pure is the Pure interpreter. The interpreter has an LLVM backend which JIT-compiles Pure programs to machine code, hence programs run blazingly fast and interfacing to C modules is easy, while the interpreter still provides a convenient, fully interactive environment for running Pure scripts and evaluating expressions.
Pure programs (a.k.a. scripts) are just ordinary text files containing Pure code. A bunch of syntax highlighting files and programming modes for various popular text editors are included in the Pure sources. There’s no difference between the Pure programming language and the input language accepted by the interpreter, except that the interpreter also understands some special commands when running in interactive mode; see the INTERACTIVE USAGE section for details.
If any source scripts are specified on the command line, they are loaded and executed, after which the interpreter exits. Otherwise the interpreter enters the interactive read-eval-print loop. You can also use the -i option to enter the interactive loop (continue reading from stdin) even after processing some source scripts. To exit the interpreter, just type the quit command or the end-of-file character (^D on Unix) at the beginning of the command line.
Options and source files are processed in the order in which they are given on the command line. Processing of options and source files ends when either the -- or the -x option is encountered. The -x option must be followed by the name of a script to be executed, which becomes the ‘‘main script’’ of the application. In either case, any remaining parameters are passed to the executing script by means of the global argc and argv variables, denoting the number of arguments and the list of the actual parameter strings, respectively. In the case of -x this also includes the script name as argv!0. The -x option is useful, in particular, to turn Pure scripts into executable programs by including a ‘‘shebang’’ like
#!/usr/local/bin/pure -x
as the first line in your main script. (This trick only works with Unix shells, though.)
On startup, the interpreter also defines the version variable, which is set to the version string of the Pure interpreter, and the sysinfo variable, which provides a string identifying the host system. These are useful if parts of your script depend on the particular version of the interpreter and the system it runs on.
If available, the prelude script prelude.pure is loaded by the interpreter prior to any other other definitions, unless the -n or --noprelude option is specified. The prelude is searched for in the directory specified with the PURELIB environment variable. If the PURELIB variable is not set, a system-specific default is used. Other source scripts specified on the command line are searched for in the current directory if a relative pathname is given. In addition, the executed program may load other scripts and libraries via a using declaration in the source, which are searched for in a number of locations, including the directories named with the -I and -L options; see the sections DECLARATIONS and C INTERFACE below for details.
If the interpreter runs in interactive mode, it may source a few additional interactive startup files immediately before entering the interactive loop, unless the --norc option is specified. First .purerc in the user’s home directory is read, then .purerc in the current working directory. These are ordinary Pure scripts which can be used to provide additional definitions for interactive usage. Finally, a .pure file in the current directory (containing a dump from a previous interactive session) is loaded if it is present. See the INTERACTIVE USAGE section for details.
When the interpreter is in interactive mode and reads from a tty, unless the --noediting option is specified, commands are read using readline(3) (providing completion for all commands listed under INTERACTIVE USAGE, as well as for symbols defined in the running program). When exiting the interpreter, the command history is stored in ~/.pure_history, from where it is restored the next time you run the interpreter.
The -v option is most useful for debugging the interpreter, or if you are interested in the code your program gets compiled to. The level argument is optional; it defaults to 1. Six different levels are implemented at this time (two more bits are reserved for future extensions). For most purposes, only the first two levels will be useful for the average Pure programmer; the remaining levels are most likely to be used by the Pure interpreter developers.
These values can be or’ed together, and, for convenience, can be specified in either decimal or hexadecimal. Thus 0xff always gives you full debugging output (which isn’t most likely be used by anyone but the Pure developers).
Note that the -v option is only applied after the prelude has been loaded. If you want to debug the prelude, use the -n option and specify the prelude.pure file explicitly on the command line. Verbose output is also suppressed for modules imported through a using clause. As a remedy, you can use the interactive show command (see the INTERACTIVE USAGE section below) to list definitions along with additional debugging information.
Pure is a fairly simple yet powerful language. Programs are basically collections of rewriting rules and expressions to be evaluated. For convenience, it is also possible to define global variables and constants, and for advanced uses Pure offers macro functions as a kind of preprocessing facility. These are all described below and in the following sections.
Here’s a first example which demonstrates how to define a simple recursive function in Pure, entered interactively in the interpreter (note that the ‘‘>’’ symbol at the beginning of each input line is the interpreter’s default command prompt):
> // my first Pure example > fact 0 = 1; > fact n::int = n*fact (n-1) if n>0; > let x = fact 10; x; 3628800
The language is free-format (whitespace is insignificant). As indicated, definitions and expressions at the toplevel have to be terminated with a semicolon. Comments have the same syntax as in C++ (using // for line-oriented and /* ... */ for multiline comments; the latter may not be nested). Lines beginning with #! are treated as comments, too; as already discussed above, on Unix-like systems this allows you to add a ‘‘shebang’’ to your main script in order to turn it into an executable program.
There are a few reserved keywords which cannot be used as identifiers: case const def else end extern if infix infixl infixr let nullary of otherwise postfix prefix private then using when with.
Pure is a terse language. You won’t see many declarations, and often your programs will read more like a collection of algebraic specifications (which in fact they are, only that the specifications are executable). This is intended and keeps the code tidy and clean.
On the surface, Pure is quite similar to other modern functional languages like Haskell and ML. But under the hood it is a much more dynamic language, more akin to Lisp. In particular, Pure is dynamically typed, so functions can be fully polymorphic and you can add to the definition of an existing function at any time:
> fact 1.0 = 1.0; > fact n::double = n*fact (n-1) if n>1; > fact 10.0; 3628800.0 > fact 10; 3628800
Like in Haskell and ML, functions and variables are often defined by pattern-matching, i.e., the left-hand side of a definition is compared to the target expression, binding the variables in the pattern to their actual values accordingly:
> foo (bar x) = x-1; > foo (bar 99); 98
In fact, due to its term rewriting semantics, Pure goes beyond most other functional languages in that it can do symbolic evaluations just as well as ‘‘normal’’ computations:
> square x = x*x; > square 4; 16 > square (a+b); (a+b)*(a+b)
Leaving aside the built-in support for some common data structures such as numbers and strings, all the Pure interpreter really does is evaluating expressions in a symbolic fashion, rewriting expressions using the equations supplied by the programmer, until no more equations are applicable. The result of this process is called a normal form which represents the ‘‘value’’ of the original expression. Keeping with the tradition of term rewriting, there’s no distinction between ‘‘defined’’ and ‘‘constructor’’ function symbols in Pure; any function symbol (or operator) also acts as a constructor if it happens to occur in a normal form term:
> (x+y)*z = x*z+y*z; x*(y+z) = x*y+x*z; > x*(y*z) = (x*y)*z; x+(y+z) = (x+y)+z; > square (a+b); a*a+a*b+b*a+b*b
Expressions are generally evaluated from left to right, innermost expressions first, i.e., using call by value semantics. Pure also has a few built-in special forms (most notably, conditional expressions, the short-circuit logical connectives && and ||, the sequencing operator $$, and the lazy evaluation operator &) which take some or all of their arguments unevaluated, using call by name. (User-defined special forms can be created with macros. More about that later.)
The Pure language provides built-in support for machine integers (32 bit), bigints (implemented using GMP), floating point values (double precision IEEE), character strings (UTF-8 encoded) and generic C pointers (these don’t have a syntactic representation in Pure, though, so they need to be created with external C functions). Truth values are encoded as machine integers (as you might expect, zero denotes false and any non-zero value true). Pure also provides some built-in support for lists and matrices, although most of the corresponding operations are actually defined in the prelude.
Second, character escapes in Pure strings have a more flexible syntax borrowed from the author’s Q language, which provides notations to specify any Unicode character. In particular, the notation \n, where n is an integer literal written in decimal (no prefix), hexadecimal (‘0x’ prefix) or octal (‘0’ prefix) notation, denotes the Unicode character (code point) #n. Since these escapes may consist of a varying number of digits, parentheses may be used for disambiguation purposes; thus, e.g. "\(123)4" denotes character #123 followed by the character ‘4’. The usual C-like escapes for special non-printable characters such as \n are also supported. Moreover, you can use symbolic character escapes of the form \&name;, where name is any of the XML single character entity names specified in the ‘‘XML Entity definitions for Characters’’, see http://www.w3.org/TR/xml-entity-names/. Thus, e.g., "\©" denotes the copyright character (code point 0x000A9).
