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2. Getting Started

A new programming language is best conveyed through some examples, and therefore it is common practice to start a language description with an introductory chapter which treats the most essential language features in a fairly informal manner. This is also the purpose of the present chapter. We first show how some of the "standard features" of the Q programming language can be employed for using the Q interpreter effectively as a sophisticated kind of "desktop calculator". The remainder of this section then adresses the question of how you can extend the Q environment by providing your own definitions in Q scripts, and describes the evaluation process and the treatment of runtime errors in the interpreter.

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2.1 Using the Interpreter

To begin with, let us show how the Q interpreter is invoked to evaluate some expressions with the built-in functions provided by the Q language. Having installed the Q programming system, you can simply invoke the interpreter from your shell by typing q. The interpreter will then start up, display its sign-on, and leaves you at its prompt:


This indicates that the interpreter is waiting for you to type an expression. Let's start with a simple one:

==> 23

Sure enough, that's correct. Whenever you type an expression, the interpreter just prints its value. Any integer is a legal value in the Q language, and the common arithmetic operations work on these values as expected. Q integers are "bignums", i.e., their size is only limited by available memory:

==> 16753418726345 * 991726534256718265234

"Real" numbers (more exactly, their approximation using 64 bit double precision floating point numbers) are provided as well. Let's try these:

==> sqrt (16.3805*5)/.05

The sqrt identifier denotes a built-in function which computes the square root of its argument. (A summary of the built-in functions can be found in Built-In Functions.)

What happens if you mistype an expression?

==> sqrt (16.3805*5)/,05
! Syntax error
>>> sqrt (16.3805*5)/,05

As you can see, the interpreter not only lets you know that you typed something wrong, but also indicates the position of the error. It is quite easy to go back now and correct the error, using the interpreter's "command history". With the up and down arrow keys you can cycle through the expressions you have typed before, edit them, and resubmit a line by hitting the carriage return key. Also note that what you typed is stored in a "history file" when you exit the interpreter, which is reloaded next time the interpreter is invoked. A number of other useful keyboard commands are provided, see Readline: (readline)Command Line Editing section `Command Line Editing' in The GNU Readline Library. In particular, you can have the command line editor "complete" function symbols with the <TAB> key. E.g., if you type in sq and press the <TAB> key, you will get the sqrt function.

Before we proceed, a few remarks about the syntax of function applications are in order. The first thing you should note is that in Q, like in most other modern functional languages, function application is simply denoted by juxtaposition:

==> sqrt 2

Multiple arguments are specified in the same fashion:

==> max 5 7

Parentheses can be used for grouping expressions as usual. In particular, if a function application appears as an argument in another function application then it must be parenthesized as follows:

==> sqrt (sqrt 2)

Another important point is that operators are in fact just functions in disguise. You can turn any operator into a prefix function by enclosing it in parentheses. Thus (+) denotes the function which adds its two arguments, and X+1 can also be written as (+) X 1; in fact, the former expression is nothing but "syntactic sugar" for the latter, see Built-In Operators. You can easily verify this in the interpreter:

==> (+) X 1

You can also have partial function applications like (*) 2 which denotes a function which doubles its argument. Moreover, Q supports so-called operator sections which allow you to specify a binary operator with only either its left or right operand. For instance, (1/) denotes the reciprocal and (+1) the "increment by 1" function:

==> (+1) 5

==> (1/) 3

The interpreter maintains a global variable environment, in which you can store arbitrary expression values. This provides a convenient means to define abbreviations for frequently-used expressions and for storing intermediate results. Variable definitions are done using the def command. For instance:

==> def X = 16.3805*5

==> X

As indicated, variable symbols usually start with a capital letter; an initial lowercase letter indicates a function symbol. This convention is valid throughout the Q language. However, it is possible to explicitly declare an uncapitalized symbol as a variable, using the var command, and then assign values to it, like so:

==> var f

==> def f = sqrt

You can also both declare a variable and initialize its value with a single var command:

==> var f = sqrt

==> f X/.05

Multiple variable declarations and definitions can be entered on a single line, using commas to separate different definitions in a def or var command, and you can separate multiple expressions and commands on the same line with semicolons:

==> def X = 16.3805*5, f = sqrt; X; f X/.05

You can also bind several variables at once by using an expression pattern as the left-hand side of a variable definition:

==> def (f, X) = (sqrt, 16.3805*5); f X/.05

(Such pattern-matching definitions only work with def; the left-hand side of a var declaration must always be a simple identifier.)

