Lecture #4, Tuesday, January 14th ================================= - Simple Parsing - The `match` Form - The `define-type` Form - The `cases` Form - Semantics (= Evaluation) - Side-note: Compositionality - Implementing an Evaluator - Implementing The AE Language - Intro to Typed Racket - Bindings & Substitution ------------------------------------------------------------------------ # Simple Parsing On to an implementation of a "parser": Unrelated to what the syntax actually looks like, we want to parse it as soon as possible --- converting the concrete syntax to an abstract syntax tree. No matter how we write our syntax: - `3+4` (infix), - `3 4 +` (postfix), - `+(3,4)` (prefix with args in parens), - `(+ 3 4)` (parenthesized prefix), we always mean the same abstract thing --- adding the number `3` and the number `4`. The essence of this is basically a tree structure with an addition operation as the root and two leaves holding the two numerals. With the right data definition, we can describe this in Racket as the expression `(Add (Num 3) (Num 4))` where `Add` and `Num` are constructors of a tree type for syntax, or in a C-like language, it could be something like `Add(Num(3),Num(4))`. Similarly, the expression `(3-4)+7` will be described in Racket as the expression: (Add (Sub (Num 3) (Num 4)) (Num 7)) Important note: "expression" was used in two *different* ways in the above --- each way corresponds to a different language, and the result of evaluating the second "expression" is a Racket value that *represents* the first expression. To define the data type and the necessary constructors we will use this: (define-type AE [Num Number] [Add AE AE] [Sub AE AE]) * Note --- Racket follows the tradition of Lisp which makes syntax issues almost negligible --- the language we use is almost as if we are using the parse tree directly. Actually, it is a very simple syntax for parse trees, one that makes parsing extremely easy. [This has an interesting historical reason... Some Lisp history --- *M-expressions* vs. *S-expressions*, and the fact that we write code that is isomorphic to an AST. Later we will see some of the advantages that we get by doing this. See also "*The Evolution of Lisp*", section 3.5.1 (especially the last sentence).] To make things very simple, we will use the above fact through a double-level approach: * we first "parse" our language into an intermediate representation --- a Racket list --- this is mostly done by a modified version of Racket's `read` function that uses curly `{}` braces instead of round `()` parens, * then we write our own `parse` function that will parse the resulting list into an instance of the `AE` type --- an abstract syntax tree (AST). This is achieved by the following simple recursive function: (: parse-sexpr : Sexpr -> AE) ;; parses s-expressions into AEs (define (parse-sexpr sexpr) (cond [(number? sexpr) (Num sexpr)] [(and (list? sexpr) (= 3 (length sexpr))) (let ([make-node (match (first sexpr) ['+ Add] ['- Sub] [else (error 'parse-sexpr "unknown op: ~s" (first sexpr))]) #| the above is the same as: (cond [(equal? '+ (first sexpr)) Add] [(equal? '- (first sexpr)) Sub] [else (error 'parse-sexpr "unknown op: ~s" (first sexpr))]) |#]) (make-node (parse-sexpr (second sexpr)) (parse-sexpr (third sexpr))))] [else (error 'parse-sexpr "bad syntax in ~s" sexpr)])) This function is pretty simple, but as our languages grow, they will become more verbose and more difficult to write. So, instead, we use a new special form: `match`, which is matching a value and binds new identifiers to different parts (try it with "Check Syntax"). Re-writing the above code using `match`: (: parse-sexpr : Sexpr -> AE) ;; parses s-expressions into AEs (define (parse-sexpr sexpr) (match sexpr [(number: n) (Num n)] [(list '+ left right) (Add (parse-sexpr left) (parse-sexpr right))] [(list '- left right) (Sub (parse-sexpr left) (parse-sexpr right))] [else (error 'parse-sexpr "bad syntax in ~s" sexpr)])) And finally, to make it more uniform, we will combine this with the function that parses a string into a sexpr so we can use strings to represent our programs: (: parse : String -> AE) ;; parses a string containing an AE expression to an AE (define (parse str) (parse-sexpr (string->sexpr str))) ------------------------------------------------------------------------ # The `match` Form The syntax for `match` is (match value [pattern result-expr] ...) The value is matched against each pattern, possibly binding names in the process, and if a pattern matches it evaluates the result expression. The simplest form of a pattern is simply an identifier --- it always matches and binds that identifier to the value: (match (list 1 2 3) [x x]) ; evaluates to the list Another simple pattern is a quoted symbol, which matches that symbol. For example: (match foo ['x "yes"] [else "no"]) will evaluate to `"yes"` if `foo` is the symbol `x`, and to `"no"` otherwise. Note that `else` is not a keyword here --- it happens to be a pattern that always succeeds, so it behaves like an else clause except that it binds `else` to the unmatched-so-far value. Many patterns look like function application --- but don't confuse them with applications. A `(list x y z)` pattern matches a list of exactly three items and binds the three identifiers; or if the "arguments" are themselves patterns, `match` will descend into the values and match them too. More specifically, this means that patterns can be nested: (match (list 1 2 3) [(list x y z) (+ x y z)]) ; evaluates to 6 (match '((1) (2) 3) [(list (list x) (list y) z) (+ x y z)]) ; also 6 There is also a `cons` pattern that matches a non-empty list and then matches the first part against the head for the list and the second part against the tail of the list. In a `list` pattern, you can use `...` to specify that the previous pattern is repeated zero or more times, and bound names get bound to the list of respective matching. One simple consequent is that the `(list hd tl ...)` pattern is exactly the same as `(cons hd tl)`, but being able to repeat an arbitrary pattern is very useful: > (match '((1 2) (3 4) (5 6) (7 8)) [(list (list x y) ...) (list x y)]) '((1 3 5 7) (2 4 6 8)) A few more useful patterns: id -- matches anything, binds `id' to it _ -- matches anything, but does not bind (number: n) -- matches any number and binds it to `n' (symbol: s) -- same for symbols (string: s) -- strings (sexpr: s) -- S-expressions (needed sometimes for Typed Racket) (and pat1 pat2) -- matches both patterns (or pat1 pat2) -- matches either pattern (careful with bindings) Note that the `foo:` patterns are all specific to our `#lang pl`, they are not part of `#lang racket` or `#lang typed/racket` The patterns are tried one by one *in-order*, and if no pattern matches the value, an error is raised. Note that `...` in a `list` pattern can follow *any* pattern, including all of the above, and including nested list patterns. Here are a few examples --- you can try them out with `#lang pl untyped` at the top of the definitions window. This: (match x [(list (symbol: syms) ...) syms]) matches `x` against a pattern that accepts only a list of symbols, and binds `syms` to those symbols. And here's an example that matches a list of any number of lists, where each of the sub-lists begins with a symbol and then has any number of numbers. Note how the `n` and `s` bindings get values for a list of all symbols and a list of lists of the numbers: > (define (foo x) (match x [(list (list (symbol: s) (number: n) ...) ...) (list 'symbols: s 'numbers: n)])) > (foo (list (list 'x 1 2 3) (list 'y 4 5))) '(symbols: (x y) numbers: ((1 2 3) (4 5))) Here is a quick example for how `or` is used with two literal alternatives, how `and` is used to name a specific piece of data, and how `or` is used with a binding: > (define (foo x) (match x [(list (or 1 2 3)) 'single] [(list (and x (list 1 _)) 2) x] [(or (list 1 x) (list 2 x)) x])) > (foo (list 3)) 'single > (foo (list (list 1 99) 2)) '(1 99) > (foo (list 1 10)) 10 > (foo (list 2 10)) 10 ------------------------------------------------------------------------ # The `define-type` Form The class language that we're using, `#lang pl`, is based on *Typed Racket*: a statically-typed dialect of Racket. It is not exactly the same as Typed Racket --- it is restricted in many ways, and extended in a few ways. (You should therefore try to avoid looking at the Typed Racket documentation and expect things to be the same in `#lang pl`.) The most important extension is `define-type`, which is the construct we will be using to create new user-defined types. In general, such definitions looks like what we just used: (define-type AE [Num Number] [Add AE AE] [Sub AE AE]) This defines a *new type* called `AE`, an `AE?