+ Abstract: ( λa M )
Variables: x, x', x'', x''', ...
(Cheat: x, y, z, x1, x2, ...)
+ Application: ( M N )
+
Each variable is a term.
For all terms M and N and variable a, the following are also terms:
+We'll tend to write ( λa M )
as just `( \a M )`.
(M N) The application of M to N
(\a M) The abstraction of a over M
+Some authors reserve the term "term" for just variables and abstracts. We won't participate in that convention; we'll probably just say "term" and "expression" indiscriminately.
Examples of terms:
+Examples of expressions:
x
(y x)
(x x)
(\x y)
(\x x)
(\x (\y x))
(x (\x x))
((\x (x x))(\x (x x)))
+ x
+ (y x)
+ (x x)
+ (\x y)
+ (\x x)
+ (\x (\y x))
+ (x (\x x))
+ ((\x (x x)) (\x (x x)))
Reduction/conversion/equality:
+The lambda calculus has an associated proof theory. For now, we can regard the proof theory as having just one rule, called the rule of "betareduction" or "betacontraction". Suppose you have some expression of the form:
Lambda terms express recipes for combining terms into new terms.
The key operation in the lambda calculus is betaconversion.
+ ((\a M) N)
((\a M) N) ~~>_beta M{a := N}
+that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "betareducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**.
The term on the left of the arrow is an application whose first
element is a lambda abstraction. (Such an application is called a
"redex".) The beta reduction rule says that a redex is
betaequivalent to a term that is constructed by replacing every
(free) occurrence of a in M by a copy of N. For example,
+The rule of betareduction permits a transition from that expression to the following:
((\x x) z) ~~>_beta z
((\x (x x)) z) ~~>_beta (z z)
((\x x) (\y y)) ~~>_beta (\y y)
+ M {a:=N}
+
+What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.
+
+What is a free occurrence?
+
+> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.
+
+> If T has the form `(M N)`, any occurrences of `a` that are bound in `M` are also bound in T, and so too any occurrences of `a` that are bound in `N`.
+
+> An occurrence of a variable is **free** if it's not bound.
Beta reduction is only allowed to replace *free* occurrences of a variable.
An occurrence of a variable a is BOUND in T if T has the form (\a N).
If T has the form (M N), and the occurrence of a is in M, then a is
bound in T just in case a is bound in M; if the occurrence of a is in
N, than a is bound in T just in case a is bound in N. An occurrence
of a variable a is FREE in a term T iff it is not bound in T.
For instance:
T = (x (\x (\y (x (y z)))))
The first occurrence of x in T is free. The second occurrence of x
immediately follows a lambda, and is bound. The third occurrence of x
occurs within a form that begins with "\x", so it is bound as well.
Both occurrences of y are bound, and the only occurrence of z is free.
+> T is defined to be `(x (\x (\y (x (y z)))))`
Lambda terms represent functions.
All (recursively computable) functions can be represented by lambda
terms (the untyped lambda calculus is Turning complete).
For some lambda terms, it is easy to see what function they represent:
+The first occurrence of `x` in `T` is free. The `\x` we won't regard as being an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free.
(\x x) the identity function: given any argument M, this function
simply returns M: ((\x x) M) ~~>_beta M.
+Here's an example of betareduction:
(\x (x x)) duplicates its argument:
((\x (x x)) M) ~~> (M M)
+ ((\x (y x)) z)
(\x (\y x)) throws away its second argument:
(((\x (\y x)) M) N) ~~> M
+betareduces to:
and so on.
+ (y z)
+
+We'll write that like this:
+
+ ((\x (y x)) z) ~~> (y z)
+
+Different authors use different notations. Some authors use the term "contraction" for a single reduction step, and reserve the term "reduction" for the reflexive transitive closure of that, that is, for zero or more reduction steps. Informally, it seems easiest to us to say "reduction" for one or more reduction steps. So when we write:
+
+ M ~~> N
+
+We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol > for onestep contraction, and the symbol >> for zeroormore step reduction. Hindley and Seldin use (triangle..sub1) and (triangle).
+
+When M and N are such that there's some P that M reduces to by zero or more steps, and that N also reduces to by zero or more steps, then we say that M and N are **betaconvertible**. We'll write that like this:
+
+ M <~~> N
+
+This is what plays the role of equality in the lambda calculus. Hankin uses the symbol `=` for this. So too do Hindley and Seldin.
+
+In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol (three bars) for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing metatheory for the lambda calculus. Instead we'll use `<~~>` as we said above. When we want to introduce a stipulative definition, we'll write it out longhand, as in:
+
+> T is defined to be `(M N)`.
