Modular arithmetic
Modular arithmetic is a form of arithmetic (a calculation technique involving the concepts of addition and multiplication) which is done on numbers with a defined congruence. This means that two numbers and are considered congruent whenever there exists an integer such that:
is called the congruence modulus. The corresponding set of integers is called the ring, where each element is uniquely represented by the remainder of its euclidean division by . Addition and multiplication on this ring have the same algebraic structure, so that a function such as a polynomial expression could receive a ring element as argument and give a consistent result.
The purpose of this task is to show, if your programming language allows it, how to redefine operators so that they can be used transparently on modular integers. You can do it either by using a dedicated library, or by implementing your own class.
You will use the following function for demonstration:
You will use as the congruence modulus and you will compute .
It is important that the function is agnostic about whether or not its argument is modular; it should behave the same way with normal and modular integers. In other words, the function is an algebraic expression that could be used with any ring, not just integers.
PARI/GP
This feature exists natively in GP: <lang parigp>Mod(3,7)+Mod(4,7)</lang>
Perl
There is a CPAN module called Math::ModInt which does the job. <lang Perl>use Math::ModInt qw(mod); sub f { my $x = shift; $x**100 + $x + 1 }; print f mod(10, 13);</lang>
- Output:
mod(1, 13)
Perl 6
There is a Panda module called Modular which works basically as Perl 5's Math::ModInt. <lang Perl 6>use Modular; sub f(\x) { x**100 + x + 1}; say f( 10 Mod 13 )</lang>
- Output:
1 「mod 13」
Prolog
Works with SWI-Prolog versin 6.4.1 and module lambda (found there : http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl ). <lang Prolog>:- use_module(library(lambda)).
congruence(Congruence, In, Fun, Out) :- maplist(Congruence +\X^Y^(Y is X mod Congruence), In, In1), call(Fun, In1, Out1), maplist(Congruence +\X^Y^(Y is X mod Congruence), Out1, Out).
fun_1([X], [Y]) :- Y is X^100 + X + 1.
fun_2(L, [R]) :- sum_list(L, R). </lang> Output :
?- congruence(13, [10], fun_1, R). R = [1]. ?- congruence(13, [10, 15, 13, 9, 22], fun_2, R). R = [4]. ?- congruence(13, [10, 15, 13, 9, 22], maplist(\X^Y^(Y is X * 13)), R). R = [0,0,0,0,0].
Racket
<lang racket>#lang racket (require racket/require
;; grab all "mod*" names, but get them without the "mod", so
;; `+' and `expt' is actually `mod+' and `modexpt'
(filtered-in (λ(n) (and (regexp-match? #rx"^mod" n)
(regexp-replace #rx"^mod" n "")))
math)
(only-in math with-modulus))
(define (f x) (+ (expt x 100) x 1)) (with-modulus 13 (f 10))
- => 1</lang>
Ruby
<lang ruby># stripped version of Andrea Fazzi's submission to Ruby Quiz #179
class Modulo
include Comparable
def initialize(n = 0, m = 13) @n, @m = n % m, m end
def to_i @n end def <=>(other_n) @n <=> other_n.to_i end
[:+, :-, :*, :**].each do |meth|
define_method(meth) { |other_n| Modulo.new(@n.send(meth, other_n.to_i), @m) }
end
def coerce(numeric) [numeric, @n] end
end
- Demo
x, y = Modulo.new(10), Modulo.new(20)
p x > y # true p x == y # false p [x,y].sort #[#<Modulo:0x000000012ae0f8 @n=7, @m=13>, #<Modulo:0x000000012ae148 @n=10, @m=13>] p x + y ##<Modulo:0x0000000117e110 @n=4, @m=13> p 2 + y # 9 p y + 2 ##<Modulo:0x00000000ad1d30 @n=9, @m=13>
p x**100 + x +1 ##<Modulo:0x00000000ad1998 @n=1, @m=13> </lang>
Tcl
Tcl does not permit overriding of operators, but does not force an expression to be evaluated as a standard expression. Creating a parser and custom evaluation engine is relatively straight-forward, as is shown here.
<lang tcl>package require Tcl 8.6 package require pt::pgen
- A simple expression parser for a subset of Tcl's expression language
- Define the grammar of expressions that we want to handle
set grammar { PEG Calculator (Expression)
Expression <- Term (' '* AddOp ' '* Term)* ;
Term <- Factor (' '* MulOp ' '* Factor)* ;
Fragment <- '(' ' '* Expression ' '* ')' / Number / Var ;
Factor <- Fragment (' '* PowOp ' '* Fragment)* ;
Number <- Sign? Digit+ ;
Var <- '$' ( 'x'/'y'/'z' ) ;
Digit <- '0'/'1'/'2'/'3'/'4'/'5'/'6'/'7'/'8'/'9' ; Sign <- '-' / '+' ; MulOp <- '*' / '/' ; AddOp <- '+' / '-' ; PowOp <- '**' ;
END; }
- Instantiate the parser class
catch [pt::pgen peg $grammar snit -class Calculator -name Grammar]
- An engine that compiles an expression into Tcl code
oo::class create CompileAST {
variable sourcecode opns
constructor {semantics} {
set opns $semantics
}
method compile {script} {
# Instantiate the parser set c [Calculator] set sourcecode $script try { return [my {*}[$c parset $script]] } finally { $c destroy }
}
method Expression-Empty args {}
method Expression-Compound {from to args} {
foreach {o p} [list Expression-Empty {*}$args] { set o [my {*}$o]; set p [my {*}$p] set v [expr {$o ne "" ? "$o \[$v\] \[$p\]" : $p}] } return $v
} forward Expression my Expression-Compound forward Term my Expression-Compound forward Factor my Expression-Compound forward Fragment my Expression-Compound
method Expression-Operator {from to args} {
list ${opns} [string range $sourcecode $from $to]
} forward AddOp my Expression-Operator forward MulOp my Expression-Operator forward PowOp my Expression-Operator
method Number {from to args} {
list ${opns} value [string range $sourcecode $from $to]
}
method Var {from to args} {
list ${opns} variable [string range $sourcecode [expr {$from+1}] $to]
}
}</lang> None of the code above knows about modular arithmetic at all, or indeed about actual expression evaluation. Now we define the semantics that we want to actually use. <lang tcl># The semantic evaluation engine; this is the part that knows mod arithmetic oo::class create ModEval {
variable mod
constructor {modulo} {set mod $modulo}
method value {literal} {return [expr {$literal}]}
method variable {name} {return [expr {[set ::$name]}]}
method + {a b} {return [expr {($a + $b) % $mod}]}
method - {a b} {return [expr {($a - $b) % $mod}]}
method * {a b} {return [expr {($a * $b) % $mod}]}
method / {a b} {return [expr {($a / $b) % $mod}]}
method ** {a b} {
# Tcl supports bignums natively, so we use the naive version return [expr {($a ** $b) % $mod}]
} export + - * / **
}
- Put all the pieces together
set comp [CompileAST new [ModEval create mod13 13]]</lang> Finally, demonstrating… <lang tcl>set compiled [$comp compile {$x**100 + $x + 1}] set x 10 puts "[eval $compiled] = $compiled"</lang>
- Output:
1 = ::mod13 + [::mod13 + [::mod13 ** [::mod13 variable x] [::mod13 value 100]] [::mod13 variable x]] [::mod13 value 1]