M ~~> N
We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol <code><big><big>→</big></big></code> for one-step contraction, and the symbol <code><big><big>↠</big></big></code> for zero-or-more step reduction. Hindley and Seldin use <code><big><big><big>⊳</big></big></big><sub>1</sub></code> and <code><big><big><big>⊳</big></big></big></code>.
+
+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 **beta-convertible**. 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. Personally, I keep confusing that with the relation to be described next, so let's use this notation instead. Note that `M <~~> N` doesn't mean that each of `M` and `N` are reducible to each other; that only holds when `M` and `N` are the same expression. (Or, with our convention of only saying "reducible" for one or more reduction steps, it never holds.)
+
+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 <code>≡</code> 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.)
+
+
+**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 (x y)))
+
+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)
+
+would abbreviate:
+
+ (\x (\y (z y)) z)
+
+**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:
+
+ (\x. x y)
+
+as:
+
+ \x. x y
+
+but you should include the parentheses in:
+
+ (\x. x y) z
+
+and:
+
+ z (\x. x y)
+
+**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)
+
+
+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)
+
+and:
+
+ (\z z)
+
+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 x. y x) (\z z)
+
+ (\x. (\z z) x)
+
+ (\z z)
+
+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 function can differentiate between them when they're supplied as an argument to it. However, these expressions are all syntactically distinct.
+
+The first two expressions are *convertible*: in particular the first reduces to the second. So they can be regarded as proof-theoretically 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 non-equivalent.
+
+There's an extension of the proof-theory 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 Monday's meeting (and further discussion is best pursued in person).
+
+
+
+Booleans and pairs
+==================
+
+Our definition of these is reviewed in [[Assignment1]].
+
+
+It's possible to do the assignment without using a Scheme interpreter, however
+you should take this opportunity to [get Scheme installed on your
+computer](/how_to_get_the_programming_languages_running_on_your_computer), and
+[get started learning Scheme](/learning_scheme). It will help you test out
+proposed answers to the assignment.
+
+
+
+
+
+
+Declarative/functional vs Imperatival/dynamic models of computation
+===================================================================
+
+Many of you, like us, will have grown up thinking the paradigm of computation is a sequence of changes. Let go of that. It will take some care to separate the operative notion of "sequencing" here from other notions close to it, but once that's done, you'll see that languages that have no significant notions of sequencing or changes are Turing complete: they can perform any computation we know how to describe. In itself, that only puts them on equal footing with more mainstream, imperatival programming languages like C and Java and Python, which are also Turing complete. But further, the languages we want you to become familiar with can reasonably be understood to be more fundamental. They embody the elemental building blocks that computer scientists use when reasoning about and designing other languages.
+
+Jim offered the metaphor: think of imperatival languages, which include "mutation" and "side-effects" (we'll flesh out these keywords as we proceeed), as the pâté of computation. We want to teach you about the meat and potatoes, where as it turns out there is no sequencing and no changes. There's just the evaluation or simplification of complex expressions.
+
+Now, when you ask the Scheme interpreter to simplify an expression for you, that's a kind of dynamic interaction between you and the interpreter. You may wonder then why these languages should not also be understood imperatively. The difference is that in a purely declarative or functional language, there are no dynamic effects in the language itself. It's just a static semantic fact about the language that one expression reduces to another. You may have verified that fact through your dynamic interactions with the Scheme interpreter, but that's different from saying that there are dynamic effects in the language itself.
+
+What the latter would amount to will become clearer as we build our way up to languages which are genuinely imperatival or dynamic.
+
+Many of the slogans and keywords we'll encounter in discussions of these issues call for careful interpretation. They mean various different things.
+
+For example, you'll encounter the claim that declarative languages are distinguished by their **referential transparency.** What's meant by this is not always exactly the same, and as a cluster, it's related to but not the same as this means for philosophers and linguists.
+
+The notion of **function** that we'll be working with will be one that, by default, sometimes counts as non-identical functions that map all their inputs to the very same outputs. For example, two functions from jumbled decks of cards to sorted decks of cards may use different algorithms and hence be different functions.
+
+It's possible to enhance the lambda calculus so that functions do get identified when they map all the same inputs to the same outputs. This is called making the calculus **extensional**. Church called languages which didn't do this **intensional**. If you try to understand that kind of "intensionality" in terms of functions from worlds to extensions (an idea also associated with Church), you may hurt yourself. So too if you try to understand it in terms of mental stereotypes, another notion sometimes designated by "intension."
+
+It's often said that dynamic systems are distinguished because they are the ones in which **order matters**. However, there are many ways in which order can matter. If we have a trivalent boolean system, for example---easily had in a purely functional calculus---we might choose to give a truth-table like this for "and":
+
+<pre><code>
+true and true = true
+true and true = true
+true and * = *
+true and false = false
+* and true = *
+* and * = *
+* and false = *
+false and true = false
+false and * = false
+false and false = false
+</code></pre>
+
+And then we'd notice that `* and false` has a different intepretation than `false and *`. (The same phenomenon is already present with the material conditional in bivalent logics; but seeing that a non-symmetric semantics for `and` is available even for functional languages is instructive.)
+
+Another way in which order can matter that's present even in functional languages is that the interpretation of some complex expressions can depend on the order in which sub-expressions are evaluated. Evaluated in one order, the computations might never terminate (and so semantically we interpret them as having "the bottom value"---we'll discuss this). Evaluated in another order, they might have a perfectly mundane value. Here's an example, though we'll reserve discussion of it until later:
+
+ (\x. y) ((\x. x x) (\x. x x))
+
+Again, these facts are all part of the metatheory of purely functional languages. But *there is* a different sense of "order matters" such that it's only in imperatival languages that order so matters.
+
+ x := 2
+ x := x + 1
+ x == 3
+
+Here the comparison in the last line will evaluate to true.
+
+ x := x + 1
+ x := 2
+ x == 3
+
+Here the comparison in the last line will evaluate to false.
+
+One of our goals for this course is to get you to understand *what is* that new
+sense such that only so matters in imperatival languages.
+
+Finally, you'll see the term **dynamic** used in a variety of ways in the literature for this course:
+
+* dynamic versus static typing
+
+* dynamic versus lexical scoping
+
+* dynamic versus static control operators
+
+* finally, we're used ourselves to talking about dynamic versus static semantics
+
+For the most part, these uses are only loosely connected to each other. We'll tend to use "imperatival" to describe the kinds of semantic properties made available in dynamic semantics, languages which have robust notions of sequencing changes, and so on.
+
+Map
+===
+
+<table>
+<tr>
+<td width=30%>Scheme (functional part)</td>
+<td width=30%>OCaml (functional part)</td>
+<td width=30%>C, Java, Pasval<br>
+Scheme (imperative part)<br>
+OCaml (imperative part)</td>
+<tr>
+<td width=30%>lambda calculus<br>
+combinatorial logic</td>
+<tr>
+<td colspan=3 align=center>--------------------------------------------------- Turing complete ---------------------------------------------------</td>
+<tr>
+<td width=30%>
+<td width=30%>more advanced type systems, such as polymorphic types
+<td width=30%>
+<tr>
+<td width=30%>
+<td width=30%>simply-typed lambda calculus (what linguists mostly use)
+<td width=30%>
+</table>
+