Operator and constant symbols must always be declared before they can be used, using corresponding prefix, postfix, infix and nullary declarations, which are discussed in section DECLARATIONS.
Note that enclosing an operator in parentheses, such as (+) or (not), turns it into an ordinary function symbol. Symbols declared as nullary denote special constant symbols which simply stand for themselves. Technically, these are just ordinary identifiers; however, the nullary attribute tells the compiler that when such an identifier occurs on the left-hand side of an equation, it is to be interpreted as a constant rather than a variable (see above).
Pure’s tuples are a bit unusual: They are constructed by just ‘‘pairing’’ things using the ‘,’ operator, for which the empty tuple acts as a neutral element (i.e., (),x is just x, as is x,()). Pairs always associate to the right, meaning that x,y,z == x,(y,z) == (x,y),z, where x,(y,z) is the normalized representation. This implies that tuples are always flat, i.e., there are no nested tuples (tuples of tuples); if you need such constructs then you should use lists instead. Also note that parentheses are generally only used to group expressions and are not part of the tuple syntax in Pure. There’s one exception to this rule, however, namely that in order to include a tuple in a bracketed list you have to put it inside parentheses. E.g., [(1,2),3,(4,5)] is a three element list consisting of the tuple 1,2, the integer 3, and another tuple 4,5. Likewise, [(1,2,3)] is list with a single element, the tuple 1,2,3.
If the interpreter was built with support for the GNU Scientific Library (GSL) then both numeric and symbolic matrices are available. The former are thin wrappers around GSL’s homogeneous arrays of double, complex double or (machine) int matrices, while the latter can contain any mixture of Pure expressions. Pure will pick the appropriate type for the data at hand. If a matrix contains values of different types, or Pure values which cannot be stored in a numeric matrix, then a symbolic matrix is created instead (this also includes the case of bigints, which are considered as symbolic values as far as matrix construction is concerned). If the interpreter was built without GSL support then symbolic matrices are the only kind of matrices supported by the interpreter.
More information about matrices and corresponding examples can be found in the EXAMPLES section below.
Matrix comprehensions work pretty much like list comprehensions, but produce matrices instead of lists. Generator clauses in matrix comprehensions alternate between row and column generation so that most common mathematical abbreviations carry over quite easily. Examples of both kinds of comprehensions can be found in the EXAMPLES section below.
The common operator symbols like +, -, *, / etc. are all declared at the beginning of the prelude, see the prelude.pure script for a list of these. Arithmetic, relational and logical operators usually follow C conventions. However, out of necessity some of Pure’s operator symbols deviate from C. Most notably, the ‘!’ symbol is Pure’s indexing operator, hence logical negation is denoted not instead (named for Haskell compatibility, but Pure’s not is a real prefix operator instead of an ordinary function symbol). Moreover, the bitwise operators are named and and or instead of ‘&’ and ‘|’, which are used for other purposes in Pure.
The sequencing operator $$ evaluates its left operand, immediately throws the result away and then goes on to evaluate the right operand which gives the result of the entire expression. This operator is useful to write imperative-style code such as the following prompt/input interaction:
> using system; > puts "Enter a number:" $$ scanf "%g"; Enter a number: 21 21.0
The & operator does lazy evaluation. This is the only postfix operator defined in the standard prelude, written as ‘x&’, where x is an arbitrary Pure expression. The & operator binds stronger than any other operation except function application. It turns its operand into a kind of parameterless anonymous closure, deferring its evaluation. These kinds of objects are also commonly known as thunks or futures. When the value of a future is actually needed (during pattern-matching, or when the value becomes an argument of a C call), it is evaluated automagically and gets memoized, i.e., the computed result replaces the thunk so that it only has to be computed once. Futures are useful to implement all kinds of lazy data structures in Pure, in particular: lazy lists a.k.a. streams. A stream is simply a list with a thunked tail, which allows it to be infinite. The Pure prelude defines many functions for creating and manipulating these kinds of objects; further details and examples can be found in the EXAMPLES section below.
> foo x = bar with bar y = x+y end; > let f = foo 99; f; #<closure bar> > f 10, f 20; 109,119
This works the same no matter what other bindings of ‘x’ may be in effect when the closure is invoked:
> let x = 77; f 10, f 20 when x = 88 end; 109,119
Global bindings of variable and function symbols work a bit differently, though. Like many languages which are to be used interactively, Pure binds global symbols dynamically, so that the bindings can be changed easily at any time during an interactive session. This is mainly a convenience for interactive usage, but works the same no matter whether the source code is entered interactively or being read from a script, in order to ensure consistent behaviour between interactive and batch mode operation.
So, for instance, you can easily bind a global variable to a new value by just entering a corresponding let command:
> foo x = c*x; > foo 99; c*99 > let c = 2; foo 99; 198 > let c = 3; foo 99; 297
This works pretty much like global variables in imperative languages, but note that in Pure the value of a global variable can only be changed with a let command at the toplevel. Thus referential transparency is unimpaired; while the value of a global variable may change between different toplevel expressions, it will always take the same value in a single evaluation.
Similarly, you can also add new equations to an existing function at any time:
> fact 0 = 1; > fact n::int = n*fact (n-1) if n>0; > fact 10; 3628800 > fact 10.0; fact 10.0 > fact 1.0 = 1.0; > fact n::double = n*fact (n-1) if n>1; > fact 10.0; 3628800.0 > fact 10; 3628800
(In interactive mode, it is even possible to completely erase a definition, see section INTERACTIVE USAGE for details.)
So, while the meaning of a local symbol never changes once its definition has been processed, toplevel definitions may well evolve while the program is being processed, and the interpreter will always use the latest definitions at a given point in the source when an expression is evaluated. This means that, even in a script file, you have to define all symbols needed in an evaluation before entering the expression to be evaluated.
In any case, the left-hand side pattern (which, as already mentioned, is always a simple expression) must not contain repeated variables (i.e., rules must be ‘‘left-linear’’), except for the anonymous variable ‘_’ which matches an arbitrary value without binding a variable symbol.
A left-hand side variable (including the anonymous variable) may be followed by one of the special type tags ::int, ::bigint, ::double, ::string, ::matrix, ::pointer, to indicate that it can only match a constant value of the corresponding built-in type. (This is useful if you want to write rules matching any object of one of these types; note that there is no way to write out all ‘‘constructors’’ for the built-in types, as there are infinitely many.)
Pure also supports Haskell-style ‘‘as’’ patterns of the form variable@pattern which binds the given variable to the expression matched by the subpattern pattern (in addition to the variables bound by pattern itself). This is convenient if the value matched by the subpattern is to be used on the right-hand side of an equation. Syntactically, ‘‘as’’ patterns are primary expressions; if the subpattern is not a primary expression, it must be parenthesized. For instance, the following function duplicates the head element of a list:
foo xs@(x:_) = x:xs;
The left-hand side of a rule can be omitted if it is the same as for the previous rule. This provides a convenient means to write out a collection of equations for the same left-hand side which discriminates over different conditions:
lhs = rhs if guard;
= rhs if guard;
...
= rhs otherwise;
For instance:
fact n = n*fact (n-1) if n>0;
= 1 otherwise;
Pure also allows a collection of rules with different left-hand sides but the same right-hand side(s) to be abbreviated as follows:
lhs |
...
lhs = rhs;
This is useful if you need different specializations of the same rule which use different type tags on the left-hand side variables. For instance:
fact n::int |
fact n::double |
fact n = n*fact(n-1) if n>0;
= 1 otherwise;
In fact, the left-hand sides don’t have to be related at all, so that you can also write something like:
foo x | bar y = x*y;
However, this is most useful when using an ‘‘as’’ pattern to bind a common variable to a parameter value after checking that it matches one of several possible argument patterns (which is slightly more efficient than using an equivalent type-checking guard). E.g., the following definition binds the xs variable to the parameter of foo, if it is either the empty list or a list starting with an integer:
foo xs@[] | foo xs@(_::int:_) = ... xs ...;
The same construct also works in case expressions, which is convenient if different cases should be mapped to the same value, e.g.:
case ans of "y" | "Y" = 1; _ = 0; end;
The factorial:
fact n = n*fact (n-1) if n>0;
= 1 otherwise;
let facts = map fact (1..10); facts;
The Fibonacci numbers:
fib n = a when a, b = fibs n end
with fibs n = 0, 1 if n<=0;
= case fibs (n-1) of
a, b = b, a+b;
end;
end;
let fibs = map fib (1..30); fibs;
It is worth noting here that in most cases Pure performs tail call optimization so that tail-recursive definitions like the following will be executed in constant stack space (see the CAVEATS AND NOTES section for more details on this).
