-Variables: `x`, `y`, `z`... -
+ succ 1 -Each variable is an expression. For any expressions M and N and variable a, the following are also expressions: +On the other hand, the functions `+`, `*`, `-`, `==`, and so on come in between their arguments, like this: -
-Abstract: `(λa M)` -
+ x < y -We'll tend to write `(λa M)` as just `(\a M)`, so we don't have to write out the markup code for the `λ`. You can yourself write `(λa M)` or `(\a M)` or `(lambda a M)`. +Functions of this latter sort are said to have an "infix" syntax. This is just a convenience for how we write them. Our language will have to keep rigorous track of which functions have infix syntax and which don't, but we'll just rely on context and our brains to make sense of this for now. Functions with the ordinary, non-infix syntax can take two arguments, as well. If we had defined the less-than relation (boolean function) in that style, we'd write it like this instead: -
-Application: `(M N)` -
+ lessthan? (x, y) -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 for expressions of any of these three forms. +or perhaps like this: -Examples of expressions: + lessthan? x y -
``` -x -(y x) -(x x) -(\x y) -(\x x) -(\x (\y x)) -(x (\x x)) -((\x (x x)) (\x (x x)))```

``` -```

+We'll get more acquainted with the difference between these next week. For now, I'll just stick to the first form. -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 **beta-reduction** or -"beta-contraction". Suppose you have some expression of the form: +Another set of operations we have are: - ((\a M) N) + and, or, not -that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "beta-reducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**. +The first two of these are infix functions that expect two boolean arguments, and gives a boolean result. The third is a function that expects only one boolean argument. Our earlier function `!=` means "doesn't equal", and: -The rule of beta-reduction permits a transition from that expression to the following: + x != y - M [a:=N] +will be just another way to write: -What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`. + not (x == y) -What is a free occurrence? +You see that you can use parentheses in the standard way. By the way, `<=` means ≤ or "less than or equals to", and `>=` means ≥. Just in case you haven't seen them written this way before. -> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`. +I've started throwing in some **variables**. We'll say variables are any expression that's written with an initial lower-case letter, then is followed by a sequence of zero or more upper- or lower-case letters, or numerals, or underscores (`_`). Then at the end you can optionally have a `?` or `!` or a sequence of `'`s, understood as "primes." Hence, all of these are legal variables: -> 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`. + x + x1 + x_not_y + xUBERANT + x' + x'' + x? + xs -> An occurrence of a variable is **free** if it's not bound. +We'll follow a *convention* of using variables with short names and a final `s` to represent collections like sequences (to be discussed below). But this is just a convention to help us remember what we're up to, not a strict rule of the language. We'll also follow a convention of only using variables ending in `?` to represent functions that return a boolean value. Thus, for example, `zero?` will be a function that expects a single number argument and returns a boolean corresponding to whether that number is `0`. `odd?` will be a function that expects a single number argument and returns a boolean corresponding to whether than number is odd. Above, I suggested we might use `lessthan?` to represent a function that expects *two* number arguments, and again returns a boolean result. -For instance: +We also conventionally reserve variables ending in `!` for a different special class of functions, that we will explain later in the course. +In fact you can think of `succ` and `pred` and `not` and the rest as also being variables; it's just that these variables have been pre-defined in our language to be bound to functions we agreed upon in advance. You can even think of `==` and `<` as being variables, too, bound to other functions. But I haven't given you parsing rules yet which would make them legal variables, because they don't start with a lower-case letter. We can make the parsing rules more liberal later. -> T is defined to be `(x (\x (\y (x (y z)))))` +Only a few simple expressions in our language aren't variables. These include the literal values, and also **keywords** like `let` and `case` and so on that we'll discuss below. You can't use `let` as a variable, else the syntax of our language would become too hard to mechanically parse. (And probably too hard for our meager brains to parse, too.) -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. +The rule for symbolic atoms is that a single quote `'` followed by any single word that could be a legal variable expresses such an atom, a different atom for each different expression. +Thus `'false` is a symbolic atom, but so too are `'x` and `'succ`. For the time