CMPSC 520: 
    FOUNDATIONS OF PROGRAMMING LANGUAGES

Administration: 
  syllabus online: http://www.cse.psu.edu/~hannan/520
  Assignments: 4-5, written, some programming
  Exams: None, but a final project/Assignment

History of Imperative Languages
  (No Foundations)

Functional & Logic Programming Languages
(Standard ML, Scheme, Prolog, lambda Prolog, Elf)

We will study:
  - Logical foundations of these languages
    including:
	o Lambda Calculi (Untyped, Implicitly typed,
	  Explicitly typed, etc.)
	o Natural Deduction
	o Sequent Calculus

A. Church (1932) develops lambda calculus as part of a general
theory of functions and logic, intended as a foundation for
mathematics.  (Inconsistent)

Kleene & Rosser (1935) proved that all recursive functions
can be represented in the lambda calculus

Turing (1937) proved that exactly the functions computable by
a Turing machine can be represented in the lambda calculus

Functional Programming - Representing computable functions
	as lambda terms

Type-Free Calculus (1932) - Every expression (considered as
	a function) can be applied to every expression

Typed Calculus (Curry 1934, Church 1940) - types assigned
	to lambda terms.
	Curry - terms are those of the type-free theory
	      - each term has a set of possible types.
	      - "Descriptive", Implicit Types, Standard ML
	Church - terms are annotated version of the type-free
		 terms;  each term has a unique type.
	       - "Prescriptive", Explicit Types, Pascal


Type-Free Lambda Calculus:

Two basic operations:
Application: F.A  or just FA
Abstraction: if M == M'[x] is an expression depending on x,
	then \x.M'[x] denotes the mapping x |-> M'[x]
	I.e., for each term x, associate the term M'[x]

  (\x.x^2 + 1)3 = 3^2+1 = 10
  (\x.M)N = M[x:=N]  (or M[N/x])  (beta)

Variables: V ::= x|y|z|...
Terms: Lambda ::= V|\V.Lambda|Lambda Lambda

We could add constants:   C ::= c|c'|...

FM1..Mn stands for...    \x1...xn.M stands for ...

Free and Bound variables:
FV(x) = {x} ...		BV(x) = {} ...

The abstraction \x.M binds the free occurrences of x in M.

Closed/Combinator
Alpha conversion/Renaming
M[x/y] ==		\x.M = \y.(M[x/y])

Substitution: M[N/x]  or M[x:=N] 
Conventions: Assume free and bound variables distinct.

Substitution Lemma:  Let M,N,L in Lambda.  
  Assume x!=y and x not free in L.  Then
  M[x:=N][y:=L] = M[y:=L][x:=N[y:=L]]

Proof: Exercise.

Beta Equality:
Eta:

Logical Axioms and Rules:
Refl, Sym, Trans, Cong1, Cong2, Xi

If M = N is provable using these rules we write \ |- M=N

Do Some Examples.

Representing Recursion in the \-calculus
THEOREM (Fixed point theorem)
 1. For all F, there exists an X such that FX = X
 2. There is a fixed point combinator Y such that
     for all F, F(YF) = YF

PROOF:
 1. Define W = \x.F(xx) and X = WW. 
    Then X = WW = (\x.F(xx))W = F(WW) = FX

 2. Define Y = \f.(\x.f(xx))(\x.f(xx))
    By proof of 1: note that
     YF = (\x.F(xx))(\x.F(xx)) = X

Corollary:  Given a term C =C[f.x] possible containing
	f and x free, then
	there exists F. for all X,
		FX =C[F,X]
	(C[F,X] = C[f:=F][x:=X])

Proof: We can construct F by assuming it has the 
required property and calculating back:
	for all X, FX = C[F,X]
	<=         Fx = C[F,x]
	<=         F  = \x.C[F,x]
	<=	   F  = (\f\x.C[f,x])F
	<=         F  = Y(\f\x.C[f,x])

This also holds for more arguments:

exists F, all xvec, F(xvec) = C(F,xvec)

Exercise: Using these results, construct terms F and G
	such that for all X,Y:
		FX = XF
		GXY = YG(YXG)


Lambda Definability

DEFINITION:  
  1. For F,M in Lambda,
	F^0(M) = M
	F^n+1 = F(F^n(M))

 2. The Church Numerals c0, c1, c2 are defined as
	cn = \f\x.f^n(x)

PROPOSITION: (J. Barkley Rosser) Define
	A+ = \nmfy.ng(mgy)
	A* = \nmg.n(mg)
	A_exp = \nm.mn

Then for all n,m in Nat,
  1. A+ cn cm = c_(n+m)
  2. A* cn cm = c_(n*m)
  3. Aexp cn cm = c_(n^m) except for m=0

Lemma:
  1. (cn x)^m(y) = x^{n*m}(y);
  2. (cn)^m(x) = c_{n^m}(x), for m>0

Proof:
  1. By induction on m.
     m=0: LHS = y = RHS
     Assume (1) correct for m.  Then
	(cn x)^{m+1}(y) = cn x((cn x)^m(y))
			= cn x(x^{n*m}(y)) by IH
			= x^n(x^{n*m}(y)) 
			= (x^{n+n*m}(y)) 
			= (x^{n*(m+1)}(y)) 

  2. By induction on m>0
     m=1: then LHS = cn x = RHS
     Assume (2) correct for m.  Then
	cn^{m+1}(x) = cn(cn^m(x))
		    = cn(c_{n^m}(x)) by IH
		    = \y.(c_{n^m}(x))^n (y)
		    = \y.x^{n^m * n}(y)
		    = (\f\y.f^{n^m * n}(y)) x
		    = c_{n^(m+1)} x


Now proof of the proposition:

  1. By induction on m.
  2. Use the lemma (1).
  3. By the lemma (2) we have for m>0

	Aexp cn cm = cm cn = \x.cn^m(x)
			   = \x.c_{n^m}x = c_{n^m}


TRUTH VALUES AND CONDITIONALS:

DEFINITION:
 1. true = \x\y.x,  false = \x\y.y
 2. If B is a Boolean (ie either true or false) then
      if B then P else Q
    can be represented by BPQ.