Another useful feature is the built-in "anonymous" variable `_', which is always set to the value of the most recent expression value printed by the interpreter:

==> _

==> 2*_

Sometimes you would also like the interpreter to "forget" about a definition. This can be done by means of an undef statement:

==> undef X; X

Besides def, undef and var, the interpreter provides a number of other special commands. The most important command for beginners certainly is the help command, which displays the online manual using the GNU info reader. You can also run this command with a keyword to be searched in the info file. For instance, to find out about all special commands provided by the interpreter, type the following:

==> help commands

(Type q when you are done reading the info file.)

Other useful commands are who which prints a list of the user-defined variables, and whos which describes the attributes of a symbol:

==> who

==> whos f sqrt

f               user-defined variable symbol
                = sqrt

sqrt            builtin function symbol
                sqrt X1

You can save user-defined variables and reload them using the save and load commands:

==> save
saving .q_vars

==> def f = 0; f

==> load
loading .q_vars

==> f

Some statistics about the most recent expression evaluation can be printed with the stats command:

==> sum [1..123456]

==> stats
0.52 secs, 246916 reductions, 246918 cells

The stats command displays the (cpu) time needed to complete an evaluation, the number of "reductions" (a.k.a. basic evaluation steps) performed by the interpreter, and the number of expression "cells" needed during the course of the computation. This information is often useful for profiling purposes.

Other commands allow you to edit and run a script directly from the interpreter, and to inspect and set various internal parameters of the interpreter; see Command Language.

Of course, the Q interpreter can also carry out computations on non-numeric data. In fact, Q provides a fairly rich collection of built-in data types, and makes it easy to define your own. For instance, strings are denoted by character sequences enclosed in double quotes. Strings can be concatenated with the `++' operator and the length of a string can be determined with `#'. Moreover, the character at a given position can be retrieved with the (zero-based) indexing operator `!':

==> "abc"++"xyz"; #"abc"; "abc"!1

The same operations also apply to lists, which are written as sequences of values enclosed in brackets:

==> [a,b,c]++[x,y,z]; #[a,b,c]; [a,b,c]!1

As in Prolog, lists are actually represented as right-recursive constructs of the form `[X|Xs]' where X denotes the head element of the list and Xs its tail, i.e., the list of the remaining elements. This representation allows a list to be traversed and manipulated in an efficient manner.

==> def [X|Xs] = [a,b,c]; X; Xs

==> [0|Xs]

Another useful list operation is "enumeration", which works with all so-called "enumeration types" (cf. Enumeration Types), as well as characters and numbers. Enumerations are constructed with the enum function, or using the familiar [X..Y] notation which is in fact only "syntactic sugar" for enum:

==> enum "a" "k"

==> ["a".."k"]

==> [0..9]

==> [0.1,0.2..1.0]

Tuples are sequences of expressions enclosed in parentheses, which are Q's equivalent of "vectors" or "arrays" in other languages. Like lists, tuples can be concatenated, indexed and enumerated, but they use an internal representation optimized for efficient storage and constant-time indexing:

==> (a,b,c)++(x,y,z); #(a,b,c); (a,b,c)!1

==> (0..9)

Q also offers built-in operations to convert between lists and tuples:

==> tuple [a,b,c]

==> list _

Q provides yet another list-like data structure, namely "lazy" lists a.k.a. streams, which are written using curly braces instead of brackets. In difference to lists, streams only produce their elements "on demand", when they are needed in the course of a computation. This makes it possible to represent large and even infinite sequences of values in an efficient manner. You can see this somewhat unusual behaviour if you try streams interactively in the interpreter. For instance, let's try to concatenate two streams:

==> {a,b,c}++{x,y,z}

You probably noticed that the tail of the resulting stream, {b,c}++{x,y,z}, has not been evaluated yet. But that is all right, because it will be evaluated as soon as we need it:

==> _!4

This lazy way of doing things becomes especially important when we work with infinite streams. We will have more to say about this in the following section.