` predicate for this type, and a few *variants* for this type: `Num`, `Add`, and `Sub` in this case. Each of these variant names is a constructor, taking in arguments with the listed types, where these types can include the newly defined type itself in (the very common) case we're defining a recursive type. The return type is always the newly defined type, `AE` here. To summarize, this definition gives us a new `AE` type, and three constructors, as if we wrote the following type declarations: * `(: Num : Number -> AE)` * `(: Add : AE AE -> AE)` * `(: Sub : AE AE -> AE)` The newly defined types are known as *"disjoint unions"*, since values in these types are disjoint --- there is no overlap between the different variants. As we will see, this is what makes this such a useful construct for our needs: the compiler knows about the variants of each newly defined type, which will make it possible for it to complain if we extend a type with more variants but not update all uses of the type. Furthermore, since the return types of these constructors are all the new type itself, there is *no way* for us to write code that expects just *one* of these variants. We will use a second form, `cases`, to handle these values. ------------------------------------------------------------------------ # The `cases` Form A `define-type` declaration defines *only* what was described above: one new type name and a matching predicate, and a few variants as constructor functions. Unlike HtDP, we don't get predicates for each of the variants, and we don't get accessor functions for the fields of the variants. The way that we handle the new kind of values is with `cases`: this is a form that is very similar to `match`, but is specific to instances of the user-defined type. > Many students find it confusing to distinguish `match` and `cases` > since they are so similar. Try to remember that `match` is for > primitive Racket values (we'll mainly use them for S-expression > values), while `cases` is for user-defined values. The distinction > between the two forms is unfortunate, and doesn't serve any purpose. > It is just technically difficult to unify the two. For example, code that handles `AE` values (as defined above) can look as follows: (cases some-ae-value [(Num n) "a number"] [(Add l r) "an addition"] [(Sub l r) "a subtraction"]) As you can see, we need to have patterns for each of the listed variants (and the compiler will throw an error if some are missing), and each of these patterns specifies bindings that will get the field values contained in a given variant object. We can also use nested patterns: (cases some-ae-value [(Num n) "a number"] [(Add (Num m) (Num n)) "a simple addition"] [(Add l r) "an addition"] [(Sub (Num m) (Num n)) "a simple subtraction"] [(Sub l r) "a subtraction"]) but this is a feature that we will not use too often. The final clause in a `cases` form can be an `else` clause, which serves as a fallback in case none of the previous clauses matched the input value. However, using an `else` like this is ***strongly discouraged!*** The problem with using it is that it effectively eliminates the advantage in getting the type-checker to complain when a type definition is extended with new variants. Using these `else` clauses, we can actually mimic all of the functionality that you expect in HtDP-style code, which demonstrates that this is equivalent to HtDP-style definitions. For example: (: Add? : AE -> Boolean) ;; identifies instances of the `Add` variant (define (Add? ae) (cases ae [(Add l r) #t] [else #f])) (: Add-left : AE -> AE) ;; get the left-hand subexpression of an addition (define (Add-left ae) (cases ae [(Add l r) l] [else (error 'Add-left "expecting an Add value, got ~s" ae)])) ... ***Important reminder:*** this is code that ***you should not write!*** Doing so will lead to code that is more fragile than just using `cases`, since you'd be losing the protection the compiler gives you in the form of type errors on occurrences of `cases` that need to be updated when a type is extended with new variants. You would therefore end up writing a bunch of boiler-plate code only to end up with lower-quality code. The core of the problem is in the prevalent use of `else` which gives up that protection. In these examples the `else` clause is justified because even if `AE` is extended with new variants, functions like `Add?` and `Add-left` should not be affected and treat the new variants as they treat all other non-`Add` instances. (And since `else` is inherent to these functions, using them in our code is inherently a bad idea.) We will, however, have a few (*very few!