Introduction
The mCRL2 language describes processes with data. The Data Library contains everything that has to do with the data part of the language. The main concepts are sorts and functions working upon these sorts. The meaning of these functions can be described by means of equational axioms. In the __mcrl2_language_reference__ these concepts are explained in more detail.
Data specifications
Data specifications contains the declaration of data types. It contains sorts, sort aliases, constructors, mappings and conditional equations.
A specification can straightforwardly be constructed by declaring the required objects and adding them to a specification. The elements that are added to the specification are not checked.
#include "mcrl2/data/data_specification.h"
#include "mcrl2/data/basic_sort.h"
#include "mcrl2/data/function_symbol.h"
#include "mcrl2/data/variable.h"
#include "mcrl2/data/data_equation.h"
using namespace mcrl2::data;
data_specification spec;
basic_sort D("D"); // sort D
spec.add_sort(D);
function_symbol m("m",D); // map m:D;
function_symbol c1("c1",D); // cons c1:D;
function_symbol c2("c2",function_sort(D,D)); // c2:D>D;
spec.add_mapping(m);
spec.add_constructor(c1);
spec.add_constructor(c2);
variable x("x", D); // var x:D;
data_application e1(c2,x);
data_equation e(variable_list({x}), sort_bool::true_(), e1, m); // eqn true > c1(x)=m;
spec.add_equation(e);
basic_sort E("E"); // sort E=D; (add a sort alias)
spec.add_alias(E,D);
For any specification it is possible to retrieve the elements that have been added to the specification as follows:
sorts_const_range s=spec.user_defined_sorts();
constructors_const_range c=spec.user_defined_constructors();
mappings_const_range m=spec.user_defined_mappings();
equations_const_range e=spec.user_defined_equations();
ltr_aliases_mapping a=spec.user_defined_aliases();
If sorts are added to a data specification, automatically certain functions, mappings and in some cases even derived sorts are added to the specification. For every sort D, there functions if then else (if:Bool#D#D>D), equality (==:D#D>Bool), inequality (!=:D#D>Bool) and inequalities (<,<=,>=,>:D#D>Bool) are added. For structured sorts (e.g. sort Tree=struct leaf(Nat)  node(Nat,Nat)) the constructors, projection functions and recognizers are added to the specification. For container sorts (List(D), Set(D) and Bag(D) for arbitrary sort D) all standard functions for these sorts are also added automatically to the specification. The same holds for standard sorts Bool (booleans), Pos (Positive numbers), Nat (Natural numbers), Int (Integers) and Real (Real numbers).
Standards sorts cannot explicitly be added to a specification. In order to indicate that certain sorts must be present in a data specification, these must be added explicitly to the context sorts. The sort Bool is always present in a specification. Sorts that occur in other sorts, constructors and mappings are automatically defined. For instance, if the sort Real and its operations should be made available in a specification, it is sufficient to add Real to the context sorts as follows:
add_context_sort(sort_real::real());
The function context_sorts gives a list of sorts added to the context.
In order to retrieve all sorts, constructors, mappings or equations in a specification (including those that are automatically generated), there are functions listed below. As generally these functions are of interest, instead of their counterparts which only define the user_defined elements, they have the shorter and more natural names.
sorts_const_range s=spec.sorts();
constructors_const_range c=spec.constructors();
mappings_const_range m=spec.mappings();
equations_const_range e=spec.equations();
ltr_aliases_mapping a=spec.aliases();
When adding sort aliases to a specification, the names of sorts are not unique anymore. When declaring in mCRL2
sort Time=Nat;
D=list(Nat);
E=struct f(E)g;
the sorts Time and Nat, as well as D and list(Nat) and even E and struct f(E)g are pairwise equal. In a specification it is not very inefficient to have different names for equal sorts. Therefore the sorts in a specification are made unique. The algorithm that is used maps every structured and container sort for which an alias is introduced to the sort alias at the left hand side. Every sort alias between basic sorts is mapped to the right hand side. In the example above every occurrence of Time is replaced by Nat, and occurrences of list(Nat) and struct f(E)g are replaced by D and E respectively.
The method sort_alias_map() delivers a mapping from sorts to sorts giving for each sort the unique name. Using the function template <class T> T normalise_sorts(T t) the sorts in each object t can be renamed to their unique representation. If this is not done, objects can be equal except for their types, and this will not be recognized. This is particularly problematic when using the rewriter. Eg. in the following process specification
sort Time=Nat;
map too_late:Time>Bool;
var t:Time;
eqn too_late(t) = t>10;
proc P(u:Time)=too_late(u) > a.delta;
init P(9);
the data specification will normalise the equation too_late(t)= t>10 to such that t has sort Nat as all occurrences of the sort Time are replaced by Nat. When this is not done in process P, the parameter u still has sort Time and too_late(u) will not be rewritten as the sorts do not match. Therefore, it is necessary to apply normalise_sorts to any object used in the context of a specification. If sort aliases are added to a data specification, all sorts in the context of this specification must be renormalised.
There are a few utility functions that help to determine the nature of sorts. The function bool is_certainly_finite(const sort_expression) indicates that a sort has a finite number of elements. This is in general an undecidable property, but in certain cases it can be determined that there are at most a finite number of elements in a sort.
The function bool is_constructor_sort(const sort_expression s) indicates whether there is a constructor with target sort s. If so, the sort is called a constructor sort.
Expressions
In this section we first introduce the basic structures of sort expressions
and data expressions. We then continue to defining the sort expressions
with operations that are predefined in the Data Library.
The code in the Data Library is inside the namespace mcrl2::data
.
Sort expressions
Except for the untyped identifiers, all expressions in the Data Library are typed. There are many different kinds of sorts in the mCRL2 language, all of which can be represented in the data library.
Type 
Meaning 

