prac5-soln-fall09

Prac5-soln-fall09 - CMSC330 Fall 2009 Practice Problems 5 Solutions 1 Lambda calculus Make all parentheses explicit in the

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Unformatted text preview: CMSC330 Fall 2009 Practice Problems 5 Solutions 1. Lambda calculus Make all parentheses explicit in the following &-expressions a. & x.xz & y.xy & ( & x.((x z) ( & y.(x y)))) b. ( & x.xz) & y.w & w.wyzx & (( & x.(x z)) ( & y.(w ( & w.((((w y) z) x)))))) c. & x.xy & x.yx & ( & x.((x y) ( & x.(y x)))) Find all free (unbound) variables in the following &-expressions d. & x.x z & y.x y & ( & x.((x z ) ( & y.(x y)))) e. ( & x. x z) & y. w & w. w y z x & (( & x.(x z )) ( & y.( w ( & w.((((w y) z ) x )))))) f. & x. x y & x. y x & ( & x.((x y ) ( & x.( y x)))) Apply -reduction to the following &-expressions as much as possible g. ( & z.z) ( & y.y y) ( & x.x a) // ¡-reduction = body[sym/replacement] ( & z. z ) ( & y.y y ) ( & x.x a) & // z[z/( & y.y y)] replace z with & y.y y ( & y. y y ) ( & x.x a ) & // y y[y/( & x.x a)] replace y with & x.x a ( & x. x a) ( & x.x a ) & // x a[x/( & x.x a)] replace x with & x.x a ( & x. x a) a & a a // x a[x/a] replace x with a h. ( & z.z) ( & z.z z) ( & z.z y) ( & z. z ) ( & z.z z ) ( & z.z y) & // ¡-reduction: replace z with & z.z z ( & z. z z ) ( & z.z y ) & // ¡-reduction: replace z with & z.z y ( & z. z y) ( & z.z y ) & // ¡-reduction: replace z with & z.z y ( & z. z y) y & y y // ¡-reduction: replace z with y i. ( & x. & y.x y y) ( & a.a) b ( & x. & y. x y y) ( & a.a ) b & // ¡-reduction: replace x with & a.a ( & y.( & a.a) y y ) b & // ¡-reduction: replace y with b ( & a. a ) b b & b b // ¡-reduction: replace a with b j. ( & x. & y.x y y) ( & y.y) y ( & x. & y .x y y ) ( & y.y) y & // ¢-conversion: rename y to a ( & x. & a. x a a) ( & y.y ) y & // ¡-reduction: replacing x with & y.y ( & a.( & y.y) a a ) y & // ¡-reduction: replacing a with y ( & y. y ) y y & y y // ¡-reduction: replacing y with y k. ( & x.x x) ( & y.y x) z y....
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This note was uploaded on 01/13/2012 for the course CMSC 330 taught by Professor Staff during the Fall '08 term at Maryland.

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Prac5-soln-fall09 - CMSC330 Fall 2009 Practice Problems 5 Solutions 1 Lambda calculus Make all parentheses explicit in the

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