*) places where we'll need to use `else` --- but this will always be done only on some specific functionality rather than a wholesale approach of defining a different interface for user-defined types. ------------------------------------------------------------------------ # Semantics (= Evaluation) > [PLAI ยง2] Back to BNF --- now, meaning. An important feature of these BNF specifications: we can use the derivations to specify *meaning* (and meaning in our context is "running" a program (or "interpreting", "compiling", but we will use "evaluating")). For example: ::= ; evaluates to the number | + ; evaluates to the sum of evaluating ; and | - ; ... the subtraction of from (... roughly!) To do this a little more formally: a. eval() = ;*** special rule: translate syntax to value b. eval( + ) = eval() + eval() c. eval( - ) = eval() - eval() Note the completely different roles of the two `+`s and `-`s. In fact, it might have been more correct to write: a. eval("") = b. eval(" + ") = eval("") + eval("") c. eval(" - ") = eval("") - eval("") or even using a marker to denote meta-holes in these strings: a. eval("$") = b. eval("$ + $") = eval("$") + eval("$") c. eval("$ - $") = eval("$") - eval("$") but we will avoid pretending that we're doing that kind of string manipulation. (For example, it will require specifying what does it mean to return `` for `$` (involves `string->number`), and the fragments on the right side mean that we need to specify these as substring operations.) Note that there's a similar kind of informality in our BNF specifications, where we assume that `` refers to some terminal or non-terminal. In texts that require more formal specifications (for example, in RFC specifications), each literal part of the BNF is usually double-quoted, so we'd get ::= | "+" | "-" An alternative popular notation for `eval(X)` is `[[X]]`: a. [[]] = b. [[ + ]] = [[]] + [[]] c. [[ - ]] = [[]] - [[]] Is there a problem with this definition? Ambiguity: eval(1 - 2 + 3) = ? Depending on the way the expression is parsed, we can get either a result of `2` or `-4`: eval(1 - 2 + 3) = eval(1 - 2) + eval(3) [b] = eval(1) - eval(2) + eval(3) [c] = 1 - 2 + 3 [a,a,a] = 2 eval(1 - 2 + 3) = eval(1) - eval(2 + 3) [c] = eval(1) - (eval(2) + eval(3)) [a] = 1 - (2 + 3) [a,a,a] = -4 Again, be very aware of confusing subtleties which are extremely important: We need parens around a sub-expression only in one side, why? --- When we write: eval(1 - 2 + 3) = ... = 1 - 2 + 3 we have two expressions, but one stands for an *input syntax*, and one stands for a real mathematical expression. In a case of a computer implementation, the syntax on the left is (as always) an AE syntax, and the real expression on the right is an expression in whatever language we use to implement our AE language. Like we said earlier, ambiguity is not a real problem until the actual parse tree matters. With `eval` it definitely matters, so we must not make it possible to derive any syntax in multiple ways or our evaluation will be non-deterministic. ------------------------------------------------------------------------ Quick exercise: We can define a meaning for ``s and then ``s in a similar way: ::= | eval(0) = 0 eval(1) = 1 eval(2) = 2 ... eval(9) = 9 eval() = eval( ) = 10*eval() + eval() Is this exactly what we want? --- Depends on what we actually want... * First, there's a bug in this code --- having a BNF derivation like ::= | is unambiguous, but makes it hard to parse a number. We get: eval(123) = 10*eval(1) + eval(23) = 10*1 + 10*eval(2) + eval(3) = 10*1 + 10*2 + 3 = 33 Changing the order of the last rule works much better: ::= | and then: eval( ) = 10*eval() + eval() * As a concrete example see how you would make it work with `107`, which demonstrates why compositionality is important. * Example for free stuff that looks trivial: if we were to define the meaning of numbers this way, would it always work? Think an average language that does not give you bignums, making the above rules fail when the numbers are too big. In Racket, we happen to be using an integer representation for the syntax of integers, and both are unlimited. But what if we wanted to write a Racket compiler in C or a C compiler in Racket? What about a C compiler in C, where the compiler runs on a 64 bit machine, and the result needs to run on a 32 bit machine? ------------------------------------------------------------------------ ## Side-note: Compositionality The example of ::= | being a language that is easier to write an evaluator for leads us to an important concept --- compositionality. This definition is easier to write an evaluator for, since the resulting language is compositional: the meaning of an expression --- for example `123` --- is composed out of the meaning of its two parts, which in this BNF are `12` and `3`. Specifically, the evaluation of ` ` is `10 *` the evaluation of the first, plus the evaluation of the second. In the ` ` case this is more difficult --- the meaning of such a number depends not only on the *meaning* of the two parts, but also on the `` *syntax*: eval( ) = eval() * 10^length() + eval() This this case this can be tolerable, since the meaning of the expression is still made out of its parts --- but imperative programming (when you use side effects) is much more problematic since it is not compositional (at least not in the obvious sense). This is compared to functional programming, where the meaning of an expression is a combination of the meanings of its subexpressions. For example, every sub-expression in a functional program has some known meaning, and these all make up the meaning of the expression that contains them --- but in an imperative program we can have a part of the code be `x++` --- and that doesn't have a meaning by itself, at least not one that contributes to the meaning of the whole program in a direct way. (Actually, we can have a well-defined meaning for such an expression: the meaning is going from a world where `x` is a container of some value N, to a world where the same container has a different value N+1. You can probably see now how this can make things more complicated. On an intuitive level --- if we look at a random part of a functional program we can tell its meaning, so building up the meaning of the whole code is easy, but in an imperative program, the meaning of a random part is pretty much useless.) ------------------------------------------------------------------------ # Implementing an Evaluator Now continue to implement the semantics of our syntax --- we express that through an `eval` function that evaluates an expression. We use a basic programming principle --- splitting the code into two layers, one for parsing the input, and one for doing the evaluation. Doing this avoids the mess we'd get into otherwise, for example: (define (eval sexpr) (match sexpr [(number: n) n] [(list '+ left right) (+ (eval left) (eval right))] [(list '- left right) (- (eval left) (eval right))] [else (error 'eval "bad syntax in ~s" sexpr)])) This is messy because it combines two very different things --- syntax and semantics --- into a single lump of code. For this particular kind of evaluator it looks simple enough, but this is only because it's simple enough that all we do is replace constructors by arithmetic operations. Later on things will get more complex, and bundling the evaluator with the parser will be more problematic. (Note: the fact that we can replace constructors with the run-time operators mean that we have a very simple, calculator-like language, and that we can, in fact, "compile" all programs down to a number.) If we split the code, we can easily include decisions like making {+ 1 {- 3 "a"}} syntactically invalid. (Which is not, BTW, what Racket does...) (Also, this is like the distinction between XML syntax and well-formed XML syntax.) An additional advantage is that by using two separate components, it is simple to replace each one, making it possible to change the input syntax, and the semantics independently --- we only need to keep the same interface data (the AST) and things will work fine. Our `parse` function converts an input syntax to an abstract syntax tree (AST). It is abstract exactly because it is independent of any actual concrete syntax that you type in, print out etc. ------------------------------------------------------------------------ # Implementing The AE Language Back to our `eval` --- this will be its (obvious) type: (: eval : AE -> Number) ;; consumes an AE and computes ;; the corresponding number which leads to some obvious test cases: (equal? 3 (eval (parse "3"))) (equal? 7 (eval (parse "{+ 3 4}"))) (equal? 6 (eval (parse "{+ {- 3 4} 7}"))) which from now on we will write using the new `test` form that the `#lang pl` language provides: (test (eval (parse "3")) => 3) (test (eval (parse "{+ 3 4}")) => 7) (test (eval (parse "{+ {- 3 4} 7}")) => 6) Note that we're testing *only* at the interface level --- only running whole functions. For example, you could think about a test like: (test (parse "{+ {- 3 4} 7}") => (Add (Sub (Num 3) (Num 4)) (Num 7))) but the details of parsing and of the constructor names are things that nobody outside of our evaluator cares about --- so we're not testing them. In fact, we shouldn't even mention `parse` in these tests, since it is not part of the public interface of our users; they only care about using it as a compiler-like black box. (This is sometimes called "integration tests".) We'll address this shortly. Like everything else, the structure of the recursive `eval` code follows the recursive structure of its input. In HtDP terms, our template is: (: eval : AE -> Number) (define (eval