being, I'll restrict myself to only talking about the symbolic atoms `'true` and `'false`. These constitute a special subclass of symbolic atoms that we call the **booleans** or truth-values. Nothing deep hangs on them being a subclass of a larger type in this way; it just seems elegant. Some other languages make booleans their own special type, not a subclass of another type. Others make them a subclass of the numbers (yuck). We will think of them this way. -Here's an example of beta-reduction: +Note that when writing a symbolic atom there is no closing `'`, just a `'` at the beginning. That's enough to make the whole word, up to the next space (or whatever) count as expressing a symbolic atom. We use the initial `'` to make it easy for us to have a rich set of symbolic atoms, as well as a rich set of variables, without getting them mixed up. Variables never begin with `'`; symbolic atoms always do. - ((\x (y x)) z) +We call these things symbolic *atoms* because they aren't collections. Thus numbers are also atoms, but not symbolic ones. And functions are also atoms, but again, not symbolic ones. -beta-reduces to: +Functions are another class of values we'll have in our language. They aren't "literal" values, though. Numbers and symbolic atoms are simple expressions in the language that evaluate to themselves. That's what we mean by calling them "literals." Functions aren't expressions in the language at all; they have to be generated from the evaluation of more complex expressions. - (y z) +(By the way, I really am serious in thinking of *the numbers themselves* as being expressions in this language; rather than some "numerals" that aren't themselves numbers. We'll talk about this down the road. For now, don't worry about it too much.) -We'll write that like this: +I said we wanted to be starting with a fragment of arithmetic, so we'll keep the function values off-stage for the moment, and also all the symbolic atoms except for `'true` and `'false`. So we've got numbers, truth-values, and some functions and relations (that is, boolean functions) defined on them. We also help ourselves to a notion of bounded quantification, as in ∀`x < M.` φ, where `M` and φ are (simple or complex) expressions that evaluate to a number and a boolean, respectively. We limit ourselves to *bounded* quantification so that the fragment we're dealing with can be "effectively" or mechanically decided. (As we extend the language, we will lose that property, but it will be a topic for later discussion exactly when that happens.) - ((\x (y x)) z) ~~> (y z) +As I mentioned in class, I will sometimes write ∀ x : ψ . φ in my informal metalanguage, where the ψ clause represents the quantifier's *restrictor*. Other people write this like `[`∀ x : ψ `]` φ, or in various other ways. My notation is meant to parallel the notation some linguists (for example, Heim & Kratzer) use in writing λ x : ψ . φ, where the ψ clause restricts the range of arguments over which the function designated by the λ-expression is defined. Later we will see the colon used in a somewhat similar (but also somewhat different) way in our programming languages. But that's foreshadowing. -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 +### Let and lambda ### -We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol `→` for one-step contraction, and the symbol `↠` for zero-or-more step reduction. Hindley and Seldin use `⊳1` and `⊳`. +So we have bounded quantification as in ∀ `x < 10.` φ. Obviously we could also make sense of ∀ `x == 5.` φ in just the same way. This would evaluate φ but with the variable `x` now bound to the value `5`, ignoring whatever it may be bound to in broader contexts. I will express this idea in a more perspicuous vocabulary, like this: `let x be 5 in` φ. (I say `be` rather than `=` because, as I mentioned before, it's too easy for the `=` sign to get used for too many subtly different jobs.) -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: +As one of you was quick to notice in class, when I shift to the `let`-vocabulary, I no longer restrict myself to just the case where φ evaluates to a boolean. I also permit myself expressions like this: - M <~~> N + let x be 5 in x + 1 -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.) +which evaluates to `6`. That's right. I am moving beyond the ∀ `x==5.` φ idea when I do this. But the rules for how to interpret this are just a straightforward generalization of our existing understanding for how to interpret bound variables. So there's nothing fundamentally novel here. -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 `≡` 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: +We can have multiple `let`-expressions embedded, as in: -> T is defined to be `(M N)`. + let y be (let x be 5 in x + 1) in 2 * y -We'll regard the following two expressions: + let x be 5 in let y be x + 1 in 2 * y - (\x (x y)) +both of which evaluate to `12`. When we have a stack of `let`-expressions as in the second example, I will write it like this: - (\z (z y)) + let + x be 5; + y be x + 1 + in 2 * 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. +It's