// tail-recursive factorial using an "accumulating parameter" fact n = loop 1 n with loop p n = if n>0 then loop (p*n) (n-1) else p; end;
Here is an example showing how constants are defined and used. Constant definitions take pretty much the same form as variable definitions with let (see above), but work more like the definition of a parameterless function whose value is precomputed at compile time:
> extern double atan(double); > const pi = 4*atan 1.0; > pi; 3.14159265358979 > foo x = 2*pi*x; > show foo foo x = 2*3.14159265358979*x; > foo 1; 6.28318530717958
primes n = sieve (2..n) with sieve [] = []; sieve (p:qs) = p : sieve [q | q = qs; q mod p]; end;
For instance:
> primes 100; [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
If you dare, you can actually have a look at the catmap-lambda-if-then-else expression the comprehension expanded to:
> show primes primes n = sieve (2..n) with sieve [] = []; sieve (p:qs) = p:sieve (catmap (\q -> if q mod p then [q] else []) qs) end;
List comprehensions are also a useful device to organize backtracking searches. For instance, here’s an algorithm for the n queens problem, which returns the list of all placements of n queens on an n x n board (encoded as lists of n pairs (i,j) with i = 1..n), so that no two queens hold each other in check.
queens n = search n 1 [] with
search n i p = [reverse p] if i>n;
= cat [search n (i+1) ((i,j):p) | j = 1..n; safe (i,j) p];
safe (i,j) p = not any (check (i,j)) p;
check (i1,j1) (i2,j2)
= i1==i2 || j1==j2 || i1+j1==i2+j2 || i1-j1==i2-j2;
end;
> fibs = 0L : 1L : zipwith (+) fibs (tail fibs) &; > fibs; 0L:1L:#<thunk 0xb5f875e8>
Note the ‘&’ on the tail of the list. This turns ‘fibs’ into a stream, which is required to prevent ‘fibs’ from recursing into samadhi. Also note that we work with bigints in this example because the Fibonacci numbers grow quite rapidly, so with machine integers the values would soon start wrapping around to negative integers.
Streams like these can be worked with in pretty much the same way as with lists. Of course, care must be taken not to invoke ‘‘eager’’ operations such as ‘#’ (which computes the size of a list) on infinite streams, to prevent infinite recursion. However, many list operations work with infinite streams just fine, and return the appropriate stream results. E.g., the ‘take’ function (which retrieves a given number of elements from the front of a list) works with streams just as well as with ‘‘eager’’ lists:
> take 10 fibs; 0L:1L:#<thunk 0xb5f87630>
Hmm, not much progress there, but that’s just how streams work (or rather they don’t, they’re lazy bums indeed!). Nevertheless, the stream computed with ‘take’ is in fact finite and we can readily convert it to an ordinary list, forcing its evaluation:
> list (take 10 fibs); [0L,1L,1L,2L,3L,5L,8L,13L,21L,34L]
(Conversely, you can also turn a list into a stream value with the ‘stream’ function. This is done less frequently, but becomes useful if you want to generate a stream from a finite list in a list comprehension.)
For interactive usage it’s often convenient to define an eager variation of ‘take’ which combines ‘take’ and ‘list’. Let’s do this now, so that we can use this operation in the following examples.
> takel n xs = list (take n xs); > takel 10 fibs; [0L,1L,1L,2L,3L,5L,8L,13L,21L,34L]
Well, this naive definition of the Fibonacci stream works, but it’s awfully slow. In fact, it takes exponential running time to determine the nth member of the sequence, because of the two recursive calls to ‘fibs’ on the right-hand side. This defect soon becomes rather annoying if we access larger members of the sequence. Just for fun, let’s measure some evaluation times with the interactive stats command:
> stats > fibs!25; fibs!26; fibs!27; 75025L 2.29s 121393L 3.75s 196418L 6.12s > stats off
It’s quite apparent that the ratios between successive running times are about the golden ratio (which is of course no accident!). So, assuming a fast computer which can produce the head element of a stream in just a nanosecond, a conservative estimate of the time needed to compute just the 128th Fibonacci number would already exceed the current age of the universe by some 29.6%, if done this way. It goes without saying that this kind of algorithm won’t even pass muster in a freshman course.
So let’s get back to the drawing board. One nice trick of the trade is to have the value of the Fibonacci stream refer to itself in its definition, rather than just the function generating it. For that we need a kind of ‘‘recursive variable definition’’ which Pure doesn’t have. (Haskellers should note here that the values of parameterless functions are never memoized in Pure, because that would wreak havoc on functions with side effects.) Fortunately, we can work around this quite easily by employing the so-called fixed point combinator, incidentally called fix in Pure and defined in the prelude as follows:
> show fix fix f = y y with y x = f (x x&) end;
(Functional programming buffs will quickly recognize this as an implementation of the normal order fixed point combinator, decorated with ‘&’ in the right place to make it work with eager evaluation. Aspiring novices may go read Wikipedia or a good book on the lambda calculus now.)
So here’s how we can define a ‘‘linear-time’’ version of the Fibonacci stream. (Note that we also define the stream as a variable now, to take full advantage of memoization.)
> clear fibs > let fibs = fix (\f -> 0L : 1L : zipwith (+) f (tail f) &); > fibs; 0L:1L:#<thunk 0xb58d8ae0> > takel 10 fibs; [0L,1L,1L,2L,3L,5L,8L,13L,21L,34L] > fibs; 0L:1L:1L:2L:3L:5L:8L:13L:21L:34L:#<thunk 0xb4ce5d30>
As you can see, the invocation of our ‘takel’ function forced the corresponding prefix of the ‘fibs’ stream to be computed. The result of the evaluation is memoized, so that this portion of the stream is now readily available in case we need to have another look at it later. By these means, possibly costly reevaluations are avoided, trading memory for execution speed.
Also, the trick we played with the fixed point combinator pays off, and in fact computing large Fibonacci numbers is a piece of cake now:
> stats > fibs!199; 173402521172797813159685037284371942044301L 0.1s > stats off
Let’s take a look at some of the other convenience operations for generating stream values. The prelude defines infinite arithmetic sequences, using inf or -inf to denote an upper (or lower) infinite bound for the sequence, e.g.:
> let u = 1..inf; let v = -1.0:-1.2..-inf; > takel 10 u; takel 10 v; [1,2,3,4,5,6,7,8,9,10] [-1.0,-1.2,-1.4,-1.6,-1.8,-2.0,-2.2,-2.4,-2.6,-2.8]
Other useful stream generator functions are ‘iterate’, ‘repeat’ and ‘cycle’, which have been adopted from Haskell. In fact, infinite arithmetic progressions are implemented in terms of ‘iterate’. The ‘repeat’ function just repeats its argument, and ‘cycle’ cycles through the elements of the given list:
> takel 10 (repeat 1); [1,1,1,1,1,1,1,1,1,1] > takel 10 (cycle [0,1]); [0,1,0,1,0,1,0,1,0,1]
Moreover, list comprehensions can draw values from streams and return the appropriate stream result:
> let rats = [m,n-m | n=2..inf; m=1..n-1; gcd m (n-m) == 1]; rats; (1,1):#<thunk 0xb5fd08b8> > takel 10 rats; [(1,1),(1,2),(2,1),(1,3),(3,1),(1,4),(2,3),(3,2),(4,1),(1,5)]
Finally, let’s rewrite our prime sieve so that it generates the infinite stream of all prime numbers:
all_primes = sieve (2..inf) with sieve (p:qs) = p : sieve [q | q = qs; q mod p] &; end;
Note that we can omit the empty list case of ‘sieve’ here, since the sieve now never becomes empty. Example:
> let P = all_primes; > takel 20 P; [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71] > P!299; 1987
You can also just print the entire stream. This will run forever, so hit Ctrl-C when you get bored.
> using system; > do (printf "%d\n") all_primes; 2 3 5 ...
(Make sure that you really use the ‘all_primes’ function instead of the P variable to print the stream. Otherwise the stream stored in P will grow with the number of elements printed until memory is exhausted. Calling ‘do’ on a fresh instance of the stream of primes allows ‘do’ to get rid of each ‘cons’ cell after having printed the corresponding stream element.)