+
+We'll regard the following two expressions:
+
+ (\x x y)
+
+ (\z z y)
+
+as syntactically equivalent, since they only involve a typographic change of a bound variable. Read Hankin section 2.3 for discussion of different attitudes one can take about this.
+
+Note that neither of those expressions are identical to:
+
+ (\x x w)
+
+because here it's a free variable that's been changed. Nor are they identical to:
+
+ (\y y y)
+
+because here the second occurrence of `y` is no longer free.
+
+There is plenty of discussion of this, and the fine points of how substitution works, in Hankin and in various of the tutorials we've linked to about the lambda calculus. We expect you have a good intuitive understanding of what to do already, though, even if you're not able to articulate it rigorously.
+
+
+Shorthand
+
+
+The grammar we gave for the lambda calculus leads to some verbosity. There are several informal conventions in widespread use, which enable the language to be written more compactly. (If you like, you could instead articulate a formal grammar which incorporates these additional conventions. Instead of showing it to you, we'll leave it as an exercise for those so inclined.)
It is easy to see that distinct lambda terms can represent the same
function. For instance, (\x x) and (\y y) both express the same
function, namely, the identity function.

Dot notation: dot means "put a left paren here, and put the right
paren as far the right as possible without creating unbalanced
parentheses". So (\x(\y(xy))) = \x\y.xy, and \x\y.(z y) x =
(\x(\y((z y) z))), but (\x\y.(z y)) x = ((\x(\y(z y))) x).
+parentheses". So:
+
+ (\x (\y (xy)))
+
+can be abbreviated as:
+
+ (\x (\y. x y))
+
+and:
+
+ (\x \y. (z y) z)
+
+would abbreviate:
+
+ (\x \y ((z y) z))
+
+This on the other hand:

+ ((\x \y. (z y) z)
Church figured out how to encode integers and arithmetic operations
using lambda terms. Here are the basics:
+would abbreviate:
0 = \f\x.fx
1 = \f\x.f(fx)
2 = \f\x.f(f(fx))
3 = \f\x.f(f(f(fx)))
...
+ ((\x (\y (z y))) z)
Adding two integers involves applying a special function + such that
(+ 1) 2 = 3. Here is a term that works for +:
+Parentheses: outermost parentheses around applications can be dropped. Moreover, applications will associate to the left, so `M N P` will be understood as `((M N) P)`. Finally, you can drop parentheses around abstracts, but not when they're part of an application. So you can abbreviate:
+ = \m\n\f\x.m(f((n f) x))
+ (\x x y)
So (+ 0) 0 =
(((\m\n\f\x.m(f((n f) x))) ;+
 \f\x.fx) ;0
 \f\x.fx) ;0
+as:
~~>_beta targeting m for beta conversion
+ \x. x y
((\n\f\x.[\f\x.fx](f((n f) x)))
 \f\x.fx)
+but you should include the parentheses in:
\f\x.[\f\x.fx](f(([\f\x.fx] f) x))
+ (\x. x y) z
\f\x.[\f\x.fx](f(fx))
+and:
\f\x.\x.[f(fx)]x
+ z (\x. x y)
\f\x.f(fx)
+Merging lambdas: an expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
+ (\x y. M)
+Similarly, `(\x (\y (\z M)))` can be abbreviated as:

+ (\x y z. M)
A concrete example: "damn" side effects
1. Sentences have truth conditions.
2. If "John read the book" is true, then
 John read something,
 Someone read the book,
 John did something to the book,
 etc.
3. If "John read the damn book",
 all the same entailments follow.
 To a first approximation, "damn" does not affect atissue truth
 conditions.
4. "Damn" does contribute information about the attitude of the speaker
 towards some aspect of the situation described by the sentence.
+Lambda terms represent functions
+
+All (recursively computable) functions can be represented by lambda
+terms (the untyped lambda calculus is Turing complete). For some lambda terms, it is easy to see what function they represent:
+
+(\x x) represents the identity function: given any argument M, this function
+simply returns M: ((\x x) M) ~~> M.
+
+(\x (x x)) duplicates its argument:
+((\x (x x)) M) ~~> (M M)
+
+(\x (\y x)) throws away its second argument:
+(((\x (\y x)) M) N) ~~> M
+
+and so on.
+
+It is easy to see that distinct lambda expressions can represent the same
+function, considered as a mapping from input to outputs. Obviously:
+ (\x x)

Old notes, no longer operative:
+and:
1. Theoretical computer science is beautiful.
+ (\z z)
 Google search for "anagram": Did you mean "nag a ram"?
 Google search for "recursion": Did you mean "recursion"?