expr) (cases expr [(Num n) ... n ...] [(Add l r) ... (eval l) ... (eval r) ...] [(Sub l r) ... (eval l) ... (eval r) ...])) In this case, filling in the gaps is very simple (: eval : AE -> Number) (define (eval expr) (cases expr [(Num n) n] [(Add l r) (+ (eval l) (eval r))] [(Sub l r) (- (eval l) (eval r))])) We now further combine `eval` and `parse` into a single `run` function that evaluates an AE string. (: run : String -> Number) ;; evaluate an AE program contained in a string (define (run str) (eval (parse str))) This function becomes the single public entry point into our code, and the only thing that should be used in tests that verify our interface: (test (run "3") => 3) (test (run "{+ 3 4}") => 7) (test (run "{+ {- 3 4} 7}") => 6) The resulting *full* code is: ;;; ---<<>>----------------------------------------------------- #lang pl #| BNF for the AE language: ::= | { + } | { - } | { * } | { / } |# ;; AE abstract syntax trees (define-type AE [Num Number] [Add AE AE] [Sub AE AE] [Mul AE AE] [Div AE AE]) (: parse-sexpr : Sexpr -> AE) ;; parses s-expressions into AEs (define (parse-sexpr sexpr) (match sexpr [(number: n) (Num n)] [(list '+ lhs rhs) (Add (parse-sexpr lhs) (parse-sexpr rhs))] [(list '- lhs rhs) (Sub (parse-sexpr lhs) (parse-sexpr rhs))] [(list '* lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))] [(list '/ lhs rhs) (Div (parse-sexpr lhs) (parse-sexpr rhs))] [else (error 'parse-sexpr "bad syntax in ~s" sexpr)])) (: parse : String -> AE) ;; parses a string containing an AE expression to an AE AST (define (parse str) (parse-sexpr (string->sexpr str))) (: eval : AE -> Number) ;; consumes an AE and computes the corresponding number (define (eval expr) (cases expr [(Num n) n] [(Add l r) (+ (eval l) (eval r))] [(Sub l r) (- (eval l) (eval r))] [(Mul l r) (* (eval l) (eval r))] [(Div l r) (/ (eval l) (eval r))])) (: run : String -> Number) ;; evaluate an AE program contained in a string (define (run str) (eval (parse str))) ;; tests (test (run "3") => 3) (test (run "{+ 3 4}") => 7) (test (run "{+ {- 3 4} 7}") => 6) (Note that the tests are done with a `test` form, which we mentioned above.) For anyone who thinks that Racket is a bad choice, this is a good point to think how much code would be needed in some other language to do the same as above. ------------------------------------------------------------------------ # Intro to Typed Racket The plan: * Why Types? * Why Typed Racket? * What's Different about Typed Racket? * Some Examples of Typed Racket for Course Programs ### Types ############################################################## - Who has used a (statically) typed language? - Who has used a typed language that's not Java? Typed Racket will be both similar to and very different from anything you've seen before. ### Why types? ######################################################### - Types help structure programs. - Types provide enforced and mandatory documentation. - Types help catch errors. --> They *will* help you. A *lot*. ### Structuring programs ############################################### - Data definitions ;; An AE is one of: ; \ ;; (make-Num Number) ; > HtDP ;; (make-Add AE AE) ; / (define-type AE ; \ [Num number?] ; > Predicates =~= contracts (PLAI) [Add AE? AE?]) ; / (has names of defined types too) (define-type AE ; \ [Num Number] ; > Typed Racket (our PL) [Add AE AE]) ; / - Data-first The structure of your program is derived from the structure of your data. You have seen this in Fundamentals with the design recipe and with templates. In this class, we will see it extensively with type definitions and the (cases ...) form. Types make this pervasive --- we have to think about our data before our code. - A language for describing data Instead of having an informal language for describing types in contract lines, and a more formal description of predicates in a `define-type` form, we will have a single, unified language for both of these. Having such a language means that we get to be more precise and more expressive (since the typed language covers cases that you would otherwise dismiss with some hand waving, like "a function"). ### Why Typed Racket? ################################################## Racket is the language we all know, and it has the benefits that we discussed earlier. Mainly, it is an excellent language for experimenting with programming languages. - Typed Racket allows us to take our Racket programs and typecheck them, so we get the benefits of a statically typed language. - Types are an important programming language feature; Typed Racket will help us understand them. [Also: the