basic_sort 
basic sort 
function_sort 
function sort 
structured_sort 
structured sort 
container_sort 
container sort 
multiple_possible_sorts 
expression matching any of multiple sorts 
unknown_sort 
unknown sort expression 
Warning
The types multiple_possible_sorts
and unknown_sort
should not occur
after type checking.
These sort expressions correspond to the grammar
S ::= Sb  Sc  S x … x S > S  Sstruct,
where Sb
is a given set of basic sorts, always including the booleans
(sort Bool
). S x … x S > S denotes the function sorts, where > is right
associative. Sc
is the set of container sorts, and Sstruct is the set of
structured sorts.
The set of container sorts Sc
is defined as follows.
Sc ::= List(S)  Set(S)  FSet(S)  Bag(S)  FBag(S)
Where FSet(S)
and FBag(S)
represent finite sets and finite bags
respectively.
The syntax of structured sorts Sstruct is defined as follows (where p is a string):
Sstruct ::= p(proj*)?p
in which proj has the following syntax:
proj ::= S  p:S
In general, structured sorts have the following form (with n
a positive number,
ki
a natural number with 1 <= i <= n
):
struct c1(pr1,1:S1,1, ..., pr1,k1:S1,k1)?is_c1 
c2(pr2,1:S2,1, ..., pr2,k2:S2,k2)?is_c2 
...
cn(prn,1:Sn,1, ..., prn,kn:Sn,kn)?is_cn;
We refer to ci
as the constructors of the structured sort. Si,j
are the
sorts of the arguments of the constructors. pri,j
are names for optional
projection functions, retrieving the corresponding argument for a constructor.
is_ci
are the names of optional recognizer functions, returning a boolean
value.
As an example of some of the introduced concepts, consider the following code snippet that constructs a structured sort
struct c1(p0:S0, S1)?is_c1 
c2(p0:S0);
The construction of this structured sort is as follows, assuming that also all of the subexpressions still need to be defined:
basic_sort s0("S0"); /* Name for the sort S0 */
basic_sort s1("S1"); /* Name for the sort S1 */
structured_sort_constructor_argument p0(s0, "p0"); /* Constructor argument p0: S0 */
structured_sort_constructor_argument p1(s1); /* Constructor argument S1 */
structured_sort_constructor_argument_vector a1; /* p0: S0, S1 */
a1.push_back(p0);
a1.push_back(p1);
structured_sort_constructor_argument_vector a2; /* p0 */
a2.push_back(p0);
structured_sort_constructor c1("c1", a1, "is_c1"); /* c1(p0:S0, S1)?is_c1 */
structured_sort_constructor c2("c2", a2); /* c2(p0:S0) */
structured_sort_constructor_vector cs; /* c1(p0:S0, S1)?is_c1  c2(p0:S0) */
cs.push_back(c1);
cs.push_back(c2);
structured_sort s(cs); /* struct c1(p0:S0, S1)?is_c1  c2(p0:S0) */
Data expressions
The class data_expression
represents expressions like true
,
[^x > 3] and [^forall n:Nat. f(n) < 5]. Each data expression d
has a type or
sort d.sort()
of type sort_expression
.
Let’s look at a simple example
that constructs the numbers two and three, and builds the expression 2 + 3:
#include "mcrl2/data/data.h"
#include "mcrl2/atermpp/aterm_init.h"
#include <cassert>
using namespace mcrl2::data;
int main(int argc, char* argv[])
{
MCRL2_ATERMPP_INIT(argc, argv)
data_expression two = sort_nat::nat(2);
data_expression three = sort_nat::nat(3);
data_expression five = sort_nat::plus(two, three);
assert(five.sort() == sort_nat::nat());
return 0;
}
Expression 
Meaning 