+both represent the same function, the identity function. However, we said above that we would be regarding these expressions as synactically equivalent, so they aren't yet really examples of *distinct* lambda expressions representing a single function. However, all three of these are distinct lambda expressions:
 Y = \f.(\x.f (x x)) (\x.f (x x))
+ (\y x. y x) (\z z)
+ (\x. (\z z) x)
1. Understanding the meaning(use) of programming languages
 helps understanding the meaning(use) of natural langauges
+ (\z z)
 1. Richard Montague. 1970. Universal Grammar. _Theoria_ 34:37598.
 "There is in my opinion no important theoretical difference
 between natural languages and the artificial languages of
 logicians; indeed, I consider it possible to comprehend the
 syntax and semantics of both kinds of languages within a
 single natural and mathematically precise theory."
+yet when applied to any argument M, all of these will always return M. So they have the same extension. It's also true, though you may not yet be in a position to see, that no other argument can differentiate between them when they're supplied as an argument to it. However, these expressions are all syntactically distinct.
 2. Similarities:
+The first two expressions are *convertible*: in particular the first reduces to the second. So they can be regarded as prooftheoretically equivalent even though they're not syntactically identical. However, the proof theory we've given so far doesn't permit you to reduce the second expression to the third. So these lambda expressions are nonequivalent.
 Function/argument structure:
 f(x)
 kill(it)
 pronominal binding:
 x := x + 1
 John is his own worst enemy
 Quantification:
 foreach x in [1..10] print x
 Print every number from 1 to 10
+There's an extension of the prooftheory we've presented so far which does permit this further move. And in that extended proof theory, all computable functions with the same extension do turn out to be equivalent (convertible). However, at that point, we still won't be working with the traditional mathematical notion of a function as a set of ordered pairs. One reason is that the latter but not the former permits uncomputable functions. A second reason is that the latter but not the former prohibits functions from applying to themselves. We discussed this some at the end of seminar (and further discussion is best pursued in person).
 3. Possible differences:
 Parentheses:
 3 * (4 + 7)
 ?It was four plus seven that John computed 3 multiplied by
 (compare: John computed 3 multiplied by four plus seven)
 Ambiguity:
 3 * 4 + 7
 Time flies like and arrow, fruit flies like a banana.
 Vagueness:
 3 * 4
 A cloud near the mountain
 Unbounded numbers of distinct pronouns:
 f(x1) + f(x2) + f(x3) + ...
 He saw her put it in ...
 [In ASL, dividing up the signing space...]


2. Standard methods in linguistics are limited.
 1. Firstorder predicate calculus
 Invented for reasoning about mathematics (Frege's quantification)
 Alethic, order insensitive: phi & psi == psi & phi
 But: John left and Mary left too /= Mary left too and John left
+Booleans and pairs
+==================
 2. Simplytyped lambda calculus
+Our definition of these is reviewed in [[Assignment1]].
 Can't express the Y combinator
3. Meaning is computation.
 1. Semantics is programming
 2. Good programming is good semantics
 1. Example
+1. Declarative vs imperatival models of computation.
+2. Variety of ways in which "order can matter."
+3. Variety of meanings for "dynamic."
+4. Schoenfinkel, Curry, Church: a brief history
+5. Functions as "firstclass values"
+6. "Curried" functions
+
+1. Beta reduction
+1. Encoding pairs (and triples and ...)
+1. Encoding booleans
 1. Programming technique
 Exceptions
 throw (raise)
 catch (handle)
 2. Application to linguistics
 presupposition
 expressives
 Develop application:
 fn application
 divide by zero
 test and repair
 raise and handle
 fn application
 presupposition failure
 build into meaning of innocent predicates?
 expressives
 throw
 handle
 resume computation

diff git a/week2.mdwn b/week2.mdwn
new file mode 100644
index 00000000..c7e82303
 /dev/null
+++ b/week2.mdwn
@@ 0,0 +1,511 @@
+1. Substitution; using alphaconversion and other strategies
+1. Conversion versus reduction
+
+1. Different evaluation strategies (call by name, call by value, etc.)