It is quite instructive to keep on playing a bit like this, to get a feeling of what the Q interpreter can do. You might also wish to consult Built-In Functions, which discusses the built-in functions, and Using Q, which shows how the interpreter is invoked and what additional capabilities it offers. To exit the interpreter when you are finished, invoke the built-in quit function:

==> quit

This function does not return a value, but immediately returns you to the operating system's command shell.

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2.2 Using the Standard Library

The standard library is a collection of useful functions and data structures which are not provided as built-ins, but are implemented by scripts which are mostly written in Q itself. Most of these scripts are included in the "prelude script" prelude.q, which is always loaded by the interpreter, so that the standard library functions are always available. See The Standard Library, for an overview of these operations. You can check which standard library scripts a.k.a. "modules" are actually loaded with the interpreter's modules command. With the standard prelude loaded, this command shows the following:

==> modules 
assert          clib*           complex         cond
error           math            prelude         rational
sort            stdlib          stream          string
tuple           typec

As you can see, there are quite a few modules already loaded, and you can use the functions provided by these modules just like any of the built-in operations. The standard library provides a lot of additional functions operating on numbers, strings and lists. For instance, you can take the sum of a list of (integer or floating point) numbers simply as follows:

==> sum [1..5]

In fact, the library provides a rather general operation, foldl, which iterates a binary function over a list, starting from a given initial value. Using the foldl function, the above example can also be written as follows:

==> foldl (+) 0 [1..5]

(There also is a foldr function which works analogously, but combines list members from right to left rather than from left to right.)

Generalizing the notion of simple enumerations of numbers like [1..5], the standard library also provides the general-purpose list-generating function while. For instance, we can generate the list of all powers of 2 in the range [1..1000] as follows:

==> while (<=1000) (2*) 1

(Recall from the previous section that (2*) is the doubling function. And (<=1000) is a predicate which checks its argument to be less than or equal to 1000.)

If we want to generate a list of a given size, we can use iter instead. So, for instance, we might compute the value of a finite geometric series as follows:

==> sum (iter 4 (/3) 1)

The map function allows you to apply a function to every member of a list. For instance, the following expression doubles each value in the list [1..5]:

==> map (2*) [1..5]

Lists can also be filtered with a given predicate:

==> filter (>=3) [1..5]

The scanl operation allows you to compute all the sums of initial segments of a list (or accumulate any other binary operation over a list):

==> scanl (+) 0 [1..5]

(Like foldl, scanl also has a sibling called scanr which collects results from right to left rather than left to right, starting at the end of the list.)

You can partition a list into an initial part with a given length and the remaining part of the list with take and drop:

==> take 3 [1..5]; drop 3 [1..5]

The takewhile and dropwhile functions have a similar effect, but partition their input list according to a given predicate:

==> takewhile (<=3) [1..5]; dropwhile (<=3) [1..5]

Another useful list operation is zip which collects pairs of corresponding elements in two input lists:

==> zip [1..5] ["a".."e"]

The effect of zip can be undone with unzip which returns a pair of lists:

==> unzip _

The zipwith function is a generalized version of zip which combines corresponding members from two lists using a given binary function:

==> zipwith (*) [1..5] [1..5]

All the operations described above - and many others - are provided by the stdlib.q script. It is instructive to take a look at this script and see how the operations are defined.

In addition, the standard library provides yet another list generating function listof which implements so-called "list comprehensions". These allow you to describe list values in much the same way as sets are commonly specified in mathematics. For instance, you can generate a list of pairs (I,J) with 1<=J<I<=5 as follows:

==> listof (I,J) (I in [1..5], J in [1..I-1])

The language provides syntactic sugar for list comprehensions so that the above example can also be written as follows:

==> [(I,J) : I in [1..5], J in [1..I-1]]

The same kind of construct also works with tuples:

==> ((I,J) : I in [1..5], J in [1..I-1])

We also have a random number generator, which is implemented by the built-in random function. Here is how we can generate a list of 5 pseudo random 32 bit integers:

==> [random : I in [1..5]]

To get random floating point values in the range [0,1] instead, we simply divide the results of random by 0xffffffff:

