CSci 4223 Principles of Programming Languages
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1 Review CSci 4223 Prciples of Programmg Languages Lecture 8 Features learned: functions, tuples, lists, let expressions, options, records, datatypes, case expressions, type synonyms, pattern matchg, exceptions Today: Tail recursion, contued Type variables Equality types Type annotations Prof. Clarkson Sprg Recursion Should now be comfortable with recursion: No harder than usg a loop (whatever that is J ) For recursive datatypes, recursion is more elegant than loops: When processg a tree (e.g., evaluate an arithmetic expression) When appg lists TAIL RECURSION Now: How to reason about efficiency of recursion e importance of tail recursion Usg an accumulator to achieve tail recursion Functional programmg idiom (no new syntax/semantics) 3 4 Call-stacks While a program runs, there is a call stack of function calls that have started but not yet returned Callg a function f pushes an stance of f on the stack When a call to f to fishes, it is popped from the stack ese stack-frames store formation like the value of local variables and what is left to do the function Example fun fact n = if n=0 then 1 else n*fact(n-1) val x = fact 3 fact 3 fact 3: 3*_ fact 3: 3*_ fact 3: 3*_ fact 2 fact 2: 2*_ fact 2: 2*_ fact 1 fact 1: 1*_ fact 0 Because of recursion, multiple stack-frames may be calls to the same function fact 3: 3*_ fact 3: 3*_ fact 3: 3*_ fact 3: 3*2 fact 2: 2*_ fact 2: 2*_ fact 2: 2*1 fact 1: 1*_ fact 1: 1*1 5 fact 0: 1 6 1
2 Example Revised e call-stacks fun fact n = let fun aux(n,acc) = if n=0 then acc else aux(n-1,acc*n) aux(n,1) val x = fact 3 fact 3 aux(3,1) aux(2,3) aux(1,6) Still recursive, more complicated, but the result of recursive calls is the result for the caller (no remag multiplication) 7 aux(1,6):_ aux(0,6) aux(1,6):_ aux(0,6):6 aux(1,6):6 aux(2,3):6 Etc. 8 An optimization It is unnecessary to keep around a stack-frame just so it can get a callee s result and return it without any further evaluation ML recognizes these tail calls the compiler and treats them differently: Pop the caller before the call, allowg callee to reuse the same stack space Implementation turns out to be as efficient as a loop Not just ML; any good implementation of functional language does tail-call optimization What really happens fact 3 fun fact n = let fun aux(n,acc) = if n=0 then acc else aux(n-1,acc*n) aux(n,1) val x = fact 3 aux(3,1) aux(2,3) aux(1,6) aux(0,6) 9 10 Moral Tail-recursive calls can be much more efficient But there s a tradeoff: makes code a little harder to read So best done when efficiency is important Another example fun sum xs = x::xs => x + sum xs Methodology to guide transformation of recursive function: Create a helper function that takes an accumulator Old base case becomes itial accumulator New base case becomes fal accumulator fun sum xs = let fun aux(xs,acc) = [] => acc x::xs => aux(xs,x+acc) aux(xs,0)
3 And another fun rev xs = [] => [] x::xs => (rev xs [x] fun rev xs = let fun aux(xs,acc) = [] => acc x::xs => aux(xs,x::acc) aux(xs,[]) List app can be REALLY efficient fun rev xs = [] => [] x::xs => (rev xs [x] For fact and sum, tail-recursion takes less space but still lear time But, non-tail recursive rev is quadratic because each recursive call uses which must traverse the first list (length-1) is almost length*length/2 Moral: especially with recursion Cons is constant-time (and fast), so the accumulator version rocks Is tail-recursive always more efficient? Precise defition of tail call NO. Sometimes recursive functions can t be evaluated a constant amount of space E.g., functions over trees In these cases, the simple recursive approach is best (You could get one recursive call to be a tail call, but rarely worth the complication) 15 If the result of f x is the immediate result for the enclosg function body, then f x is a tail call Can defe this notion more precisely A tail call is a function call tail position If an expression is not tail position, then no subexpressions are In fun f p = e, the body e is tail position If if e1 then e2 else e3 is tail position, then e2 and e3 are tail position (but e1 is not). (Similar for case-expressions) If let b1 bn e is tail position, then e is tail position (but no bdg expressions are) Function-call arguments are not tail position 16 What the difference? Length of a list: fun len (xs: t list) = _::xs => 1 + len xs TYPE VARIABLES & EQUALITY TYPES 17 fun len (xs: strg list) = _::xs => 1 + len xs No algorithmic difference! Would be silly to have to write function for every kd of list type 18 3