+1. Strongly normalizing vs weakly normalizing vs nonnormalizing; ChurchRosser Theorem(s)
+1. Lambda calculus compared to combinatorial logic+1. Churchlike encodings of numbers, defining addition and multiplication +1. Defining the predecessor function; alternate encodings for the numbers +1. Homogeneous sequences or "lists"; how they differ from pairs, triples, etc. +1. Representing lists as pairs +1. Representing lists as folds +1. Typical higherorder functions: map, filter, fold
+1. Recursion exploiting the foldlike representation of numbers and lists ([[!wikipedia Deforestation (computer science)]], [[!wikipedia Zipper (data structure)]]) +1. General recursion using omega + +1. Eta reduction and "extensionality" ?? +Undecidability of equivalence + +There is no algorithm which takes as input two lambda expressions and outputs TRUE or FALSE depending on whether or not the two expressions are equivalent. This was historically the first problem for which undecidability could be proven. As is common for a proof of undecidability, the proof shows that no computable function can decide the equivalence. Church's thesis is then invoked to show that no algorithm can do so. + +Church's proof first reduces the problem to determining whether a given lambda expression has a normal form. A normal form is an equivalent expression which cannot be reduced any further under the rules imposed by the form. Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. Building on earlier work by Kleene and constructing a GÃ¶del numbering for lambda expressions, he constructs a lambda expression e which closely follows the proof of GÃ¶del's first incompleteness theorem. If e is applied to its own GÃ¶del number, a contradiction results. + + + +1. The Y combinator(s); more on evaluation strategies
+1. Introducing the notion of a "continuation", which technique we'll now already have used a few times
+
+
+
+alphaconvertible
+
+syntactic equality `===`
+contract/reduce/`~~>`
+convertible `<~~>`
+
+normalizing
+ weakly normalizable
+ strongly normalizable
+ "normal order" reduction vs "applicative order"
+ eval strategy choices
+
+ Reduction strategies For more details on this topic, see Evaluation
+ strategy.
+
+ Whether a term is normalising or not, and how much work needs to be
+ done in normalising it if it is, depends to a large extent on the reduction
+ strategy used. The distinction between reduction strategies relates to the
+ distinction in functional programming languages between eager evaluation and
+ lazy evaluation.
+
+ Full beta reductions Any redex can be reduced at any time. This means
+ essentially the lack of any particular reduction strategyâwith regard to
+ reducibility, "all bets are off". Applicative order The leftmost, innermost
+ redex is always reduced first. Intuitively this means a function's arguments
+ are always reduced before the function itself. Applicative order always
+ attempts to apply functions to normal forms, even when this is not possible.
+ Most programming languages (including Lisp, ML and imperative languages like C
+ and Java) are described as "strict", meaning that functions applied to
+ nonnormalising arguments are nonnormalising. This is done essentially using
+ applicative order, call by value reduction (see below), but usually called
+ "eager evaluation". Normal order The leftmost, outermost redex is always
+ reduced first. That is, whenever possible the arguments are substituted into
+ the body of an abstraction before the arguments are reduced. Call by name As
+ normal order, but no reductions are performed inside abstractions. For example
+ Î»x.(Î»x.x)x is in normal form according to this strategy, although it contains
+ the redex (Î»x.x)x. Call by value Only the outermost redexes are reduced: a
+ redex is reduced only when its right hand side has reduced to a value (variable
+ or lambda abstraction). Call by need As normal order, but function applications
+ that would duplicate terms instead name the argument, which is then reduced
+ only "when it is needed". Called in practical contexts "lazy evaluation". In
+ implementations this "name" takes the form of a pointer, with the redex
+ represented by a thunk.
+
+ Applicative order is not a normalising strategy. The usual
+ counterexample is as follows: define Î© = ÏÏ where Ï = Î»x.xx. This entire
+ expression contains only one redex, namely the whole expression; its reduct is
+ again Î©. Since this is the only available reduction, Î© has no normal form
+ (under any evaluation strategy). Using applicative order, the expression KIÎ© =
+ (Î»xy.x) (Î»x.x)Î© is reduced by first reducing Î© to normal form (since it is the
+ leftmost redex), but since Î© has no normal form, applicative order fails to
+ find a normal form for KIÎ©.
+
+ In contrast, normal order is so called because it always finds a
+ normalising reduction if one exists. In the above example, KIÎ© reduces under
+ normal order to I, a normal form. A drawback is that redexes in the arguments
+ may be copied, resulting in duplicated computation (for example, (Î»x.xx)
+ ((Î»x.x)y) reduces to ((Î»x.x)y) ((Î»x.x)y) using this strategy; now there are two
+ redexes, so full evaluation needs two more steps, but if the argument had been
+ reduced first, there would now be none).
+
+ The positive tradeoff of using applicative order is that it does not
+ cause unnecessary computation if all arguments are used, because it never
+ substitutes arguments containing redexes and hence never needs to copy them
+ (which would duplicate work). In the above example, in applicative order
+ (Î»x.xx) ((Î»x.x)y) reduces first to (Î»x.xx)y and then to the normal order yy,
+ taking two steps instead of three.