development of Typed Racket is happening here in Northeastern, and will benefit from your feedback.] ### How is Typed Racket different from Racket ########################## - Typed Racket will reject your program if there are type errors! This means that it does that at compile-time, *before* any code gets to run. - Typed Racket files start like this: #lang typed/racket ;; Program goes here. but we will use a variant of the Typed Racket language, which has a few additional constructs: #lang pl ;; Program goes here. - Typed Racket requires you to write the contracts on your functions. Racket: ;; f : Number -> Number (define (f x) (* x (+ x 1))) Typed Racket: #lang pl (: f : Number -> Number) (define (f x) (* x (+ x 1))) [In the "real" Typed Racket the preferred style is with prefix arrows: #lang typed/racket (: f (-> Number Number)) (define (f x) : Number (* x (+ x 1))) and you can also have the type annotations appear inside the definition: #lang typed/racket (define (f [x : Number]) : Number (* x (+ x 1))) but we will not use these form.] - As we've seen, Typed Racket uses types, not predicates, in `define-type`. (define-type AE [Num Number] [Add AE AE]) versus (define-type AE [Num number?] [Add AE? AE?]) - There are other differences, but these will suffice for now. ### Examples ########################################################### (: digit-num : Number -> (U Number String)) (define (digit-num n) (cond [(<= n 9) 1] [(<= n 99) 2] [(<= n 999) 3] [(<= n 9999) 4] [else "a lot"])) (: fact : Number -> Number) (define (fact n) (if (zero? n) 1 (* n (fact (- n 1))))) (: helper : Number Number -> Number) (define (helper n acc) (if (zero? n) acc (helper (- n 1) (* acc n)))) (: fact : Number -> Number) (define (fact n) (helper n 1)) (: fact : Number -> Number) (define (fact n) (: helper : Number Number -> Number) (define (helper n acc) (if (zero? n) acc (helper (- n 1) (* acc n)))) (helper n 1)) (: every? : (All (A) (A -> Boolean) (Listof A) -> Boolean)) ;; Returns false if any element of lst fails the given pred, ;; true if all pass pred. (define (every? pred lst) (or (null? lst) (and (pred (first lst)) (every? pred (rest lst))))) (define-type AE [Num Number] [Add AE AE] [Sub AE AE]) ;; the only difference in the following definition is ;; using (: : ) instead of ";; : " (: parse-sexpr : Sexpr -> AE) ;; parses s-expressions into AEs (define (parse-sexpr sexpr) (match sexpr [(number: n) (Num n)] [(list '+ left right) (Add (parse-sexpr left) (parse-sexpr right))] [(list '- left right) (Sub (parse-sexpr left) (parse-sexpr right))] [else (error 'parse-sexpr "bad syntax in ~s" sexpr)])) ### More interesting examples ########################################## * Typed Racket is designed to be a language that is friendly to the kind of programs that people write in Racket. For example, it has unions: (: foo : (U String Number) -> Number) (define (foo x) (if (string? x) (string-length x) ;; at this point it knows that `x' is not a ;; string, therefore it must be a number (+ 1 x))) This is not common in statically typed languages, which are usually limited to only *disjoint* unions. For example, in OCaml you'd write this definition: type string_or_number = Str of string | Int of int ;; let foo x = match x with Str s -> String.length s | Int i -> i+1 ;; And use it with an explicit constructor: foo (Str "bar") ;; foo (Int 3) ;; * Note that in the Typed Racket case, the language keeps track of information that is gathered via predicates --- which is why it knows that one `x` is a String, and the other is a Number. * Typed Racket has a concept of subtypes --- which is also something that most statically typed languages lack. In fact, the fact that it has (arbitrary) unions means that it must have subtypes too, since a type is always a subtype of a union that contains this type. * Another result of this feature is that there is an `Any` type that is the union of all other types. Note that you can always use this type since everything is in it --- but it gives you the *least* information about a value. In other words, Typed Racket gives you a choice: *you* decide which type to use, one that is very restricted but has a lot of information about its values to a type that is very permissive but has almost no useful information. This is in contrast to other type system (HM systems) where there is always exactly one correct type. To demonstrate, consider the identity function: (define (id x) x) You could use a type of `(: id : Integer -> Integer)` which is very restricted, but you know that the function always returns an integer value. Or you can make it very permissive