But Pure’s facilities for matrix and list processing also make it easy to roll your own, if desired. First, the prelude provides matrix versions of the common list operations like map, fold, zip etc., which provide a way to implement common matrix operations. E.g., multiplying a matrix x with a scalar a amounts to mapping the function \x->a*x to x, which can be done as follows:
> a * x::matrix = map (\x->a*x) x if not matrixp a;
> 2*{1,2,3;4,5,6};
{2,4,6;8,10,12}
Likewise, matrix addition and other element-wise operations can be realized using zipwith, which combines corresponding elements of two matrices using a given binary function:
> x::matrix + y::matrix = zipwith (+) x y;
> {1,2,3;4,5,6}+{1,2,1;3,2,3};
{2,4,4;7,7,9}
Second, matrix comprehensions make it easy to express a variety of algorithms which would typically be implemented using ‘for’ loops in conventional programming languages. To illustrate the use of matrix comprehensions, here is how we can define an operation to create a square identity matrix of a given dimension:
> eye n = {i==j | i = 1..n; j = 1..n};
> eye 3;
{1,0,0;0,1,0;0,0,1}
Note that the i==j term is just a Pure idiom for the Kronecker symbol. Another point worth mentioning here is that the generator clauses of matrix comprehensions alternate between row and column generation automatically. (More precisely, the last generator, which varies most quickly, always yields a row, the next-to-last one a column of these row vectors, and so on.) This makes matrix comprehensions resemble customary mathematical notation very closely.
As a slightly more comprehensive example (no pun intended!), here is a definition of matrix multiplication in Pure. The building block here is the ‘‘dot’’ product of two vectors which can be defined as follows:
> dot x::matrix y::matrix = foldl (+) 0 [x!i*y!i | i=0..#x-1];
> dot {1,2,3} {1,0,1};
4
(For the sake of simplicity, this doesn’t do much error checking; if the two vectors aren’t the same size then you’ll get an ‘out_of_bounds’ exception with the definition above.)
The general matrix product now boils down to a simple matrix comprehension which just computes the dot product of all rows of x with all columns of y (the rows and cols functions are prelude operations found in matrices.pure):
> x::matrix * y::matrix = {dot u v | u = rows x; v = cols y};
> {0,1;1,0;1,1}*{1,2,3;4,5,6};
{4,5,6;1,2,3;5,7,9}
Well, that was easy. So let’s take a look at a more challenging example, Gaussian elimination, which can be used to solve systems of linear equations. The algorithm brings a matrix into ‘‘row echelon’’ form, a generalization of triangular matrices. The resulting system can then be solved quite easily using back substitution. Here is a Pure implementation of the algorithm:
gauss_elimination x::matrix = p,x
when n,m = dim x; p,_,x = foldl step (0..n-1,0,x) (0..m-1) end;
// One pivoting and elimination step in column j of the matrix:
step (p,i,x) j
= if max_x==0 then p,i,x
else
// updated row permutation and index:
transp i max_i p, i+1,
{// the top rows of the matrix remain unchanged:
x!!(0..i-1,0..m-1);
// the pivot row, divided by the pivot element:
{x!(i,l)/x!(i,j) | l=0..m-1};
// subtract suitable multiples of the pivot row:
{x!(k,l)-x!(k,j)*x!(i,l)/x!(i,j) | k=i+1..n-1; l=0..m-1}}
when
n,m = dim x; max_i, max_x = pivot i (col x j);
x = if max_x>0 then swap x i max_i else x;
end with
pivot i x = foldl max (0,0) [j,abs (x!j)|j=i..#x-1];
max (i,x) (j,y) = if x<y then j,y else i,x;
end;
The real meat is in the pivoting and elimination step (’step’ function) which is iterated over all columns of the input matrix. In each step, x is the current matrix, i the current row index, j the current column index, and p keeps track of the current permutation of the row indices performed during pivoting. The algorithm returns the updated matrix x, row index i and row permutation p.
Please refer to any good textbook on numeric mathematics for a closer description of the algorithm. But here is a brief rundown of what happens in each elimination step: First we find the pivot element in column j of the matrix. (We’re doing partial pivoting here, i.e., we only look for the element with the largest absolute value in column j, starting at row i. That’s usually good enough to achieve numerical stability.) If the pivot is zero then we’re done (the rest of the pivot column is already zeroed out). Otherwise, we bring it into the pivot position (swapping row i and the pivot row), divide the pivot row by the pivot, and subtract suitable multiples of the pivot row to eliminate the elements of the pivot column in all subsequent rows. Finally we update i and p accordingly and return the result.
In order to complete the implementation, we still need the following little helper functions to swap two rows of a matrix (this is used in the pivoting step) and to apply a transposition to a permutation (represented as a list):
swap x i j = x!!(transp i j (0..n-1),0..m-1) when n,m = dim x end; transp i j p = [p!tr k | k=0..#p-1] with tr k = if k==i then j else if k==j then i else k end;
Finally, let us define a convenient print representation of double matrices a la Octave (the meaning of the __show__ function is explained in the CAVEATS and NOTES section):
using system; __show__ x::matrix = strcat [printd j (x!(i,j))|i=0..n-1; j=0..m-1] + "\n" with printd 0 = sprintf "\n%10.5f"; printd _ = sprintf "%10.5f" end when n,m = dim x end if dmatrixp x;
Example:
> let x = dmatrix {2,1,-1,8; -3,-1,2,-11; -2,1,2,-3};
> x; gauss_elimination x;
2.00000 1.00000 -1.00000 8.00000
-3.00000 -1.00000 2.00000 -11.00000
-2.00000 1.00000 2.00000 -3.00000
[1,2,0],
1.00000 0.33333 -0.66667 3.66667
0.00000 1.00000 0.40000 2.60000
0.00000 0.00000 1.00000 -1.00000
Whereas the macro facilities of most programming languages simply provide a kind of textual substitution mechanism, Pure macros operate on symbolic expressions and are implemented by the same kind of rewriting rules that are also used to define ordinary functions in Pure. In difference to these, macro rules start out with the keyword def, and only simple kinds of rules without any guards or multiple left-hand and right-hand sides are permitted.
Syntactically, a macro definition looks just like a variable or constant definition, using def in lieu of let or const, but they are processed in a different way. Macros are substituted into the right-hand sides of function, constant and variable definitions. All macro substitution happens before constant substitutions and the actual compilation step. Macros can be defined in terms of other macros (also recursively), and will be expanded using the leftmost-innermost reduction strategy (i.e., macro calls in macro arguments are expanded before the macro gets applied to its parameters).
> def succ x = x+1; > foo x::int = succ (succ x); > show foo foo x::int = x+1+1;
Rules like these can be useful to help the compiler generate better code. Note that a macro may have the same name as an ordinary Pure function, which is essential if you want to optimize calls to an existing function, as in the previous example.
A somewhat more practical example is the following rule from the prelude, which eliminates saturated instances of the right-associative function application operator:
def f $ x = f x;
Like in Haskell, this low-priority operator is handy to write cascading function calls. With the above macro rule, these will be ‘‘inlined’’ as ordinary function applications automagically. Example:
> foo x = bar $ bar $ 2*x; > show foo foo x = bar (bar (2*x));
Here is slightly more tricky rule from the prelude, which optimizes the case of ‘‘throwaway’’ list comprehensions. This is useful if a list comprehension is evaluated solely for its side effects.
def void (catmap f x) = do f x;
Note that the ‘void’ function simply throws away its argument and returns () instead. The ‘do’ function applies a function to every member of a list (like ‘map’), but throws away all intermediate results and just returns (), which is much more efficient if you don’t need those results anyway. These are both defined in the prelude.