+
+ Most purely functional programming languages (notably Miranda and its
+ descendents, including Haskell), and the proof languages of theorem provers,
+ use lazy evaluation, which is essentially the same as call by need. This is
+ like normal order reduction, but call by need manages to avoid the duplication
+ of work inherent in normal order reduction using sharing. In the example given
+ above, (Î»x.xx) ((Î»x.x)y) reduces to ((Î»x.x)y) ((Î»x.x)y), which has two redexes,
+ but in call by need they are represented using the same object rather than
+ copied, so when one is reduced the other is too.
+
+
+
+
+ Strict evaluation Main article: strict evaluation
+
+ In strict evaluation, the arguments to a function are always evaluated
+ completely before the function is applied.
+
+ Under Church encoding, eager evaluation of operators maps to strict evaluation
+ of functions; for this reason, strict evaluation is sometimes called "eager".
+ Most existing programming languages use strict evaluation for functions. [edit]
+ Applicative order
+
+ Applicative order (or leftmost innermost) evaluation refers to an evaluation
+ strategy in which the arguments of a function are evaluated from left to right
+ in a postorder traversal of reducible expressions (redexes). Unlike
+ callbyvalue, applicative order evaluation reduces terms within a function
+ body as much as possible before the function is applied. [edit] Call by value
+
+ Callbyvalue evaluation (also referred to as passbyvalue) is the most common
+ evaluation strategy, used in languages as different as C and Scheme. In
+ callbyvalue, the argument expression is evaluated, and the resulting value is
+ bound to the corresponding variable in the function (frequently by copying the
+ value into a new memory region). If the function or procedure is able to assign
+ values to its parameters, only its local copy is assigned â that is, anything
+ passed into a function call is unchanged in the caller's scope when the
+ function returns.
+
+ Callbyvalue is not a single evaluation strategy, but rather the family of
+ evaluation strategies in which a function's argument is evaluated before being
+ passed to the function. While many programming languages (such as Eiffel and
+ Java) that use callbyvalue evaluate function arguments lefttoright, some
+ evaluate functions and their arguments righttoleft, and others (such as
+ Scheme, OCaml and C) leave the order unspecified (though they generally require
+ implementations to be consistent).
+
+ In some cases, the term "callbyvalue" is problematic, as the value which is
+ passed is not the value of the variable as understood by the ordinary meaning
+ of value, but an implementationspecific reference to the value. The
+ description "callbyvalue where the value is a reference" is common (but
+ should not be understood as being callbyreference); another term is
+ callbysharing. Thus the behaviour of callbyvalue Java or Visual Basic and
+ callbyvalue C or Pascal are significantly different: in C or Pascal, calling
+ a function with a large structure as an argument will cause the entire
+ structure to be copied, potentially causing serious performance degradation,
+ and mutations to the structure are invisible to the caller. However, in Java or
+ Visual Basic only the reference to the structure is copied, which is fast, and
+ mutations to the structure are visible to the caller. [edit] Call by reference
+
+ In callbyreference evaluation (also referred to as passbyreference), a
+ function receives an implicit reference to the argument, rather than a copy of
+ its value. This typically means that the function can modify the argument
+ something that will be seen by its caller. Callbyreference therefore has the
+ advantage of greater time and spaceefficiency (since arguments do not need to
+ be copied), as well as the potential for greater communication between a
+ function and its caller (since the function can return information using its
+ reference arguments), but the disadvantage that a function must often take
+ special steps to "protect" values it wishes to pass to other functions.
+
+ Many languages support callbyreference in some form or another, but
+ comparatively few use it as a default; Perl and Visual Basic are two that do,
+ though Visual Basic also offers a special syntax for callbyvalue parameters.
+ A few languages, such as C++ and REALbasic, default to callbyvalue, but offer
+ special syntax for callbyreference parameters. C++ additionally offers
+ callbyreferencetoconst. In purely functional languages there is typically
+ no semantic difference between the two strategies (since their data structures
+ are immutable, so there is no possibility for a function to modify any of its
+ arguments), so they are typically described as callbyvalue even though
+ implementations frequently use callbyreference internally for the efficiency
+ benefits.
+
+ Even among languages that don't exactly support callbyreference, many,
+ including C and ML, support explicit references (objects that refer to other
+ objects), such as pointers (objects representing the memory addresses of other
+ objects), and these can be used to effect or simulate callbyreference (but
+ with the complication that a function's caller must explicitly generate the
+ reference to supply as an argument). [edit] Call by sharing
+
+ Also known as "call by object" or "call by objectsharing" is an evaluation
+ strategy first named by Barbara Liskov et al. for the language CLU in 1974[1].
+ It is used by languages such as Python[2], Iota, Java (for object
+ references)[3], Ruby, Scheme, OCaml, AppleScript, and many other languages.
