2012-01-20 - Semantics (= Evaluation) - Implementing an Evaluator - Implementing The AE Language - Introduction to Typed Racket - Bindings & Substitution - Adding Bindings to AE: The WAE Language ======================================================================== >>> Semantics (= Evaluation) [[[ PLAI Chapter 2 ]]] 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: moves syntax into a 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 where more formality is required (for example, in RFC specifications), each literal part of the BNF is usually marked with double quotes, 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 get either 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 case, 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() * 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 comment on compositionality The example of ::= | being a language that is easier to 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 face, "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. 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) ;; to convert 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. ======================================================================== >>> Introduction 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?]) ; / (actually, the PLAI form has names 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 2500 and 2510 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 "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 Typed Racket you can also have the type annotations appear inside the definition: #lang pl (define: (f [x : Number]) : Number (* x (+ x 1))) but we will not use this 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) ;; to convert 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. * 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 `ann' form 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 ======================================================================== >>> Adding Bindings to AE: The WAE Language [[[ PLAI Chapter 3 ]]] To add this to our language, we start with the BNF. We now call our language `WAE' (With+AE): ::= | { + } | { - } | { * } | { / } | { with { } } | Note that we had to introduce two new rules: one for introducing an identifier, and one for using it. This is common in many language specifications, for example `define-type' introduces a new type, and it comes with `cases' that allows us to destruct its instances. For we need to use some form of identifiers, the natural choice in Racket is to use symbols. We can therefore write the corresponding type definition: (define-type WAE [Num Number] [Add WAE WAE] [Sub WAE WAE] [Mul WAE WAE] [Div WAE WAE] [Id Symbol] [With Symbol WAE WAE]) The parser is easily extended to produce these syntax objects: (: parse-sexpr : Sexpr -> WAE) ;; to convert s-expressions into WAEs (define (parse-sexpr sexpr) (match sexpr [(number: n) (Num n)] [(symbol: name) (Id name)] [(list 'with (list (symbol: name) named) body) (With name (parse-sexpr named) (parse-sexpr body))] [(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)])) But note that this parser is inconvenient -- if any of these expressions: {* 1 2 3} {foo 5 6} {with x 5 {* x 8}} {with {5 x} {* x 8}} would result in a "bad syntax" error, which is not very helpful. To make things better, we can add another case for `with' expressions that are malformed, and give a more specific message in that case: (: parse-sexpr : Sexpr -> WAE) ;; to convert s-expressions into WAEs (define (parse-sexpr sexpr) (match sexpr [(number: n) (Num n)] [(symbol: name) (Id name)] [(list 'with (list (symbol: name) named) body) (With name (parse-sexpr named) (parse-sexpr body))] [(cons 'with more) (error 'parse-sexpr "bad `with' syntax in ~s" sexpr)] [(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)])) and finally, to group all of the parsing code that deals with `with' expressions (both valid and invalid ones), we can use a single case for both of them: (: parse-sexpr : Sexpr -> WAE) ;; to convert s-expressions into WAEs (define (parse-sexpr sexpr) (match sexpr [(number: n) (Num n)] [(symbol: name) (Id name)] [(cons 'with more) ;; go in here for all sexpr that begin with a 'with (match sexpr [(list 'with (list (symbol: name) named) body) (With name (parse-sexpr named) (parse-sexpr body))] [else (error 'parse-sexpr "bad `with' syntax in ~s" sexpr)])] [(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)])) And now we're done with the syntactic part of the `with' extension. (Quick question -- why would we indent `With' like a normal function in code like this: (With 'x (Num 2) (Add (Id 'x) (Num 4))) instead of an indentation that looks like a `let': (With 'x (Num 2) (Add (Id 'x) (Num 4))) ?) ========================================================================