Let’s see how this rule transforms a list comprehension if we ‘‘voidify’’ it:
> using system; > f = [printf "%g\n" (2^x+1) | x=1..5; x mod 2]; > g = void [printf "%g\n" (2^x+1) | x=1..5; x mod 2]; > show f g f = catmap (\x -> if x mod 2 then [printf "%g0 (2^x+1)] else []) (1..5); g = do (\x -> if x mod 2 then [printf "%g0 (2^x+1)] else []) (1..5);
Ok, so the ‘catmap’ got replaced with a ‘do’ which is just what we need to make this code go essentially as fast as a ‘for’ loop in conventional programming languages (up to constant factors, of course). Here’s how it looks like when we run the ‘g’ function:
> g; 3 9 33 ()
It’s not all roses, however, since the above macro rule will only get rid of the outermost ‘catmap’ if the list comprehension binds multiple variables:
> u = void [puts $ str (x,y) | x=1..2; y=1..3]; > show u u = do (\x -> catmap (\y -> [puts (str (x,y))]) (1..3)) (1..2);
If you’re bothered by this, you’ll have to apply ‘void’ recursively, creating a nested list comprehension which expands to a nested ‘do’:
> v = void [void [puts $ str (x,y) | y=1..3] | x=1..2]; > show v v = do (\x -> [do (\y -> [puts (str (x,y))]) (1..3)]) (1..2);
(It would be nice to have this handled automatically, but the left-hand side of a macro definition must be a simple expression, and thus it’s not possible to write a macro which descends recursively into the lambda argument of ‘catmap’.)
> def foo (bar x) = foo x+1; > def foo x = x; > baz = foo (bar (bar (bar x))); > show baz baz = x+1+1+1;
Note that, technically, Pure macros are just as powerful as (unconditional) term rewriting systems and thus they are Turing-complete. This implies that a badly written macro may well send the Pure compiler into an infinite recursion, which results in a stack overflow at compile time. See the CAVEATS AND NOTES section at the end of this manual for information on how to deal with these by setting the PURE_STACK environment variable.
Our definition of ‘timex’ employs the system function ‘clock’ to report the cpu time in seconds needed to evaluate a given expression, along with the computed result:
> using system; > def timex x = (clock-t0)/CLOCKS_PER_SEC,y when t0 = clock; y = x end; > sum = foldl (+) 0L; > timex $ sum (1L..100000L); 0.43,5000050000L
(Note that the above definition of ‘timex’ wouldn’t work as an ordinary function definition, since by virtue of Pure’s basic eager evaluation strategy the x parameter would have been evaluated already before it is passed to ‘timex’, making ‘timex’ always return a zero time value. Try it.)
Pure macros also have their limitations. Specifically, the left-hand side of a macro rule must be a simple expression, just like in ordinary function definitions. This restricts the kinds of expressions which can be rewritten by a macro. But Pure macros are certainly powerful enough for most common preprocessing purposes, while still being robust and easy to use.
private foo bar; foo (bar x) = x+1; // foo and bar symbols are private here
Note that to declare multiple symbols in a single declaration, you just list them all with whitespace in between. The same applies to the other types of symbol declarations discussed below.
Ten different precedence levels are available for user-defined operators, numbered 0 (lowest) thru 9 (highest). On each precedence level, you can declare (in order of increasing precedence) infix (binary non-associative), infixl (binary left-associative), infixr (binary right-associative), prefix (unary prefix) and postfix (unary postfix) operators. For instance:
infixl 6 + - ; infixl 7 * / div mod ;
One thing worth noting here is that unary minus plays a special role in the syntax. Like in Haskell, unary minus is the only prefix operator symbol which is also used as an infix operator, and it always has the same precedence as binary minus (whose precedence may be chosen freely in the prelude). Thus, with the standard prelude, -x+y will be parsed as (-x)+y, whereas -x*y is the same as -(x*y). Also note that the notation ‘(-)’ always denotes the binary minus operator; the unary minus operation can be denoted using the built-in ‘neg’ function.
nullary [] (); private nullary nil;
As explained in the PURE OVERVIEW section, nullary symbols are like ordinary identifiers, but are treated as constants rather than variables when they occur on the left-hand side of an equation.
Examples for all types of symbol declarations can be found in the prelude which declares a bunch of standard (arithmetic, relational, logical) operator symbols as well as the list and pair constructors ‘:’ and ‘,’ and the empty list and tuple constants ‘[]’ and ‘()’.
Note that the using clause also has an alternative form which allows dynamic libraries to be loaded, this will be discussed in the C INTERFACE section.
To facilitate modular development, each script also has a separate namespace for private symbols which are only visible in the script which declares them (see the explanation of the private declaration above). This makes it possible to hide away internal operations, prevent name clashes between symbols of different modules, and keep the public namespace tidy and clean. A private declaration shadows a public symbol with the same print name, but this takes effect only after the private declaration of the symbol, so you can easily define yourself an alias for the public symbol before that, e.g.:
public_foo = foo; private foo; foo x = public_foo (bar x);
The script name in a using clause can be specified either as a string denoting the proper filename (possibly including path and/or filename extension), or as an identifier. In the latter case, the .pure filename extension is added automatically. In both cases, the interpreter performs a search to locate the script, unless an absolute pathname was given. It first searches the directory of the script containing the using clause (or the current working directory if the clause was read from standard input, as is the case, e.g., in an interactive session), then the directories named in -I options on the command line (in the given order), then the colon-separated list of directories in the PURE_INCLUDE environment variable, and finally the directory named by the PURELIB environment variable. Note that the current working directory is not searched by default (unless the using clause is read from standard input), but of course you can force this by adding the option -I. to the command line, or by including ‘.’ in the PURE_INCLUDE variable.
For the purpose of comparing script names, the interpreter always uses the canonicalized full pathname of the script, following symbolic links to the destination file (albeit only one level). Thus different scripts with the same basename, such as foo/utils.pure and bar/utils.pure can both be included in the same program (unless they link to the same file). The resolution of symbolic links also makes it possible to have an executable script with a shebang line in its own directory, to be executed via a symbolic link placed on the system PATH. In this case the script search performed in using clauses will use the real script directory and thus other required scripts can be located in the script directory. This is the recommended practice for installing standalone Pure applications in source form which are to be run directly from the shell.
> catch error (throw hello_world); error hello_world
Exceptions are also generated by the runtime system if the program runs out of stack space, when a guard does not evaluate to a truth value, and when the subject term fails to match the pattern in a pattern-matching lambda abstraction, or a let, case or when construct. These types of exceptions are reported using the symbols stack_fault, failed_cond and failed_match, respectively, which are declared as constant symbols in the standard prelude. You can use catch to handle these kinds of exceptions just like any other. For instance:
> fact n = if n>0 then n*fact(n-1) else 1; > catch error (fact foo); error failed_cond > catch error (fact 100000); error stack_fault
(You’ll only get the latter kind of exception if the interpreter does stack checks, see the discussion of the PURE_STACK environment variable in the CAVEATS AND NOTES section.)
Note that unhandled exceptions are reported by the interpreter with a corresponding error message:
> fact foo; <stdin>:2.0-7: unhandled exception ’failed_cond’ while evaluating ’fact foo’
Exceptions also provide a way to handle asynchronous signals. Most standard termination signals (SIGINT, SIGTERM, etc.) are set up during startup of the interpreter to produce corresponding Pure exceptions of the form signal SIG where SIG is the signal number. Pure’s system module provides symbolic constants for common POSIX signals and also defines the operation trap which lets you rebind any signal to a signal exception. For instance, the following lets you handle the SIGQUIT signal:
> using system; > trap SIG_TRAP SIGQUIT;
You can also use trap to just ignore a signal or revert to the system’s default handler (which might take different actions depending on the type of signal, see signal(7) for details):
> trap SIG_IGN SIGQUIT; // signal is ignored > trap SIG_DFL SIGQUIT; // reinstalls the default signal handler
Last but not least, exceptions can also be used to implement non-local value returns. For instance, here’s a variation of our n queens algorithm which only returns the first solution. Note the use of throw in the recursive search routine to bail out with a solution as soon as we found one. The value thrown there is caught in the main routine. Also note the use of ‘void’ in the second equation of ‘search’. This effectively turns the list comprehension into a simple loop which suppresses the normal list result and just returns () instead. Thus, if no value gets thrown then the function regularly returns with () to indicate that there is no solution.
queens1 n = catch reverse (search n 1 []) with
search n i p = throw p if i>n;
= void [search n (i+1) ((i,j):p) | j = 1..n; safe (i,j) p];
safe (i,j) p = not any (check (i,j)) p;
check (i1,j1) (i2,j2)
= i1==i2 || j1==j2 || i1+j1==i2+j2 || i1-j1==i2-j2;
end;
E.g., let’s compute a solution for a standard 8x8 board:
> queens 8; [(1,1),(2,5),(3,8),(4,6),(5,3),(6,7),(7,2),(8,4)]
> extern double sin(double); > sin 0.3; 0.29552020666134
Multiple prototypes can be given in one extern declaration, separating them with commas:
extern double sin(double), double cos(double), double tan(double);
For clarity, the parameter types can also be annotated with parameter names, e.g.:
extern double sin(double x);
Parameter names in prototypes only serve informational purposes and are for the human reader; they are effectively treated as comments by the compiler.