==> map (/0xffffffff) _

Lists can be sorted using quicksort (this one comes from sort.q):

==> qsort (<) _

As already mentioned, the Q language also provides a "lazy" list data structure called "streams". Streams are like lists but can actually be infinite because their elements are only produced "on demand". The standard library includes a module stream.q which implements a lot of useful stream operations. Most list operations carry over to streams accordingly. For instance, if we want to create a geometric series like the one generated with iter above, but we do not know how many elements will be needed in advance, we can employ the stream generation function iterate:

==> iterate (/3) 1
{1|iterate (/3) ((/3) 1)}

The {|} stream constructor works in the same way as the [|] list constructor, but is "special" in that it does not evaluate its arguments, i.e., the head and the tail of the stream. (Otherwise the call to iterate would never terminate.)

To get all members of the series we can apply the scanl function:

==> def S = scanl (+) 0 _; S
{0|scanl (+) (0+1) (iterate (/3) ((/3) 1))}

Now we can extract any number of initial values of the series using the take operation, and convert the resulting stream to an ordinary list with the list function. For instance, if we are interested in the first five values of the series, we proceed as follows:

==> list (take 5 S)

Note that the stream S is really infinite; if we want, we can extract any value in the series:

==> S!9999

Let's see how many iterations are actually required to reach the limit 1.5 with an error of at most 1e-15:

==> #takewhile (>1e-15) (map (1.5-) S)

This means that the sum of the first 31 series terms is needed to get an accuracy of 15 digits (which is the best that we can hope for with 64 bit floating point values). We can readily verify this using iter:

==> sum (iter 31 (/3) 1)

The standard library also implements stream enumerations and comprehensions which work like the corresponding list and tuple constructs but are evaluated in a lazy fashion, and the interpreter provides syntactic sugar for these. In particular, the notation {X..} or {X,Y..} can be used to specify an infinite arithmetic sequence. For instance, here is another way to define the infinite geometric series from above:

==> def S = scanl (+) 0 {1/3^N : N in {0..}}

==> list (take 5 S)

It is worth noting here that streams are just one simple example of a lazy data structure, which happen to play an important role in many useful algorithms so that it makes sense to provide them as a builtin. The Q language makes it rather easy to define your own lazy data structures using so-called special forms. In fact, special forms provide a uniform means to define both lazy data constructors and macro-like functions which take their arguments unevaluated, employing "call by name" parameter passing. This will be discussed in detail in Special Forms.

Special forms are an integrated part of Q which adds a great amount of expressive power to the language. In particular, they allow specialized constructs such as a "short-circuit" conditional expressions to be defined in the standard library, just like "ordinary" functions, rather than having to provide them as builtins. One example is the conditional expression if X then Y else Z which is nothing but syntactic sugar for the standard library function ifelse:

==> ifelse (5>0) "positive" "negative"

==> if 5>0 then "positive" else "negative"

Another special form which is useful in many situations is lambda (as of Q 7.1, this one is actually provided as a builtin), which allows you to create little "anonymous" functions on the fly. The customary notation \X.Y is provided as a shorthand for lambda X Y. These constructs are also called lambda abstractions, or simply lambdas. For instance, the following interpreter command defines the well-known factorial function using a lambda abstraction and assigns the resulting function object to the variable fac:

==> var fac = \N.if N>0 then N*fac (N-1) else 1

==> fac
\X1 . if X1>0 then X1*fac (X1-1) else 1

==> map fac [1..10]

Lambdas taking multiple arguments can be created like this:

==> \X Y.(1-X)*Y
\X1 . \X2 . (1-X1)*X2

==> _ 0.9 0.5

The lambda arguments can also be patterns to be matched against the actual parameters. For instance:

==> \(X,Y).(1-X)*Y
\(X1,X2) . (1-X1)*X2

==> _ (0.9,0.5)

Besides string and list processing functions and general utility functions of the kind sketched out above, the standard library also includes a collection of efficient "container" data structures which are useful in many applications. For instance, so-called hashed dictionaries (also known as "hashes" or "associative arrays") are implemented by the hdict.q script. The following expressions show how to create a dictionary object from a list, and how to access this dictionary using the index (!), keys and vals operations. (Note that here we have to load the hdict.q module explicitly, as it is not automatically included via the prelude. We could also import the stdtypes.q module instead, to get hold of all the container types.)

