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Lambda Calculus

Lambda Calculus. Programming Language Principles Lecture 11. Prepared by Manuel E. Bermúdez, Ph.D. Associate Professor University of Florida. Lambda Calculus. To obtain the "value" of an RPAL program: 1. Transduce RPAL source to an AST. 2. Standardize the AST into ST.

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Lambda Calculus

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  1. Lambda Calculus Programming Language Principles Lecture 11 Prepared by Manuel E. Bermúdez, Ph.D. Associate Professor University of Florida

  2. Lambda Calculus • To obtain the "value" of an RPAL program: 1. Transduce RPAL source to an AST. 2. Standardize the AST into ST. 3. Linearize the ST into a lambda-expression. 4. Evaluate the lambda-expression, using the CSE machine (more later)

  3. First, need some theory. RPAL’s flattener grammar: use bracket tree notation: <‘root’ s1 … sn> RPAL → E E → <'' E E > => E E → <'' V E > => '' V '.' E → <id:x> => x → <integer:i> => i → <string:s> => s → 'true' => 'true' → 'false' => 'false' . . . end RPAL.

  4. Result of Tree Flattening: a string • An Applicative Expression (AE), or well-formed formula (WFF). • The interpretation of the AE is ambiguous. • Need parentheses to disambiguate.

  5. Disambiguation Rules • Function application is left associative. • If an expression of the formx.M occurs in a larger expression, then M is extended as far as possible (i.e. to the end of the entire expression or to the next unmatched right parenthesis).

  6. Example:  x.  y.+xy 2 3 is equivalent to  x.( y.+xy 2 3) • Must be parenthesized to obtain the intended expression: ( x.  y.+xy) 2 3.

  7. Definition • Let M and N be-expressions. Anoccurrence of x in a-expression is freeif it can be proved so via the following three rules: • The occurrence of x in-expression "x" is free. • Any free occurrence of x in either M or N is free in M N. • Any free occurrence of x in M is freeiny.M, if x and y are different.

  8. Practical, Equivalent Definition • An occurrence of x in M is free if thepath from x to the root of M includes nonode with x on its left. • Definition: • An occurrence of x in a-expression M is said to be boundif it is not free in M.

  9. Examples: a - a occurs free x - x occurs free a x - a and x both occur free ( x.ax)x - a occurs free; x occurs both free and bound ( x.  y.x+y) y 3 - first occurrence of y is not free, second occurrence is free, in the entire expression.

  10. More definitions • Definition: • In an expression of the form x.M, x is the bound variable, and M is the body. • Definition: • The scope of an identifier x, in an expression of the form x.M, consists of all free occurrences of x in M.

  11. Axiom Delta Let M and N be AE's that do not contain -expressions. Then M => N if Val(M) = Val(N). We say that M and N are delta-convertible, pronounced “M delta reduces to N”. Val: Value obtained from ordinary evaluation. • Example: +35 => 8.

  12. Axiom Alpha Let x and y be names, and M be an AE with no free occurrences of y. Then, in anycontext, x.M => y.subst[y,x,M] subst[y,x,M] means "substitute y for x in M". Axiom Alpha used to rename the bound variable. • Example:x.+x3 => y.+y3

  13. Axiom Beta • Let x be a name, and M and N be AE's. Then, in any context, (x.M) N => subst[N,x,M]. • Called a "beta-reduction", used to apply a function to its argument. • Pronounced “beta reduces to”. • Need to define the “subst” function.

  14. Definition (subst): • Let M and N be AE's, and x be a name. • Then subst[N,x,M], also denoted as [N/x]M, means: • If M is an identifier, then 1.1. if M=x, then return N. 1.2.if M is not x, then return M. • If M is of the form X Y, then return ([N/x]X) ([N/x]Y).

  15. Definition (subst[N,x,M], cont’d) • If M is of the form y.Y, then 3.1. if y=x then return y.Y. 3.2.if y is not x then 3.2.1.if x does not occur free in Y, then return y.Y. 3.2.2.if y does not occur free in N, then return y.[N/x]Y. 3.2.3. if x occurs free in Y, and y occurs free in N, then returnw.[N/x] ([w/y]Y), for any w that does not occur free in either N or Y.

  16. Examples • [3/x]( x.+x2) =  x.+x2 (by 3.1) • [3/x]( y.y) =  y.y (by 3.2.1) • [3/x]( y.+xy) =  y.[3/x](+xy) =  y.+3y (by 3.2.2 and 2) • [y/x]( y.+xy) =  z.[y/x]([z/y](+xy)) =  z.[y/x](+xz) =  z.+yz (by 3.2.3, 2, and 2)

  17. Definition • An AE M is said to be "directly convertible“ to an AE N, denotedM => N, if one of these three holds: • M => N, M => N, M => N. • Definition: • Two AE's M and N are said to beequivalent if M =>* N.

  18. Definition • An AE M is in normal form if either • M does not contain any ’s, or • M contains at least one  , and 2.1. M is of the form X Y, X and Y are in normal form, and X is not a -expression. 2.2. M is of the form  x.N, and N is in normal form. Shorter version: M is in normal form if no beta-reductions apply.

  19. Definition • Given an AE M, a reduction sequence on M is a finite sequence of AE's E0 , E1 , ..., En, such that M = E0 => E1 ... => En. • Definition: • A reduction sequence is said to terminate if its last AE is in normal form.

  20. Definitions • Two AE's M and N are be congruent if M <=>* N. • A reduction sequence is in normal order if in each reduction, the left-most  is reduced. • A reduction sequence is PL order if for each beta-reduction of the form ( x.M) N => subst[N,x,M], N is normal form, unless N is a -expression.

  21. Examples • Normal Order: ( y.y)[( x.+x3)2] => ( x.+x3) 2 =>(+23) => 5 • PL order: ( y.y)[( x.+x3)2] => ( y.y)(+23) => ( y.y)5 => 5

  22. PL Order “faster” than Normal Order • PL Order: (x.x+x+x) ((y.y+1)2) => (x.x+x+x) 3 => 3+3+3 Cost: 2 ’s. • Normal Order: (x.x+x+x) ((y.y+1)2) =>((y.y+1)2) + ((y.y+1)2) + ((y.y+1)2) =>,  ,  3+3+3 Cost: 4 ’s.

  23. PL Order “riskier” than Normal Order Normal Order: (x.1) ( (x.x x) (x.x x) ) => 1 Terminates. PL Order: (x.1) ( (x.x x) (x.x x) ) => (x.1) ( (x.x x) (x.x x) ) => (x.1) ( (x.x x) (x.x x) ) . . . Doesn’t terminate !

  24. Theorem (Church-Rosser) • All sequences of reductions on an AE that terminate, do so on congruent AE's. • If there exists a sequence of reductions on an AE that terminates, then reduction in normal order also terminates.

  25. Conclusions • Some AE's can be reduced to normal form, but some cannot. • Still stuck with the classic halting problem. • If an AE can be reduced to normal form, then that normal form is unique to within a choice of bound variables. • If a normal form exists, a reduction to (some) normal form can be obtained in a finite number of steps using normal order.

  26. Lambda Calculus Programming Language Principles Lecture 11 Prepared by Manuel E. Bermúdez, Ph.D. Associate Professor University of Florida

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