The interpreter makes sure that the parameters in a call match; if not, the call is treated as a normal form expression by default, which gives you the opportunity to extend the external function with your own Pure equations (see below). The range of supported C types is a bit limited right now (void, bool, char, short, int, long, float, double, as well as arbitrary pointer types, i.e.: void*, char*, etc.), but in practice these should cover most kinds of calls that need to be done when interfacing to C libraries.
Single precision float arguments and return values are converted from/to Pure’s double precision floating point numbers automatically.
A variety of C integer types (char, short, int, long) are provided which are converted from/to the available Pure integer types in a straightforward way. One important thing to note here is that the ‘long’ type always denotes 64 bit integers, even if the corresponding C type is actually 32 bit (as it usually is on most contemporary systems). All integer parameters take both Pure ints and bigints as actual arguments; truncation or sign extension is performed as needed, so that the C interface behaves as if the argument was ‘‘cast’’ to the C target type. Returned integers use the smallest Pure type capable of holding the result (i.e., int for the C char, short and int types, bigint for long a.k.a. 64 bit integers).
Pure considers all integers as signed quantities, but it is possible to pass unsigned integers as well (if necessary, you can use a bigint to pass positive values which are too big to fit into a machine int). Also note that when an unsigned integer is returned by a C routine which is too big to fit into the corresponding signed integer type, it will ‘‘wrap around’’ and become negative. In this case, depending on the target type, you can use the ubyte, ushort, uint and ulong functions provided by the prelude to convert the result back to an unsigned quantity.
Concerning the pointer types, char* is for string arguments and return values which need translation between Pure’s internal utf-8 representation and the system encoding, while void* is for any generic kind of pointer (including strings, which are not translated when passed/returned as void*). Any other kind of pointer (except expr* and the GSL matrix pointer types, which are discussed below) is effectively treated as void* right now, although in a future version the interpreter may keep track of the type names for the purpose of checking parameter types.
The expr* pointer type is special; it indicates a Pure expression parameter or return value which is just passed through unchanged. All other types of values have to be ‘‘unboxed’’ when they are passed as arguments (i.e., from Pure to C) and ‘‘boxed’’ again when they are returned as function results (from C to Pure). All of this is handled by the runtime system in a transparent way, of course.
The matrix pointer types dmatrix*, cmatrix* and imatrix* can be used to pass double, complex double and int matrices to GSL functions taking pointers to the corresponding GSL types (gsl_matrix, gsl_matrix_complex and gsl_matrix_int) as arguments or returning them as results. Note that there is no marshalling of Pure’s symbolic matrix type, as these aren’t supported by GSL anyway. Also note that matrices are always passed by reference. If you need to pass a matrix as an output parameter of a GSL matrix routine, you can either create a zero matrix or a copy of an existing matrix. The prelude provides various operations for that purpose (in particular, see the dmatrix, cmatrix, imatrix and pack functions in matrices.pure). For instance, here is how you can quickly wrap up GSL’s double matrix addition function in a way that preserves value semantics:
> extern int gsl_matrix_add(dmatrix*, dmatrix*);
> x::matrix + y::matrix = gsl_matrix_add x y $$ x when x = pack x end;
> let x = dmatrix {1,2,3}; let y = dmatrix {2,3,2}; x; y; x+y;
{1.0,2.0,3.0}
{2.0,3.0,2.0}
{3.0,5.0,5.0}
Most GSL matrix routines can be wrapped in this fashion quite easily. A ready-made GSL interface providing access to all of GSL’s numeric functions will be provided in the future.
As already mentioned, it is possible to augment an external C function with ordinary Pure equations, but in this case you have to make sure that the extern declaration of the function comes first. For instance, we might want to extend our imported sin function with a rule to handle integers:
> extern double sin(double); > sin 0.3; 0.29552020666134 > sin 0; sin 0 > sin x::int = sin (double x); > sin 0; 0.0
Sometimes it is preferable to replace a C function with a wrapper function written in Pure. In such a case you can specify an alias under which the original C function is known to the Pure program, so that you can still call the C function from the wrapper. An alias is introduced by terminating the extern declaration with a clause of the form ‘‘= alias’’. For instance:
> extern double sin(double) = c_sin; > sin x::double = c_sin x; > sin x::int = c_sin (double x); > sin 0.3; sin 0; 0.29552020666134 0.0
External C functions are resolved by the LLVM runtime, which first looks for the symbol in the C library and Pure’s runtime library (or the interpreter executable, if the interpreter was linked statically). Thus all C library and Pure runtime functions are readily available in Pure programs. Other functions can be provided by adding them to the runtime, or by linking them statically into the runtime or the interpreter executable. Better yet, you can just ‘‘dlopen’’ shared libraries at runtime with a special form of the using clause:
using "lib:libname[.ext]";
For instance, if you want to call the GMP functions directly from Pure:
using "lib:libgmp";
After this declaration the GMP functions will be ready to be imported into your Pure program by means of corresponding extern declarations.
Shared libraries opened with using clauses are searched for in the same way as source scripts (see section DECLARATIONS above), using the -L option and the PURE_LIBRARY environment variable in place of -I and PURE_INCLUDE. If the library isn’t found by these means, the interpreter will also consider other platform-specific locations searched by the dynamic linker, such as the system library directories and LD_LIBRARY_PATH on Linux. The necessary filename suffix (e.g., .so on Linux or .dll on Windows) will be supplied automatically when needed. Of course you can also specify a full pathname for the library if you prefer that. If a library file cannot be found, or if an extern declaration names a function symbol which cannot be resolved, an appropriate error message is printed.
The prelude also offers support operations for the implementation of list and matrix comprehensions, as well as the customary list operations like head, tail, drop, take, filter, map, foldl, foldr, scanl, scanr, zip, unzip, etc., which make list programming so much fun in modern FPLs. In Pure, these also work on strings as well as matrices, although, for reasons of efficiency, these data structures are internally represented as different kinds of array data structures.
Besides the prelude, Pure’s standard library also comprises a growing number of additional library modules which we can only mention in passing here. In particular, the math.pure module provides additional mathematical functions as well as Pure’s complex and rational number data types. Common container data structures like sets and dictionaries are implemented in the set.pure and dict.pure modules, among others. Moreover, the (beginnings of a) system interface can be found in the system.pure module. In particular, this module also provides operations to do basic C-style I/O, including printf and scanf. More stuff will likely be provided in future releases.
The input language is just the same as for source scripts, and hence individual definitions and expressions must be terminated with a semicolon before they are processed. For instance, here is a simple interaction which defines the factorial and then uses that definition in some evaluations. Input lines begin with ‘‘>’’, which is the interpreter’s default command prompt:
> fact 1 = 1; > fact n = n*fact (n-1) if n>1; > let x = fact 10; x; 3628800 > map fact (1..10); [1,2,6,24,120,720,5040,40320,362880,3628800]
As indicated, in interactive mode the normal forms of toplevel expressions are printed after each expression is entered. We also call this the read-eval-print loop. Normal form expressions are usually printed in the same form as you’d enter them. However, there are a few special kinds of objects like closures (anonymous and local functions), thunks (‘‘lazy’’ values to be evaluated when needed) and pointers which don’t have a textual representation in the Pure syntax and will be printed in the format #<object description> by default. It is also possible to override the print representation of any kind of expression by means of the __show__ function, see the CAVEATS AND NOTES section for details.
When running interactively, the interpreter also accepts a number of special commands useful for interactive purposes. Here is a quick rundown of the currently supported operations:
This command only supports a subset of the show options, type dump -h for a description of these. Also note that by default the dump command only writes interactive definitions to the output file, which is equivalent to the -t1 option of the show command. Presumably this is the most common usage, but using the -t option, you can select any definitions level just as with show. In particular, dump -t saves only the definitions made after the most recent save command, and dump -t0 can be used to dump the entire program (including the definitions of the prelude), which can be useful for debugging purposes.
