Lambda Calculus

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)

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First, need some theory.

• RPALs flattener grammar use bracket tree
  notation ltroot s1 sngt
• RPAL ? E
• E ? lt'?' E E gt gt E E
• ? lt'?' V E gt gt '?' V '.' E
• ? ltidxgt gt x
• ? ltintegerigt gt i
• ? ltstringsgt gt s
• ? 'true' gt 'true'
• ? 'false' gt 'false'
• . . .
• end RPAL.

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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.

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Disambiguation Rules

1. Function application is left associative.
2. 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).

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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.

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Definition

• Let M and N be ?-expressions. An occurrence of x
  in a ?-expression is free if 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 free in ?y.M, if
  x and y are different.

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Practical, Equivalent Definition

• An occurrence of x in M is free if the path 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 bound if it is not free in M.

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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.xy) y 3
• - first occurrence of y is not free,
• second occurrence is free,
• in the entire expression.

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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.

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Axiom Delta

• Let M and N be AE's that do not contain
• ?-expressions.
• Then M gt? 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 gt? 8.

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Axiom Alpha

• Let x and y be names, and M be an AE with no free
  occurrences of y. Then, in any context, ? x.M
  gt? ? y.substy,x,M
• substy,x,M means "substitute y for x in M".
• Axiom Alpha used to rename the bound variable.
• Example ?x.x3 gt? ? y.y3

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Axiom Beta

• Let x be a name, and M and N be AE's. Then, in
  any context, (?x.M) N gt? substN,x,M.
• Called a "beta-reduction", used to apply a
  function to its argument.
• Pronounced beta reduces to.
• Need to define the subst function.

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Definition (subst)

• Let M and N be AE's, and x be a name.
• Then substN,x,M, also denoted as N/xM,
  means
• If M is an identifier, then
• 1.1. if Mx, then return N.
• 1.2. if M is not x, then return M.
• If M is of the form X Y, then return (N/xX)
  (N/xY).

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Definition (substN,x,M, contd)

• If M is of the form ? y.Y, then
• 3.1. if yx 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/xY.
• 3.2.3. if x occurs free in Y,
• and y occurs free in N, then
• return ?w.N/x (w/yY),
• for any w that does not occur
• free in either N or Y.

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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)

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Definition

• An AE M is said to be "directly convertible to
  an AE N, denoted M gt N, if one of these three
  holds
• M gt? N, M gt? N, M gt? N.
• Definition
• Two AE's M and N are said to be equivalent if M
  gt N.

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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.

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Definition

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

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Definitions

• Two AE's M and N are be congruent if
• M ltgt? 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 gt?
  substN,x,M, N is normal form, unless N is a
  ?-expression.

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Examples

• Normal Order
• (? y.y)(? x.x3)2 gt? (? x.x3) 2
• gt? (23)
• gt? 5
• PL order
• (? y.y)(? x.x3)2 gt? (? y.y)(23)
• gt? (? y.y)5
• gt? 5

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PL Order faster than Normal Order

• PL Order
• (?x.xxx) ((?y.y1)2)
• gt? (?x.xxx) 3
• gt? 333 Cost 2 ?s.
• Normal Order
• (?x.xxx) ((?y.y1)2)
• gt? ((?y.y1)2) ((?y.y1)2)
  ((?y.y1)2)
• gt?, ? , ? 333
• Cost 4 ?s.

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PL Order riskier than Normal Order

• Normal Order
• (?x.1) ( (?x.x x) (?x.x x) )
• gt? 1
• Terminates.
• PL Order
• (?x.1) ( (?x.x x) (?x.x x) )
• gt? (?x.1) ( (?x.x x) (?x.x x) )
• gt? (?x.1) ( (?x.x x) (?x.x x) )
• . . .
• Doesnt terminate !

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Theorem (Church-Rosser)

1. All sequences of reductions on an AE that
   terminate, do so on congruent AE's.
2. If there exists a sequence of reductions on an AE
   that terminates, then reduction in normal order
   also terminates.

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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.

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Lambda Calculus
Programming Language Principles Lecture 11

• Prepared by
• Manuel E. Bermúdez, Ph.D.
• Associate Professor
• University of Florida