+ However, the term "call by sharing" is not in common use; the terminology is
+ inconsistent across different sources. For example, in the Java community, they
+ say that Java is passbyvalue, whereas in the Ruby community, they say that
+ Ruby is passbyreference, even though the two languages exhibit the same
+ semantics. Callbysharing implies that values in the language are based on
+ objects rather than primitive types.
+
+ The semantics of callbysharing differ from callbyreference in that
+ assignments to function arguments within the function aren't visible to the
+ caller (unlike byreference semantics)[citation needed]. However since the
+ function has access to the same object as the caller (no copy is made),
+ mutations to those objects within the function are visible to the caller, which
+ differs from callbyvalue semantics.
+
+ Although this term has widespread usage in the Python community, identical
+ semantics in other languages such as Java and Visual Basic are often described
+ as callbyvalue, where the value is implied to be a reference to the object.
+ [edit] Call by copyrestore
+
+ Callbycopyrestore, callbyvalueresult or callbyvaluereturn (as termed
+ in the Fortran community) is a special case of callbyreference where the
+ provided reference is unique to the caller. If a parameter to a function call
+ is a reference that might be accessible by another thread of execution, its
+ contents are copied to a new reference that is not; when the function call
+ returns, the updated contents of this new reference are copied back to the
+ original reference ("restored").
+
+ The semantics of callbycopyrestore also differ from those of
+ callbyreference where two or more function arguments alias one another; that
+ is, point to the same variable in the caller's environment. Under
+ callbyreference, writing to one will affect the other; callbycopyrestore
+ avoids this by giving the function distinct copies, but leaves the result in
+ the caller's environment undefined (depending on which of the aliased arguments
+ is copied back first).
+
+ When the reference is passed to the callee uninitialized, this evaluation
+ strategy may be called callbyresult. [edit] Partial evaluation Main article:
+ Partial evaluation
+
+ In partial evaluation, evaluation may continue into the body of a function that
+ has not been applied. Any subexpressions that do not contain unbound variables
+ are evaluated, and function applications whose argument values are known may be
+ reduced. In the presence of sideeffects, complete partial evaluation may
+ produce unintended results; for this reason, systems that support partial
+ evaluation tend to do so only for "pure" expressions (expressions without
+ sideeffects) within functions. [edit] Nonstrict evaluation
+
+ In nonstrict evaluation, arguments to a function are not evaluated unless they
+ are actually used in the evaluation of the function body.
+
+ Under Church encoding, lazy evaluation of operators maps to nonstrict
+ evaluation of functions; for this reason, nonstrict evaluation is often
+ referred to as "lazy". Boolean expressions in many languages use lazy
+ evaluation; in this context it is often called short circuiting. Conditional
+ expressions also usually use lazy evaluation, albeit for different reasons.
+ [edit] Normal order
+
+ Normalorder (or leftmost outermost) evaluation is the evaluation strategy
+ where the outermost redex is always reduced, applying functions before
+ evaluating function arguments. It differs from callbyname in that
+ callbyname does not evaluate inside the body of an unapplied
+ function[clarification needed]. [edit] Call by name
+
+ In callbyname evaluation, the arguments to functions are not evaluated at all
+ â rather, function arguments are substituted directly into the function body
+ using captureavoiding substitution. If the argument is not used in the
+ evaluation of the function, it is never evaluated; if the argument is used
+ several times, it is reevaluated each time. (See Jensen's Device.)
+
+ Callbyname evaluation can be preferable over callbyvalue evaluation because
+ callbyname evaluation always yields a value when a value exists, whereas
+ callbyvalue may not terminate if the function's argument is a nonterminating
+ computation that is not needed to evaluate the function. Opponents of
+ callbyname claim that it is significantly slower when the function argument
+ is used, and that in practice this is almost always the case as a mechanism
+ such as a thunk is needed. [edit] Call by need
+
+ Callbyneed is a memoized version of callbyname where, if the function
+ argument is evaluated, that value is stored for subsequent uses. In a "pure"
+ (effectfree) setting, this produces the same results as callbyname; when the
+ function argument is used two or more times, callbyneed is almost always
+ faster.
+
+ Because evaluation of expressions may happen arbitrarily far into a
+ computation, languages using callbyneed generally do not support
+ computational effects (such as mutation) except through the use of monads and
+ uniqueness types. This eliminates any unexpected behavior from variables whose
+ values change prior to their delayed evaluation.
+
+ This is a kind of Lazy evaluation.
+
+ Haskell is the most wellknown language that uses callbyneed evaluation.
+
+ R also uses a form of callbyneed. [edit] Call by macro expansion
+
+ Callbymacroexpansion is similar to callbyname, but uses textual
+ substitution rather than captureavoiding substitution. With uncautious use,
+ macro substitution may result in variable capture and lead to undesired
+ behavior. Hygienic macros avoid this problem by checking for and replacing
+ shadowed variables that are not parameters.
