) }

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A proof that 3SAT ≤ P TQBF.

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A CFG for the language { | is in cnf }, where variable x i is represented by the string x i.

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A proof that TQBF is not regular.

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A proof that A TM is PSPACE-Hard.

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A proof that { M | L(M) ∈ P } is undecidable, using the assumption P NP.

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FINAL JEOPARDY TRUE

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CSCI 2670 Introduction to Theory of Computing November 29, 2005.

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