The default output file is .pure in the current directory, which is then reloaded automatically the next time the interpreter starts up in interactive mode in the same directory. This provides a quick-and-dirty means to save an interactive session and have it restored later. Please note that this isn’t perfect; in order to properly handle extern and using declarations and other special cases such as thunks and pointers stored in variables, you’ll probably have to prepare a corresponding .purerc file yourself, see ‘‘Startup Files’’ below.
A different filename can be specified with the -F option. You can then edit that file and use it as a starting point for an ordinary script or a .purerc file, or you can just run the file with the run command (see below) to restore the definitions in a subsequent interpreter session.
You can also specify a subset of symbols to be saved. Shell glob patterns can be used if the -g option is given. Options may be combined; e.g., dump -Ffg foo.pure foo* is just the same as dump -F foo.pure -f -g foo*, and dumps all functions whose names start with ‘foo’ to the file ‘foo.pure’.
Note that these special commands are only recognized at the beginning of the interactive command line (they are not reserved keywords of the Pure language). Thus it’s possible to ‘‘escape’’ identifiers looking like commands by entering a space at the beginning of the line. However, the compiler also warns you about identifiers which might be mistaken as command names, so that you can avoid this kind of problem.
Some commands which are especially important for effective operation of the interpreter are discussed in more detail in the following sections.
The interpreter first looks for a .purerc file in the user’s home directory (as given by the HOME environment variable) and then for a .purerc file in the current working directory. These are just ordinary Pure scripts which may contain any additional definitions that you need. The .purerc file in the home directory is for global definitions which should always be available when running interactively, while the .purerc file in the current directory can be used for project-specific definitions.
Finally, you can also have a .pure initialization file in the current directory, which is created by the dump command (see above) and is loaded after the .purerc files if it is present.
The interpreter processes all these files in the same way as with the run command (see above). When invoking the interpreter, you can specify the --norc option on the command line if you wish to skip these initializations.
If none of the -c, -f, -m and -v options are specified, then all kinds of symbols (constants, functions, macros and variables) are printed, otherwise only the specified categories will be listed.
Note that some of the options (in particular, -a and -d) may produce excessive amounts of information. By setting the PURE_MORE environment variable accordingly, you can specify a shell command to be used for paging, usually more(1) or less(1) .
For instance, to list all definitions in all loaded scripts (including the prelude), simply say:
> show
This may produce quite a lot of output, depending on which scripts are loaded. The following command will only show summary information about the variable symbols along with their current values (using the ‘‘long format’’):
> show -lv argc var argc = 0; argv var argv = []; sysinfo var sysinfo = "i686-pc-linux-gnu"; version var version = "0.8"; 4 variables
If you’re like me then you’ll frequently have to look up how some operations are defined. No sweat, with the Pure interpreter there’s no need to dive into the sources, the show command can easily do it for you. For instance, here’s how you can list the definitions of all ‘‘fold-left’’ operations from the prelude in one go:
> show -g foldl* foldl f a x::matrix = foldl f a (list x); foldl f a s::string = foldl f a (chars s); foldl f a [] = a; foldl f a (x:xs) = foldl f (f a x) xs; foldl1 f x::matrix = foldl1 f (list x); foldl1 f s::string = foldl1 f (chars s); foldl1 f (x:xs) = foldl f x xs;
Each save command introduces a new temporary level, and each subsequent clear command ‘‘pops’’ the definitions on the current level (including any definitions read using the run command) and returns you to the previous one (if any). This gives you a ‘‘stack’’ of up to 255 temporary environments which enables you to ‘‘plug and play’’ in a (more or less) safe fashion, without affecting the rest of your program. Example:
> foo (x:xs) = x+foo xs; > foo [] = 0; > show -t foo (x:xs) = x+foo xs; foo [] = 0; > foo (1..10); 55 > clear This will clear all temporary definitions at level #1. Continue (y/n)? y > show foo > foo (1..10); foo [1,2,3,4,5,6,7,8,9,10]
(Please note that the clear command only works in this way when invoked without arguments. Otherwise the symbols given as arguments will be purged unconditionally, at all levels.)
We’ve seen already that normally, if you enter a sequence of equations, they will be recorded in the order in which they were written. However, it is also possible to override definitions in lower levels with the override command:
> foo (x:xs) = x+foo xs; > foo [] = 0; > show foo foo (x:xs) = x+foo xs; foo [] = 0; > foo (1..10); 55 > save save: now at temporary definitions level #2 > override > foo (x:xs) = x*foo xs; > show foo foo (x:xs) = x*foo xs; foo (x:xs) = x+foo xs; foo [] = 0; > foo (1..10); 0
Note that the equation ‘foo (x:xs) = x*foo xs;’ was inserted before the previous ‘foo (x:xs) = x+foo xs;’ rule, which is at level #1.
Even in override mode, new definitions will be added after other definitions at the current level. This allows us to just continue adding more high-priority definitions overriding lower-priority ones:
> foo [] = 1; > show foo foo (x:xs) = x*foo xs; foo [] = 1; foo (x:xs) = x+foo xs; foo [] = 0; > foo (1..10); 3628800
Again, the new equation was inserted above the existing lower-priority rules, but below our previous ‘foo (x:xs) = x*foo xs;’ equation entered at the same level. As you can see, we have now effectively replaced our original definition of ‘foo’ with a version that calculates list products instead of sums, but of course we can easily go back one level to restore the previous definition:
> clear This will clear all temporary definitions at level #2. Continue (y/n)? y clear: now at temporary definitions level #1 clear: override mode is on > show foo foo (x:xs) = x+foo xs; foo [] = 0; > foo (1..10); 55
Note that clear reminded us that override mode is still enabled (save will do the same if override mode is on while pushing a new definitions level). To turn it off again, use the underride command. This will revert to the normal behaviour of adding new equations below existing ones:
> underride
The short answer is that I simply liked the name, and there wasn’t any programming language named ‘‘Pure’’ yet (quite a feat nowadays), so there’s one now. :)
__show__ is just an ordinary Pure function expected to return a string with the desired custom representation of a normal form value given as the function’s single argument. This function is not defined by default, so you are free to add any rules that you want. The interpreter prints the strings returned by __show__ just as they are. It will not check whether they conform to Pure syntax and/or semantics, or modify them in any way.
Custom print representations are most useful for interactive purposes, if you’re not happy with the default print syntax of some kinds of objects. One particularly useful application of __show__ is to change the format of numeric values. Here are some examples:
> using system; > __show__ x::double = sprintf "%0.6f" x; > 1/7; 0.142857 > __show__ x::int = sprintf "0x%0x" x; > 1786; 0x6fa > using math; > __show__ (x::double+:y::double) = sprintf "%0.6f+%0.6fi" (x,y); > cis (-pi/2); 0.000000+-1.000000i
The prelude function str, which returns the print representation of any Pure expression, calls __show__ as well:
> str (1/7); "0.142857"
Conversely, you can call the str function from __show__, but in this case it always returns the default representation of an expression. This prevents the expression printer from going recursive, and allows you to define your custom representation in terms of the default one. E.g., the following rule removes the ‘L’ suffixes from bigint values:
> __show__ x::bigint = init (str x); > fact n = foldl (*) 1L (1..n); > fact 30; 265252859812191058636308480000000
Of course, your definition of __show__ can also call __show__ itself recursively to determine the custom representation of an object.
One case which needs special consideration are thunks (futures). The printer will never use __show__ for those, to prevent them from being forced inadvertently. In fact, you can use __show__ to define custom representations for thunks, but only in the context of a rule for other kinds of objects, such as lists. For instance:
> nullary ...; > __show__ (x:xs) = str (x:...) if thunkp xs; > 1:2:(3..inf); 1:2:3:...