okay to also write it all inline, like so: `let x be 5; y be x + 1 in 2 * y`. The `;` represents that we have a couple of `let`-bindings coming in sequence. The earlier bindings in the sequence are considered to be in effect for the later right-hand expressions in the sequence. Thus in: -Note that neither of those expressions are identical to: + let x be 0 in (let x be 5; y be x + 1 in 2 * y) - (\x (x w)) +The `x + 1` that is evaluated to give the value that `y` gets bound to uses the (more local) binding of `x` to `5`, not the (previous, less local) binding of `x` to `0`. By the way, the parentheses in that displayed expression were just to focus your attention. It would have parsed and meant the same without them. -because here it's a free variable that's been changed. Nor are they identical to: +Now we can allow ourselves to introduce λ-expressions in the following way. If a λ-expression is applied to an argument, as in: `(`λ `x.` φ`) M`, for any (simple or complex) expressions φ and `M`, this means the same as: `let x be M in` φ. That is, the argument to the λ-expression provides (when evaluated) a value for the variable `x` to be bound to, and then the result of the whole thing is whatever φ evaluates to, under that binding to `x`. - (\y (y y)) +If we restricted ourselves to only that usage of λ-expressions, that is when they were applied to all the arguments they're expecting, then we wouldn't have moved very far from the decidable fragment of arithmetic we began with. -because here the second occurrence of `y` is no longer free. +However, it's tempting to help ourselves to the notion of (at least partly) *unapplied* λ-expressions, too. If I can make sense of what: -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. +`(`λ `x. x + 1) 5` +means, then I can make sense of what: -Shorthand ---------- +`(`λ `x. x + 1)` -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.) +means, too. It's just *the function* that waits for an argument and then returns the result of `x + 1` with `x` bound to that argument. +This does take us beyond our (first-order) fragment of arithmetic, at least if we allow the bodies and arguments of λ-expressions to be any expressible value, including other λ-expressions. But we're having too much fun, so why should we hold back? -**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: +So now we have a new kind of value our language can work with, alongside numbers and booleans. We now have function values, too. We can bind these function values to variables just like other values: - (\x (\y (x y))) +`let id be` λ `x. x; y be id 5 in y` -can be abbreviated as: +evaluates to `5`. In reaching that result, the variable `id` was temporarily bound to the identity function, that expects an argument, binds it to the variable `x`, and then returns the result of evaluating `x` under that binding. - (\x (\y. x y)) +This is what is going on, behind the scenes, with all the expressions like `succ` and `+` that I said could really be understood as variables. They have just been pre-bound to certain agreed-upon functions rather than others. -and: - (\x (\y. (z y) z)) +### Containers ### -would abbreviate: +So far, we've only been talking about *atomic* values. Our language will also have some *container* values, that have other values as members. One example are **ordered sequences**, like: - (\x (\y ((z y) z))) + [10, 20, 30] -This on the other hand: +This is a sequence of length 3. It's the result of *cons*ing the value `10` onto the front of the shorter, length-2 sequence `[20, 30]`. In this made-up language, we'll represent the sequence-consing operation like this: - (\x (\y. z y) z) + 10 & [20, 30] -would abbreviate: +If you want to know why we call it "cons", that's because this is what the operation is called in Scheme, and they call it that as shorthand for "constructing" the longer list (they call it a "list" rather than a "sequence") out of the components `10` and `[20, 30]`. The name is a bit unfortunate, though, because other structured values besides lists also get "constructed", but we don't say "cons" about them. Still, this is the tradition. Let's just take "cons" to be a nonsense label with an interesting back-history. - (\x (\y (z y)) z) +The sequence `[20, 30]` in turn is the result of: -**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: + 20 & [30] - (\x. x y) +and the sequence `[30]` is the result of consing `30` onto the empty sequence `[]`. Note that the sequence `[30]` is not the same as the number `30`. The former is a container value, with one element. The latter is an atomic value, and as such won't have any elements. If you try to do this: -as: + [30] + 1 - \x. x y +it won't work. We haven't discussed what happens with illegal expressions like that, or like `'true + 1`. For the time being, I'll just say these "don't work", or that they "crash". We'll discuss the variety of ways these illegalities might be handled later. -but you should include the parentheses in: +Also, if you try to do this: - (\x. x y) z + 20 & 30 -and: +it won't work. The consing operator `&` always requires a container (here, a sequence) on its right-hand side. And `30` is not a container. - z (\x. x y) +We've said that: -**Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as: + [10, 20, 30] - (\x y. M) +is the same as; -Similarly, `(\x (\y (\z M)))` can be abbreviated as: + 10 & (20 & (30 & [])) - (\x y z. M) +and the latter can also be written without the parentheses. Our language knows that `&` should always be understood as "implicitly associating to the right", that is, that: + 10 & 20 & 30 & [] -Lambda terms represent functions --------------------------------- +should be interpreted like the expression displayed before. Other operators like `-` should be understood as "implicitly associating to the left." If we write: -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: + 30 - 2 - 1 -> `(\x x)` represents the identity function: given any argument `M`, this function -simply returns `M`: `((\x x) M) ~~> M`. +we presumably want it to be understood as: -> `(\x (x x))` duplicates its argument: -`((\x (x x)) M) ~~> (M M)` + (30 - 2) - 1 -> `(\x (\y x))` throws away its second argument: -`(((\x (\y x)) M) N) ~~> M` +not as: -and so on. + 30 - (2 - 1) -It is easy to see that distinct lambda expressions can represent the same -function, considered as a mapping from input to outputs. Obviously: +Other operators don't implicitly associate at all. For example, you may understand the expression: - (\x x) + 10 < x < 20 -and: +because we have familiar conventions about what it means. But what it means is not: - (\z z) + (10 < x) < 20 -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: +The result of the parenthesized expression is either `'true` or `'false`, assuming `x` evaluates to a number. But `'true < 20` doesn't mean anything, much less what we expect `10 < x < 20` to mean. So `<` doesn't implicitly associate to the left. Neither does it implicitly associate to the right. If you want expressions like `10 < x < 20` to be meaningful, they will need their own special rules. - (\y x. y x) (\z z) +Sequences are containers that keep track of the order of their arguments, and also those arguments' multiplicity (how many times each one appears). Other containers might also keep track of these things, and more structural properties too, or they might keep track of less. Let's say we also have **set containers** too, like this: - (\x. (\z z) x) + {10, 20, 30} - (\z z) +Whereas the sequences `[10, 20, 10]`, `[10, 20]`, and `[20, 10]` are three different sequences, `{10, 20, 10}`, `{10, 20}`, and `{20, 10}` would just be different ways of expressing a single set. -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. +We can let the `&` operator do extra-duty, and express the "consing" relation for sets, too: -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. + 10 & {20} -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). +evaluates to `{10, 20}`, and so too does: + 10 & {10, 20} +As I mentioned in class, we'll let `&&` express the operation by which two sequences are appended or concatenated to each other: -Booleans and pairs -================== + [10, 20] && [30, 40, 50] -Our definition of these is reviewed in [[Assignment1]]. +evaluates to `[10, 20, 30, 40, 50]`. For sets, we'll let `and` and `or` and `-` do extra duty, and express set intersection, set union, and set subtraction, when their arguments are sets. If the arguments of `and` and `or` are booleans, on the other hand, or the arguments of `-` are numbers, then they express the functions we were understanding them to express before. +In addition to sequences, there's another kind of expression that might initially be confused with them. We might call these **tuples** or **multivalues**. They are written surrounded by parentheses rather than square brackets. Here's an example: -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. +`(0, 'true,` λ`x. x)` +That's a multivalue or tuple with 3 elements (also called a "triple"). +In the programming languages and other formal systems we'll be looking at, tuples and sequences are usually understood and handled differently. This is because we apply different assumptions to them. In the case of a sequence, it's assumed that they will have homogeneously-typed elements, and that their length will be irrelevant to their own type. So you can have the sequence: + [20, 30] +and the sequence: + [30] -Declarative/functional vs Imperatival/dynamic models of computation -=================================================================== +and even the sequence: -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. +and these will all be of the same type, namely a sequence of numbers. You can have sequences with other types of elements, too, for example a sequence of booleans: -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. + ['true, 'false, 'true] -What the latter would amount to will become clearer as we build our way up to languages which are genuinely imperatival or dynamic. +or a sequence of sequences of numbers: -Many of the slogans and keywords we'll encounter in discussions of these issues call for careful interpretation. They mean various different