+
+
+
+
+ Eager evaluation or greedy evaluation is the evaluation strategy in most
+ traditional programming languages.
+
+ In eager evaluation an expression is evaluated as soon as it gets bound to a
+ variable. The term is typically used to contrast lazy evaluation, where
+ expressions are only evaluated when evaluating a dependent expression. Eager
+ evaluation is almost exclusively used in imperative programming languages where
+ the order of execution is implicitly defined by the source code organization.
+
+ One advantage of eager evaluation is that it eliminates the need to track and
+ schedule the evaluation of expressions. It also allows the programmer to
+ dictate the order of execution, making it easier to determine when
+ subexpressions (including functions) within the expression will be evaluated,
+ as these subexpressions may have sideeffects that will affect the evaluation
+ of other expressions.
+
+ A disadvantage of eager evaluation is that it forces the evaluation of
+ expressions that may not be necessary at run time, or it may delay the
+ evaluation of expressions that have a more immediate need. It also forces the
+ programmer to organize the source code for optimal order of execution.
+
+ Note that many modern compilers are capable of scheduling execution to better
+ optimize processor resources and can often eliminate unnecessary expressions
+ from being executed entirely. Therefore, the notions of purely eager or purely
+ lazy evaluation may not be applicable in practice.
+
+
+
+ In computer programming, lazy evaluation is the technique of delaying a
+ computation until the result is required.
+
+ The benefits of lazy evaluation include: performance increases due to avoiding
+ unnecessary calculations, avoiding error conditions in the evaluation of
+ compound expressions, the capability of constructing potentially infinite data
+ structures, and the capability of defining control structures as abstractions
+ instead of as primitives.
+
+ Languages that use lazy actions can be further subdivided into those that use a
+ callbyname evaluation strategy and those that use callbyneed. Most
+ realistic lazy languages, such as Haskell, use callbyneed for performance
+ reasons, but theoretical presentations of lazy evaluation often use
+ callbyname for simplicity.
+
+ The opposite of lazy actions is eager evaluation, sometimes known as strict
+ evaluation. Eager evaluation is the evaluation behavior used in most
+ programming languages.
+
+ Lazy evaluation refers to how expressions are evaluated when they are passed as
+ arguments to functions and entails the following three points:[1]
+
+ 1. The expression is only evaluated if the result is required by the calling
+ function, called delayed evaluation.[2] 2. The expression is only evaluated to
+ the extent that is required by the calling function, called shortcircuit
+ evaluation. 3. The expression is never evaluated more than once, called
+ applicativeorder evaluation.[3]
+
+ Contents [hide]
+
+ * 1 Delayed evaluation
+ o 1.1 Control structures
+ * 2 Controlling eagerness in lazy languages 3 Other uses 4 See also 5
+ * References 6 External links
+
+ [edit] Delayed evaluation
+
+ Delayed evaluation is used particularly in functional languages. When using
+ delayed evaluation, an expression is not evaluated as soon as it gets bound to
+ a variable, but when the evaluator is forced to produce the expression's value.
+ That is, a statement such as x:=expression; (i.e. the assignment of the result
+ of an expression to a variable) clearly calls for the expression to be
+ evaluated and the result placed in x, but what actually is in x is irrelevant
+ until there is a need for its value via a reference to x in some later
+ expression whose evaluation could itself be deferred, though eventually the
+ rapidlygrowing tree of dependencies would be pruned in order to produce some
+ symbol rather than another for the outside world to see.
+
+ Some programming languages delay evaluation of expressions by default, and some
+ others provide functions or special syntax to delay evaluation. In Miranda and
+ Haskell, evaluation of function arguments is delayed by default. In many other
+ languages, evaluation can be delayed by explicitly suspending the computation
+ using special syntax (as with Scheme's "delay" and "force" and OCaml's "lazy"
+ and "Lazy.force") or, more generally, by wrapping the expression in a thunk.
+ The object representing such an explicitly delayed evaluation is called a
+ future or promise. Perl 6 uses lazy evaluation of lists, so one can assign
+ infinite lists to variables and use them as arguments to functions, but unlike
+ Haskell and Miranda, Perl 6 doesn't use lazy evaluation of arithmetic operators
+ and functions by default.
+
+ Delayed evaluation has the advantage of being able to create calculable
+ infinite lists without infinite loops or size matters interfering in
+ computation. For example, one could create a function that creates an infinite
+ list (often called a stream) of Fibonacci numbers. The calculation of the nth
+ Fibonacci number would be merely the extraction of that element from the
+ infinite list, forcing the evaluation of only the first n members of the list.