==> import hdict

==> def D = hdict [(foo,99),(bar,-1),(gnu,foo)]; D!gnu

==> keys D

==> vals D

This completes our little tour through the standard library. To find out more, please refer to The Standard Library, or take a look at the scripts themselves.

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2.3 Writing a Script

Now that we have taken the first big hurdle and are confident that we can actually get the Q interpreter to evaluate us some expressions, let us try to define a new function for use in the interpreter. That is, we are going to write our first own Q "program" or "script".

In the Q language, there are actually two different ways to create new functions. As we have already seen in the previous sections, little "anonymous" functions are often computed on the fly by deriving them from existing functions using partial applications or lambda abstractions. The other way, which is often more convenient when the functions to be defined get more complicated, is the definition of named functions by the means of "equations", which works in a way similar to algebraic definitions in mathematics. We will discuss how to do this in the following.

Having exited the Q interpreter, invoke your favourite text editor and enter the following simple script which implements the factorial function:

fac N                   = N*fac(N-1) if N>0;
                        = 1 otherwise;

(Note that in order to define functions equationally, you really have to write a script file. For technical reasons there currently is no way to enter an equational definition directly in the interpreter.)

Save the script in the file fac.q and then restart the interpreter as follows (here and in the following `$' denotes the command shell prompt):

$ q fac.q

You can also edit the script from within the interpreter (using vi or the editor named by the EDITOR environment variable) and then restart the interpreter with the new script, as follows:

==> edit fac.q

==> run fac.q

In any case, now the interpreter should know about the definition of fac and we can use it like any of the built-in operations:

==> map fac [1..10]

For instance, what is the number of 30-element subsets of a 100-element set?

==> fac 100 div (fac 30*fac 70)

As another example, let us write down Newton's algorithm for computing roots of a function. Type in the following script and save it to the file newton.q:

newton DX DY F          = until (satis DY F) (improve DX F);
satis DY F X            = abs (F X) < DY;
improve DX F X          = X - F X / derive DX F X;
derive DX F X           = (F (X+DX) - F X) / DX;

Restart the interpreter with the newton.q script and try the following. Note that the target function here is \Y.Y^3-X which becomes zero when Y equals the cube root of X.

==> var eps = .0001, cubrt = \X.newton eps eps (\Y.Y^3-X) X

==> cubrt 8

Well, this is not that precise, but we can do better:

==> def eps = 1e-12

==> cubrt 8

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2.4 Definitions

As mentioned in the previous section, in the Q language function definitions take the form of equations which specify how a given expression pattern, the left-hand side of the equation, is transformed into a new expression, the right-hand side of the equation. In this section we elaborate on this some more to give you an idea about how these equational definitions work. Our explanations will necessarily be a bit sketchy at this point, but the concepts we briefly touch on in this section will be explained in much more detail later.

The left-hand side of an equation may introduce "variables" (denoted by capitalized identifiers, as in Prolog) which play the role of formal parameters in conventional programming languages. For instance, the following equation defines a function sqr which squares its (numeric) argument:

sqr X                   = X*X;

An equation may also include a condition part, as in the definition of the factorial function from the previous section:

fac N                   = N*fac(N-1) if N>0;
                        = 1 otherwise;

As indicated above, several equations for the same left-hand side can be factored to the left, omitting repetition of the left-hand side. Furthermore, the otherwise keyword may be used to denote the "default" alternative.

The left-hand side of an equation may actually be an arbitrary compound expression pattern to be matched in the expression to be evaluated. For instance, as we have already seen, the expressions [] and [X|Xs] denote, respectively, the empty list and a list starting with head element X and continuing with a list Xs of remaining elements, just as in Prolog. We can use these patterns to define a function add which adds up the members of a list as follows:

add []                  = 0;
add [X|Xs]              = X+add Xs;

Thus no explicit operations for "extracting" the members from a compound data object such as a list are required; the components are simply retrieved by "pattern matching". This works in variable definitions as well. For instance, in the interpreter you can bind variables to the head element and the tail of a list using a single definition as follows:

==> def [X|Xs] = [1,2,3]; X; Xs

Pattern matching also works if the arguments take the form of function applications. For instance, the following definition implements an insertion operation on binary trees:

insert nil Y            = bin Y nil nil;
insert (bin X T1 T2) Y  = bin Y T1 T2 if X=Y;
                        = bin X (insert T1 Y) T2 if X>Y;
                        = bin X T1 (insert T2 Y) if X<Y;

Note the symbols nil and bin which act as "constructors" for the binary tree data structure in this example. (Q actually has no rigid distinction between "function applications" and "data constructors". This is unnecessary since function applications can be "values" in their own right, as discussed below.)

