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    <p>There are two properties that need to be satisfied for it to be NP Complete.</p> <p>A decision problem C is NP-complete if:</p> <ol> <li>C is in NP, and</li> <li>Every problem in NP is reducible to C in polynomial time.</li> </ol> <p>We have established 1. 2 results from the fact that every problem in NP is reducible to 3SAT and 3SAT is reducible to the current problem.</p> <p>Therefore it is NP-complete.</p> <p><em>(edit) Answer to the comment below:</em></p> <p>I will prove that SAT is reducible to the current problem, and since 3SAT is reducible to SAT, the result follows.</p> <p>Input formula is the conjunction of the following expressions:<p> (x<sub>1</sub> V x<sub>2</sub> V x<sub>3</sub> V ... x<sub>n</sub> V y<sub>1</sub>)<p> (x<sub>1</sub> V x<sub>2</sub> V x<sub>3</sub> V ... x<sub>n</sub> V y<sub>2</sub>)<p> (x<sub>1</sub> V x<sub>2</sub> V x<sub>3</sub> V ... x<sub>n</sub> V y<sub>3</sub>)<p> .<p> .<p> .<p> (x<sub>1</sub> V x<sub>2</sub> V x<sub>3</sub> V ... x<sub>n</sub> V y<sub>64</sub>)<p></p> <p>where each y<sub>i</sub> is a boolean based on what the order of the operators applied between all the x<sub>i</sub>'s is. i.e., y<sub>i</sub> can take a total of 4x4x4x4x1 values (assuming that only +, -, x, / are the operators and = is always the last operator; this can be changed if the operator set is modified to include other operators)</p> <p>If none of the expressions is true, then the complete expression will evaluate to FALSE, and there is no way to check unless we substitute all possible values, i.e, x<sub>1</sub> through x<sub>n</sub> as the n numbers and y<sub>1</sub> through y<sub>64</sub> as the various ways in which the operators can be applied (This takes care of order)</p> <p>This conversion is in POLY-time, and the given boolean formula is satisfiable iff the mathematical expression is valid, etc.</p> <p>Anyone notice a flaw?</p>
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    1. POIs this "Valid mathematical expression" problem P, or NP?
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    1. USSuvesh Pratapa
    1. USSuvesh Pratapa
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    1. COWould you care to show how 3SAT is reducible to this? I'm honestly not coming up with a way. How would you model a 3SAT problem as a sequence of integers and a set of operations?
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      1. USDavid Thornley
 

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