+
+ For example, in Haskell, the list of all Fibonacci numbers can be written as
+
+ fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
+
+ In Haskell syntax, ":" prepends an element to a list, tail returns a list
+ without its first element, and zipWith uses a specified function (in this case
+ addition) to combine corresponding elements of two lists to produce a third.
+
+ Provided the programmer is careful, only the values that are required to
+ produce a particular result are evaluated. However, certain calculations may
+ result in the program attempting to evaluate an infinite number of elements;
+ for example, requesting the length of the list or trying to sum the elements of
+ the list with a fold operation would result in the program either failing to
+ terminate or running out of memory. [edit] Control structures
+
+ Even in most eager languages if statements evaluate in a lazy fashion.
+
+ if a then b else c
+
+ evaluates (a), then if and only if (a) evaluates to true does it evaluate (b),
+ otherwise it evaluates (c). That is, either (b) or (c) will not be evaluated.
+ Conversely, in an eager language the expected behavior is that
+
+ define f(x,y) = 2*x set k = f(e,5)
+
+ will still evaluate (e) and (f) when computing (k). However, userdefined
+ control structures depend on exact syntax, so for example
+
+ define g(a,b,c) = if a then b else c l = g(h,i,j)
+
+ (i) and (j) would both be evaluated in an eager language. While in
+
+ l' = if h then i else j
+
+ (i) or (j) would be evaluated, but never both.
+
+ Lazy evaluation allows control structures to be defined normally, and not as
+ primitives or compiletime techniques. If (i) or (j) have side effects or
+ introduce run time errors, the subtle differences between (l) and (l') can be
+ complex. As most programming languages are Turingcomplete, it is of course
+ possible to introduce lazy control structures in eager languages, either as
+ builtins like C's ternary operator ?: or by other techniques such as clever
+ use of lambdas, or macros.
+
+ Shortcircuit evaluation of Boolean control structures is sometimes called
+ "lazy". [edit] Controlling eagerness in lazy languages
+
+ In lazy programming languages such as Haskell, although the default is to
+ evaluate expressions only when they are demanded, it is possible in some cases
+ to make code more eagerâor conversely, to make it more lazy again after it has
+ been made more eager. This can be done by explicitly coding something which
+ forces evaluation (which may make the code more eager) or avoiding such code
+ (which may make the code more lazy). Strict evaluation usually implies
+ eagerness, but they are technically different concepts.
+
+ However, there is an optimisation implemented in some compilers called
+ strictness analysis, which, in some cases, allows the compiler to infer that a
+ value will always be used. In such cases, this may render the programmer's
+ choice of whether to force that particular value or not, irrelevant, because
+ strictness analysis will force strict evaluation.
+
+ In Haskell, marking constructor fields strict means that their values will
+ always be demanded immediately. The seq function can also be used to demand a
+ value immediately and then pass it on, which is useful if a constructor field
+ should generally be lazy. However, neither of these techniques implements
+ recursive strictnessâfor that, a function called deepSeq was invented.
+
+ Also, pattern matching in Haskell 98 is strict by default, so the ~ qualifier
+ has to be used to make it lazy. [edit]
+
+
+
+
+confluence/ChurchRosser
+
+
+"combinators", useful ones:
+
+Useful combinators
+I
+K
+omega
+true/getfirst/K
+false/getsecond
+makepair
+S,B,C,W/dup,Omega
+
+(( combinatorial logic ))
+
+composition
+nary[sic] composition
+"foldbased"[sic] representation of numbers
+defining some operations, not yet predecessor
+ iszero,succ,add,mul,...?
+
+lists?
+ explain differences between list and tuple (and stream)
+ FIFO queue,LIFO stack,etc...
+"pairbased" representation of lists (1,2,3)
+nil,cons,isnil,head,tail
+
+explain operations like "map","filter","fold_left","fold_right","length","reverse"
+but we're not yet in position to implement them because we don't know how to recurse
+
+Another way to do lists is based on model of how we did numbers
+"foldbased" representation of lists
+One virtue is we can do some recursion by exploiting the foldbased structure of our implementation; don't (yet) need a general method for recursion
+
+Go back to numbers, how to do predecessor? (a few ways)
+For some purposes may be easier (to program,more efficient) to use "pairbased" representation of numbers
+("More efficient" but these are still base1 representations of numbers!)
+In this case, too you'd need a general method for recursion
+(You could also have a hybrid, pairandfold based representation of numbers, and a hybrid, pairandfold based representation of lists. Works quite well.)
+
+Recursion
+Even if we have foldbased representation of numbers, and predecessor/equal/subtraction, some recursive functions are going to be out of our reach
+Need a general method, where f(n) doesn't just depend on f(n1) (or