DEFINITION:  For M,N in Lambda, write
		[M,N] = \z.zMN
  Then:
	[M,N]true = M
	[M,N]false = N
  Hence, [M,N] provides an ordered pair.

DEFINITION: 
  1. A numeric function is a map f:N^p -> N for some p.
  2. A numeric function f with p arguments is called
     lambda definable if for some combinator F,
	Fc_{n1}...c_{np} = c_{f(n1,...,np)}	(*)
     for all n1,...,np in N.  If (*) holds, then f is said
     to be lambda-defined by F.

DEFINITION:
  1. The initial function are the numeric functions
     U_r^i, S+, Z defined by:
	U_r^i(x1,...,x_r) = x_i, 1<=i<=r
		    S+(n) = n+1
		     Z(n) = 0

  2. Let P(n) be a numeric relation.
	      mu m.P(m)
     denotes the least number m such that P(m) holds if
     there is such a number; otherwise undefined.

DEFINITION: 
	Given a function g of m arguments and m functions
	h1,...,hm, each of m arguments, the composition
		f o (g1,...,gm)
	of f and g1,...,gm is the function h of m arguments
	such that for all x1,...,xm in N,
	  h(x1,...,xm) = g(h1(x1,...,xm),...hm(x1,...,xm))


DEFINITION:
	Given a function g of m arguments and a function h
	of m+2 arguments, the function f of m+1 arguments
	is defined by primitive recursion from f and g iff
	the following hold:
	  for all x1,...,xm,y in N,
		f(0,x1,...,xm) = g(x1,...,xm)
		f(k+1,x1,...,xm) = h(f(k,x1,...,xm),k,x1,...,xm)

  NOTE: in the special case of n=0:
		f(0) = m
		f(k+1) = h(f(k),k)

	Examples: Factorial, Addition
	
DEFINITION:
	Given a function g of m+1 arguments, the function f of
	m>0 arguments is defined by minimalization from g iff
	the following holds:
	  for all x1,...,xm in N,
	     (1) f(x1,...,xm) is defined iff there is some
		 y in N such that g(x1,...,xm,z) is defined
		 for all z<=y and g(x1,...,xm,y) = 0;
	     (2) If f(x1,...,xm) is defined then f(x1,...,xm)
		 is equal to the least y satisfying (1). I.e.,
		 f(x1,...,xm) = y iff g(x1,...,xm) = 0 and
		 for all z<y, g(x1,.,,,xn,z) is defined
		 and non-zero.

DEFINITION:
	The class R of recursive functions is the smallest
	class of numeric functions that contain all the
	initial functions and is closed under composition,
	primitive recursion and minimalization.

We now prove that all recursive functions are lambda definiable
(due to Kleene, 1936).


LEMMA: The initial functions are lambda-definable
PROOF:
	U_p^i = \x1...\xp.xi
	   S+ = \n\f\x.f(n f x)
	   Z  = \x.c0

LEMMA: The \-definable functions are closed under composition.
PROOF: Let g,h1,...,hm be \-defined by G,H1,...Hm respectively.
	Then f(x1,...,xm) = g(h1(x1,..,xm),...hm(x1,...,xm)
	is \-defined by
	  F = \x1...\xm.G (H1 x1...xm)... (Hm x1...xm)

LEMMA: The lambda definable functions are closed under
	primitive recursion.
PROOF:
	Let f be defined by
		f(0,x1,...,xm) = g(x1,...,xm)
		f(k+1,x1,...,xm) = h(f(k,x1,...,xm),k,x1,...,xm)
	where g and h are \-defined by G and H respectively.

	For simplicity, we consider the case of m=0.  (m>0
	follows similarly) So G = c_{f(0)}.

	Define
		T = \p.[S+ (p true), H (p false) (p true)]
		  = \p.(\z.z(\n\f\x.f(nfx) (p \x\y.x))
		            (H (p \x\y.y) (p \x\y.x)))

	Then for all k one has
	  T([c_k,c_{f(k)} = [fS+ c_k, H c_{f(k)} c_k]
			  = [c_{k+1},c_{f(k+1)}]

	By induction on k it follows that
	  [c_k,c_{f(k)}] = T^k[c_0,c_{f(0)}]

	Therefore,
		c_{f(k)} = c_k T [c_0, c_{f(0)}] false

	and f can be \-defined by
		F = \k.k T [c_0,G] false

LEMMA: The \-definable functions are closed under 
       minimalization.
PROOF: 
	Let f be defined by f(x1,..,xm) = mu k[g(x1,...xm,k)=0].
	Let g be \-defined by G.
	Define
		zero = \n.n(true false)true
	Then
		zero c0 = true
		zero c_{n+1} = false

	By results from fixed points (Given a term C[f,x] there
	exists F. for all X, FX = C[F,X])
	 there exists a term H such that
		H x1...xm y = if (zero(G x1...xm y)) 
				then y
				else H x1...xm (S+ y)

	Let F = \x1...\xm.H x1...xm c0.
	Then F \-defines f:

		F c_x1...c_xm = H c_x1...c_xm c0
			      = c0 if G c_x1...c_xm c0 = c0
			      = H c_x1...c_xm c1 else;
			      = c1 if G c_x1...c_xm c1 = c0
			      = H c_x1...c_xm c2 else;
			      = c2 if G c_x1...c_xm c2 = c0
				...

THEOREM: All recursive functions are \-definable.