data_expression 
any data expression 
function_symbol 
function symbol 
variable 
variable 
abstraction 
expression with variable binding 
lambda 
lambda abstraction 
forall 
universal quantification 
exists 
existential quantification 
where_clause 
where clause 
application 
function application 
identifier 
untyped identifier (not to be used after type checking) 
Warning
The expression identifier
should not occur after type checking, as it entails an
untyped sort expression, whereas all libraries and tools in the toolset in
general assume fully typed expressions.
An overview of all data expressions in the Data Library is given in the table
above. More detailed, data expressions are divided into function symbols, represented
by the class function_symbol
, variables, represented by variable
,
abstractions, represented by the class abstraction
, where clauses,
represented by where_clause
, and applications of expressions to expressions,
represented by application
. Furthermore, when used in the initial phases
of parsing and type checking, the use of untyped identifiers, represented
by identifier
is allowed.
Abstractions provide a mechanism for variable binding. As such, they are
further subdivided into lambda abstraction, represented by lambda
,
and universal and existential quantifications, represented by
forall
and exists
respectively.
More formally, data expressions e
, with sort expression S
and variable names
x
correspond to the following grammar:
 e ::= x  n  e(e, …, e)  lambda x:S, …, x:S . e 
forall x:S, …, x:S. e  exists x:S, …, x:S. e  e whr x = e, …, x = e end
Here
e(e,...,e)
denotes application of data expressions,lambda x:S, ..., x:S . e
denotes lambda abstraction,forall x:S, ..., x:S . e
andexists x:S, ..., x:S . e
denote universal and existential quantification.
Predefined sorts
The mCRL2 language has a number of predefined sorts, given in the table below:
Expression 
Sort 

sort_bool::bool_() 
booleans 
sort_pos::pos() 
positive numbers 
sort_nat::nat() 
natural numbers 
sort_int::int_() 
integers 
sort_real::real() 
real numbers 
Furthermore, a number of container sorts is predefined. Assuming that s is a sort expression, all container sorts are given in the table below:
Expression 
Type 