with a `(: id : Any -> Any)`, but then you know nothing about the result --- in fact, `(+ 1 (id 2))` will throw a type error. It *does* return `2`, as expected, but the type checker doesn't know the type of that `2`. If you wanted to use this type, you'd need to check that the result is a number, eg: (let ([x (id 123)]) (if (number? x) (+ x 10) 999)) This means that for this particular function there is no good *specific* type that we can choose --- but there are *polymorphic* types. These types allow propagating their input type(s) to their output type. In this case, it's a simple "my output type is the same as my input type": (: id : (All (A) A -> A)) This makes the output preserve the same level of information that you had on its input. * Another interesting thing to look at is the type of `error`: it's a function that returns a type of `Nothing` --- a type that is the same as an *empty* union: `(U)`. It's a type that has no values in it --- it fits `error` because it *is* a function that doesn't return any value, in fact, it doesn't return at all. In addition, it means that an `error` expression can be used anywhere you want because it is a subtype of anything at all. * An `else` clause in a `cond` expression is almost always needed, for example: (: digit-num : Number -> (U Number String)) (define (digit-num n) (cond [(<= n 9) 1] [(<= n 99) 2] [(<= n 999) 3] [(<= n 9999) 4] [(> n 9999) "a lot"])) (and if you think that the type checker should know what this is doing, then how about (> (* n 10) (/ (* (- 10000 1) 20) 2)) or (>= n 10000) for the last test?) * In some rare cases you will run into one limitation of Typed Racket: it is difficult (that is: a generic solution is not known at the moment) to do the right inference when polymorphic functions are passed around to higher-order functions. For example: (: call : (All (A B) (A -> B) A -> B)) (define (call f x) (f x)) (call rest (list 4)) In such cases, we can use `inst` to *instantiate* a function with a polymorphic type to a given type --- in this case, we can use it to make it treat `rest` as a function that is specific for numeric lists: (call (inst rest Number) (list 4)) In other rare cases, Typed Racket will infer a type that is not suitable for us --- there is another form, `ann`, that allows us to specify a certain type. Using this in the `call` example is more verbose: (call (ann rest : ((Listof Number) -> (Listof Number))) (list 4)) However, these are going to be rare and will be mentioned explicitly whenever they're needed. ------------------------------------------------------------------------ # Bindings & Substitution We now get to an important concept: substitution. Even in our simple language, we encounter repeated expressions. For example, if we want to compute the square of some expression: {* {+ 4 2} {+ 4 2}} Why would we want to get rid of the repeated sub-expression? * It introduces a redundant computation. In this example, we want to avoid computing the same sub-expression a second time. * It makes the computation more complicated than it could be without the repetition. Compare the above with: with x = {+ 4 2}, {* x x} * This is related to a basic fact in programming that we have already discussed: duplicating information is always a bad thing. Among other bad consequences, it can even lead to bugs that could not happen if we wouldn't duplicate code. A toy example is "fixing" one of the numbers in one expression and forgetting to fix the corresponding one: {* {+ 4 2} {+ 4 1}} Real world examples involve much more code, which make such bugs very difficult to find, but they still follow the same principle. * This gives us more expressive power --- we don't just say that we want to multiply two expressions that both happen to be `{+ 4 2}`, we say that we multiply the `{+ 4 2}` expression by *itself*. It allows us to express identity of two values as well as using two values that happen to be the same. So, the normal way to avoid redundancy is to introduce an identifier. Even when we speak, we might say: "let x be 4 plus 2, multiply x by x". (These are often called "variables", but we will try to avoid this name: what if the identifier does not change (vary)?) To get this, we introduce a new form into our language: {with {x {+ 4 2}} {* x x}} We expect to be able to reduce this to: {* 6 6} by substituting 6 for `x` in the body sub-expression of `with`. A little more complicated example: {with {x {+ 4 2}} {with {y {* x x}} {+ y y}}} [add] = {with {x 6} {with {y {* x x}} {+ y y}}} [subst]= {with {y {* 6 6}} {+ y y}} [mul] = {with {y 36} {+ y y}} [subst]= {+ 36 36} [add] = 72