4 Type variables to the rescue Another polymorphic function Use type variable to stand place of an arbitrary type: fun len (xs: 'a list) = _::xs => 1 + len xs What is type of list app? fun app (xs,ys) = [] => ys x::xs => x :: app(xs,ys) Just like we use variables to stand place of arbitrary values Creates a polymorphic function ( poly =many, morph =form) Closely related to generics Java Might look like, but is rather less related to, templates C++ You might expect strg list * strg list -> strg list But REPL reports 'a list * 'a list -> 'a list is is okay [such as on your homework]: why? More general e generalization rule for types e type 'a list * 'a list -> 'a list is more general than the type strg list * strg list -> strg list Polymorphic type can be used as any less general type, such as t list * t list -> t list Easy rule you (and the type-checker) can apply without thkg: A type t1 is more general than the type t2 if you can take t1, replace its type variables consistently, and get t2 Example: Replace each 'a with t * t Example: Replace each 'a with bool and each 'b with bool Example: Replace each 'a with bool and each 'b with t But it is not more general than the type t list * strg list -> t list Must replace type variables consistently Equivalence rules for types Equality types Can combe the more general rule with rules for equivalence Use of type synonyms does not matter Order of field names does not matter Example, given type foo = t * t the type {quux : 'b, bar : t * 'a, baz : 'b} is more general than {quux : strg, bar : foo, baz : strg} which is equivalent to {bar : t*t, baz : strg, quux : strg} You might also see type variables with a second quote Example: ''a list * ''a -> bool ese are equality types that arise from usg the = operator e = operator works on lots of types: t, strg, tuples contag all equality types, But not all types: function types, real, e rules for more general are exactly the same except you have to replace an equality-type variable with a type that can be used with = A strange feature of ML because = is special
5 Example (* a * a -> strg *) fun same_thg(x, y) = if x=y then "yes" else "no" (* t -> strg *) fun is_three x = if x=3 then "yes" else "no" (You can ignore the warng about callg polyequal ) TYPE INFERENCE We stopped writg types How does compiler fer? Started off semester writg: fun pow (x : t, y : t) = if y=0 then 1 else x * pow(x,y-1) But recently we ve been leavg off argument types: fun pow (x, y ) = if y=0 then 1 else x * pow(x,y-1) Here s the idea: fun pow (x, y ) = if y=0 then 1 else x * pow(x,y-1) y must be t, sce compared with = to 0 pow must return t, sce then branch returns 1 x must be t, sce multiplied with an t (defe more carefully later lecture) Either way, compiler has been ferrg that return type is t Selectors complicate type ference Pattern matchg simplifies type ference What is the type of argument to this function? What about this function? fun sum_triple t = case t of (x,y,z) => x+y+z fun sum_triple t = #1 t + #2 t + #3 t Error: unresolved flex record (need to know the names of ALL the fields this context). Which is why we had to write fun sum_triple (t:t*t*t) = #1 t + #2 t + #3 t 29 Pattern gives enough hts to compiler to fer type: fun sum_triple t = case triple of (x,y,z) => x+y+z at s why we don t need to write down types anymore And also why we started off writg down types You can add type annotations anywhere you like Instead of e write e:t e :t is an annotation or constrat used by type ference And that s all that the argument types we wrote down ever were 30 5
6 Functions are values FIRST-CLASS FUNCTIONS Can use them anywhere we use values Arguments, results, parts of tuples, bound to variables, carried by datatype constructors or exceptions, First-class citizens of language, afforded all the rights of any other values Functions can take functions as arguments Functions can return functions as results so functions are higher-order is is not a new language feature; just a consequence of choice we made long ago when we said functions are values Example Can reuse n_times rather than defg many similar functions Computes f(f( f(x))) where number of calls is n fun n_times (f,n,x) = if n=0 then x else f (n_times(f,n-1,x)) fun double x = x + x fun crement x = x + 1 val x1 = n_times(double,4,7) val x2 = n_times(crement,4,7) val x3 = n_times(tl,2,[4,8,12,16,20]) fun double_n_times (n,x) = n_times(double,n,x) fun nth_tail (n,x) = n_times(tl,n,x) 33 6
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