things. + [[10, 20], [], [30]] -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. +An excellent question that came up in class is "How do we tell whether `[]` expresses the empty sequence of numbers or the empty sequence of something else?" We will discuss that question in later weeks. It's central to some of the developments we'll be exploring. For now, just put that question on a mental shelf and assume that somehow this just works out right. -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. +Now whereas sequences expect homogenously-typed elements, and their length is irrelevant to their own type, mulivalues or tuples are the opposite in both respects. They may have elements of heterogenous type, as our example: -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." +`(0, 'true,` λ`x. x)` -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": +did. They need not, but they may. Also, the type of a multivalue or tuple does depend on its length, and moreover on the specific types of each of its elements. A tuple of length 2 (also called a "pair") whose first element is a number and second element is a boolean is a different type of thing that a tuple whose first element is a boolean and whose second element is a number. Most functions expecting the first as an argument will "crash" if you give them the second instead. -
``````
-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
-``````
+Earlier I said that we can call these things "multivalues or tuples". Here I'll make a technical comment, that in fact I'll understand these slightly differently. Really I'll understand the bare expression `(10, x)` to express a multivalue, and to express a tuple proper, you'll have to write `Pair (10, x)` or something like that. The difference between these is that only the tuple proper is a single value that can be bound to a single variable. The multivalue isn't a single value at all, but rather a plurality of values. This is a bit subtle, and other languages we're looking at this term don't always make this distinction. But the result is that they have to say complicated things elsewhere. If we permit ourselves this fine distinction here, many other things downstream will go more smoothly than they do in the languages that don't make it. Ours is just a made-up language, but I've thought this through carefully, so humor me. We haven't yet introduced the apparatus to make sense of expressions like `Pair (10, x)`, so for the time being I'll just restrict myself to multivalues, not to tuples proper. The result will be that while we can say: -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.) + let x be [10, 20] in ... -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: +that is, sequences are first-class values in our language, we can't say: - (\x. y) ((\x. x x) (\x. x x)) + let x be (10, 'true) in ... -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. +or even: - x := 2 - x := x + 1 - x == 3 + let x be (10, 20) in ... -Here the comparison in the last line will evaluate to true. +However, intuitively it ought to make sense to say: - x := x + 1 - x := 2 - x == 3 + let (x, y) be (10, 'true) in ... -Here the comparison in the last line will evaluate to false. +That should just bind the variable `x` to the value `10` and the variable `y` to the value `'true`, and go on to evaluate the rest of the expression with those bindings in place. In this particular example, we could equally have said: -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. + let x be 10; y be 'true in ... -Finally, you'll see the term **dynamic** used in a variety of ways in the literature for this course: +but in other examples it will be substantially more convenient to be able to bind `x` and `y` simultaneously. Here's an example: -* dynamic versus static typing +`let` +  `f be` λ `x. (x, 2*x)` +  `(x, y) be f 10` +`in [x, y]` -* dynamic versus lexical scoping +which evaluates to `[10, 20]`. Note that we have the function `f` returning two values, rather than just one, just by having its body evaluate to a multivalue rather than to a single value. -* dynamic versus static control operators +It's a little bit awkward to say `let (x, y) be ...`, so I propose we instead always say `let (x, y) match ...`. (This will be even more natural as we continue generalizing what we've done here, as we will in the next section.) For consistency, we'll say `match` instead of `be` in all cases, so that we write even this: -* finally, we're used ourselves to talking about dynamic versus static semantics + let + x match 10 + in ... -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. +rather than: -Map -=== + let + x be 10 + in ... - - - - - - - - - - - -
 Scheme (functional part) OCaml (functional part) C, Java, Pasval -Scheme (imperative part) -OCaml (imperative part) lambda calculus -combinatorial logic --------------------------------------------------- Turing complete --------------------------------------------------- - more advanced type systems, such as polymorphic types - - - simply-typed lambda calculus (what linguists mostly use) - -
```-	`∀x. (F x or ∀x (not (F x)))`