The Q interpreter treats equations as "rewrite rules" which are used to reduce expressions to "normal forms". Evaluation normally uses the standard "leftmost-innermost" rule, also known as "applicative order" or "call by value" evaluation. Using this strategy, expressions are evaluated from left to right, innermost expressions first; thus the arguments of a function application (and also the function object itself) are evaluated before the function is applied to its arguments.

An expression is in normal form if it cannot be evaluated any further by applying matching equations (including some predefined rules which are used to implement the built-in operations, see Equations and Expression Evaluation). A normal form expression simply stands for itself, and constitutes a "value" in the Q language. The built-in objects of the Q language (such as numbers and strings) are always in normal form, but also compound objects like lists (given that their elements are in normal form) and any other symbol or function application which does not match a built-in or user-defined equation.

This means that Q is essentially an "exception-free" language. That is, an "error condition" such as "wrong argument type" does not raise an exception by default, but simply causes the offending expression to be returned "as is". For instance:

==> hd []
hd []

==> sin (X+1)
sin (X+1)

You may be disconcerted by the fact that expressions like hd [] and sin (X+1) actually denote legal values in the Q language. However, this is one of Q's key features as a term rewriting programming language, and it adds a great amount of flexibility to the language. Most importantly, it allows expressions, even expressions involving variables, to be evaluated in a symbolic fashion. For instance, you might add rules for algebraic identities and symbolic differentiation to your script and have the interpreter simplify expressions according to these rules. On the other hand, the Q language also allows you to refine the definition of any built-in or user-defined operation and thus it is a simple matter to add an "error rule" like the following to your script when needed:

hd []                   = error "hd: empty list";

(The error function is defined in the standard library script error.q, see Diagnostics and Error Messages.) In order to implement more elaborate error handling, you can also use Q's built-in throw and catch functions to raise and respond to exceptions on certain error conditions. This is discussed in Exception Handling.

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2.5 Runtime Errors

There are a few error conditions which may cause the Q interpreter to abort the evaluation with a runtime error message. A runtime error arises when the interpreter runs out of memory or stack space, when the user aborts an evaluation with the interrupt key (usually Ctl-C), and when the condition part of an equation does not evaluate to a truth value (true or false). (All these error conditions can also be handled by the executing script, see Exception Handling.)

You can use the break on command to tell the interpreter that it should fire up the debugger when one of the latter two conditions arises. After Ctl-C, you can then resume evaluation in the debugger, see Debugging. This is not possible if the interpreter stumbles across an invalid condition part; in this case the debugger is invoked merely to report the equation which caused the error, and to inspect the call chain of the offending rule. Hitting the carriage return key will return you to the interpreter's evaluation loop.

For instance, reload the fac.q script from Writing a Script:

fac N                   = N*fac(N-1) if N>0;
                        = 1 otherwise;

Now enable the debugger with the break command and type some "garbage" like fac fac:

==> break on; fac fac
! Error in conditional
  0>  fac.q, line 1: fac fac  ==>  fac*fac (fac-1) if fac>0
(type ? for help)

What happened? Well, the debugger informs us that the error occurred when the interpreter tried to apply the first equation,

fac N                   = N*fac(N-1) if N>0;

to the expression fac fac. In fact this is no big surprise since we cannot expect the interpreter to know how to compare a symbol, fac, with a number, 0, which it tried when evaluating the condition part of the above equation. So let us return to the interpreter's prompt by hitting the carriage return key:

: <CR>


The interpreter now waits for us to type the next expression. (For a summary of debugger commands please refer to Debugging. You can also type ? or help while in the debugger to obtain a list of available commands.)

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