Another case which needs special consideration are numeric matrices. For efficiency, the expression printer will always use the default representation for these, unless you override the representation of the matrix as a whole. E.g., the following rule for double matrices mimics Octave’s default output format (for the sake of simplicity, this isn’t perfect, but you get the idea):
> __show__ x::matrix =
> strcat [printd j (x!(i,j))|i=0..n-1; j=0..m-1] + "\n"
> with printd 0 = sprintf "\n%10.5f"; printd _ = sprintf "%10.5f" end
> when n,m = dim x end if dmatrixp x;
> {1.0,1/2;1/3,4.0};
1.00000 0.50000
0.33333 4.00000
Finally, by just purging the definition of the __show__ function you can easily go back to the standard print syntax:
> clear __show__ > 1/7; 1786; cis (-pi/2); 0.142857142857143 1786 6.12303176911189e-17+:-1.0
Note that if you have a set of definitions for the __show__ function which should always be loaded at startup, you can put them into the interpreter’s interactive startup files, see INTERACTIVE USAGE.
a@foo x y = a,x,y; a@(foo x) y = a,x,y; a@(foo x y) = a,x,y;
This is because the spine of a function application is not available when the function is called at runtime. ‘‘As’’ patterns in pattern bindings (case, when) are not affected by this restriction since the entire value to be matched is available at runtime. For instance:
> case bar 99 of y@(bar x) = y,x+1; end; bar 99,100
> foo x = a [] x with a xs (x@_ y) = a (y:xs) x; a xs x = x:xs end; > foo (a b c d); [a,b,c,d]
This may seem a little awkward, but as a matter of fact the ‘‘head = function’’ rule is quite useful since it covers the common cases without forcing the programmer to declare ‘‘constructor’’ symbols (except nullary symbols). On the other hand, generic rules operating on arbitrary function applications are not all that common, so having to ‘‘escape’’ a variable using the anonymous ‘‘as’’ pattern trick is a small price to pay for that convenience.
Sometimes you may also run into the complementary problem, i.e., to match a function argument against a given function. Consider this code fragment:
foo x = x+1; foop f = case f of foo = 1; _ = 0 end;
You might expect ‘foop’ to return true for ‘foo’, and false on all other values. Better think again, because in reality ‘foop’ will always return true! In fact, the Pure compiler will warn you about the second rule of the case expression not being used at all:
> foop 99; warning: rule never reduced: _ = 0; 1
This happens because a non-nullary symbol on the left-hand side of a rule, which is not the head symbol of a function application, is always considered to be a variable, even if that symbol is defined as a global function elsewhere. So ‘foo’ isn’t a literal name in the above case expression, it’s a variable! (As a matter of fact, this is rather useful, since otherwise a rule like ‘f g = g+1’ would suddenly change meaning if you happen to add a definition like ‘g x = x-1’ somewhere else in your program, which certainly isn’t desirable.)
Fortunately, the syntactic equality operator ‘===’ defined in the prelude comes to the rescue here. Just define ‘foop’ as follows:
> foop f = f===foo; > foop foo, foop 99; 1,0
Another pitfall is that with and when clauses are tacked on to the end of the expression they belong to, which mimics mathematical language but may be unfamilar if you’re more accustomed to programming languages from the Algol/Pascal/C family. If you want to figure out what is actually going on there, it’s usually best to read nested scopes ‘‘in reverse’’ (proceeding from the rightmost/outermost to the leftmost/innermost clause).
Also note that since with and when are part of the expression, not the rule syntax, these clauses cannot span both the right-hand side and the guard of a rule. Usually it’s easy to work around this with conditional and case expressions, though.
fact n::int |
fact n = n*fact(n-1) if n>0;
= 1 otherwise;
(This obviously becomes unwieldy if you have to deal with several numeric arguments of different types, however, so in this case it is usually better to just use a polymorphic rule.)
Also note that int (the machine integers) and bigint (the GMP ‘‘big’’ integers) are really different kinds of objects, and thus if you want to define a function operating on both kinds of integers, you’ll also have to provide equations for both. This also applies to equations matching against constant values of these types; in particular, a small integer constant like ‘0’ only matches machine integers, not bigints; for the latter you’ll have to use the ‘‘big L’’ notation ‘0L’.
> extern double atan(double); > const pi = 4*atan 1.0; > foo x = 2*pi*x; > show foo foo x = 2*3.14159265358979*x;
(If you take a look at the disassembled code for this function, you will find that the value 2*3.14159265358979 = 6.28318530717959 has actually been computed at compile time.)
Note that constant definitions differ from parameterless macros in that the right-hand side of the definition is in fact evaluated at compile time. E.g., compare the above with the following macro definition:
> clear pi foo > def pi = 4*atan 1.0; > foo x = 2*pi*x; > show foo foo x = 2*(4*atan 1.0)*x;
The LLVM backend also eliminates dead code automagically, which enables you to employ a constant computed at runtime to configure your code for different environments, without any runtime penalties:
const win = index sysinfo "mingw32" >= 0;
check boy = bad boy if win;
= good boy otherwise;
In this case the code for one of the branches of ‘check’ will be completely eliminated, depending on the outcome of the configuration check.
On the other hand, constant definitions are somewhat limited in scope compared to variable definitions, since the bound value must be usable at compile time, so that it can be substituted into other definitions. Thus, while there is no a priori restriction on the computations you can perform to obtain the value of the constant, the value must not be a pointer object (other than the null pointer), or an anonymous closure (which also rules out local functions, because these cannot be referred to by their names at the toplevel), or an aggregate value containing any such values.
Constants also differ from variables in that they cannot be redefined (that’s their purpose after all) and will only take effect on subsequent definitions. E.g.:
> const c = 2; > foo x = c*x; > show foo foo x = 2*x; > foo 99; 198 > const c = 3; <stdin>:5.0-8: symbol ’c’ is already defined as a constant
Well, in fact this not the full truth because in interactive mode it is possible to redefine constants after all, if the old definition is first purged with the clear command. However, this won’t affect any other existing definitions:
> clear c > const c = 3; > bar x = c*x; > show foo bar foo x = 2*x; bar x = 3*x;
(You’ll also have to purge any existing definition of a variable if you want to redefine it as a constant, or vice versa, since Pure won’t let you redefine an existing constant or variable as a different kind of symbol. The same also holds if a symbol is currently defined as a function or a macro.)
You also have to be careful when passing generic pointer values to external C routines, since currently there is no type checking for these; any pointer type other than char* and expr* is effectively treated as void*. This considerably simplifies lowlevel programming and interfacing to C libraries, but also makes it very easy to have your program segfault all over the place. Therefore it is highly recommended that you wrap your lowlevel code in Pure routines and data structures which do all the checks necessary to ensure that only the right kind of data is passed to C routines.
Specifically, the prelude goes to great lengths to implement all standard list operations in a way that properly deals with streams (a.k.a. list futures). What this all boils down to is that all list operations which can reasonably be expected to operate in a lazy way on streams, will do so. (Exceptions are inherently eager operations such as ‘#’, reverse and foldl.) Only those portions of an input stream will be traversed which are strictly required to produce the result. For most purposes, this works just like in fully lazy FPLs such as Haskell. However, there are some notable differences:
Fortunately, Pure normally does proper tail calls (if LLVM provides that feature on the platform at hand), so most tail-recursive definitions should work fine in limited stack space. For instance, the following little program will loop forever if your platform supports the required optimizations:
loop = loop;
This also works if your definition involves function parameters, guards and multiple equations, of course. Moreover, conditional expressions (if-then-else) are tail-recursive in both branches, and the sequence operator $$ is tail-recursive in its second operand. Note, however, that the logical operators && and || are not tail-recursive in Pure, because they are required to always yield a proper truth value (0 or 1), which wouldn’t be possible with tail call semantics. (The rationale behind this design decision is that it allows the compiler to generate much better code for logical expressions.)
There is one additional restriction in the current implementation, namely that a tail call will be eliminated only if the call is done directly, i.e., through an explicit call, not through a (global or local) function variable. Otherwise the call will be handled by the runtime system which is written in C and can’t do proper tail calls because C can’t (at least not in a portable way). This also affects mutually recursive global function calls, since there the calls are handled in an indirect way, too, through an anonymous global variable. (This is done so that a global function definition can be changed at any time during an interactive session, without having to recompile the entire program.) However, mutual tail recursion does work with local functions, so it’s easy to work around this limitation.
loop = check $$ loop; check = ();
To handle signals while the above loop is executing, you can add an exception handler like the following:
loop = catch handle check $$ loop with handle (signal k) = catch handle (...) end;
(Note the ‘catch handle’ around the signal processing code which is needed for safety because another signal may arrive while the signal handler is being executed.)
Of course, in a real application the ‘check’ function would most likely have to do some actual processing, too. In that case you’d probably want the ‘loop’ function to carry around some ‘‘state’’ argument to be processed by the ‘check’ routine, which then returns an updated state value for the next iteration. This can be implemented as follows:
loop x = loop (catch handle (check x)) with handle (signal k) = catch handle (...) end; check x = ...;