sort_list::list(s) 
lists 
sort_set::set_(s) 
sets 
sort_fset::fset(s) 
finite sets 
sort_bag::bag(s) 
bags 
sort_fbag::fbag(s) 
finite bags 
Note that the source code for all predefined sorts is generated from specification files.
Operations on data expressions
Default operations
For all sorts, a number of operations is available by default. The corresponding functions can be found in standard.h.
Let b
be a data expressions of sort Bool
, and let x
and y
be two data expressions with the same sort. Then the following
operations are supported:
Expression 
Syntax 
Meaning 

equal_to(x, y) 
x == y 
equality 
not_equal_to(x, y) 
x != y 
inequality 
if_(b, x, y) 
if(b,x,y) 
conditional expression 
less(x,y) 
x < y 
less than 
less_equal(x,y) 
x <= y 
less than or equal to 
greater(x,y) 
x > y 
greater than 
greater_equal(x,y) 
x >= y 
greater than or equal to 
For the predefined sorts, the most important operations are also available by default.
Note
In all definitions of operations on predefined sorts, elements of which the syntax starts with @ cannot directly be entered by the user when writing an mCRL2 specification. The @ means that the specified operation is implementation specific. Printing such an expression as feedback to the user should be prevented at all times.
Booleans
All standard operations for the Booleans are available in bool.h, and can be
found in the namespace data::sort_bool
. First of all
the two constants true
and false
can be constructed.
Expression 
Syntax 
Meaning 

true() 
true 
true 
false() 
false 
false 
Furthermore the following functions are available on Booleans (for details
about the allowed types also see bool.spec). Let b
and c
be Boolean expressions.
Expression 
Syntax 
Meaning 

not_(b,c) 
!b 
negation 
and_(b,c) 
b && c 
conjunction 
or_(b,c) 
b  c 
disjunction 
implies(b,c) 
b => c 
implication 
Positive numbers
All standard operations for positive numbers are available in pos.h, and can
be found in the namespace data::sort_pos
. The positive numbers have two
constructors, facilitating an encoding with size logarithmic in the number
that is represented.
Let b
be a Boolean expression, and p
be a positive expression.
Expression 
Syntax 
Meaning 

c1() 
@1 
1 
cdub(b,p) 
@cDub(b,p) 
2*p + b 
Furthermore the standard operations are available on Positive numbers.
Let b
and c
be Boolean expressions, and p
, q
, and r
be positive
numbers.
Expression 
Syntax 
Meaning 

max(p,q) 
max(p,q) 
maximum 
min(p,q) 
min(p,q) 
minimum 
abs(p) 
abs(p) 
absolute value 
succ(p) 
succ(p) 
successor 
plus(p,q) 
p+q 
addition 
add_with_carry(b,p,q) 
@addc(b,p,q) 
addition with carry (p + q + b) 
Natural numbers
All standard operations for natural numbers are available in nat.h, and can
be found in the namespace data::sort_nat
. The natural numbers have two
constructors, representing 0
and a positive number interpreted as a
natural number.
Let p
be a positive expression.
Expression 
Syntax 
Meaning 

c0() 
@0 
0 
cnat(p) 
@cNat(p) 
p interpreted as natural number 
Furthermore the standard operations are available on Natural numbers.
Let b
and c
be Boolean expressions, p
, q
be positive numbers,
and n
, m
, u
, v
be natural numbers.
Expression 
Syntax 
Meaning 

Pos2Nat(p) 
Pos2Nat(p) 
explicit conversion of 
Nat2Pos(n) 
Nat2Pos(n) 
explicit conversion of 
max(p,n) 
max(p,n) 
maximum 
max(n,p) 
max(n,p) 
maximum 
max(m,n) 
max(m,n) 
maximum 
min(m,n) 
min(m,n) 
minimum 
abs(n) 
abs(n) 
absolute value 
succ(n) 
succ(n) 
successor 
pred(n) 
pred(n) 
predecessor 
dub(b,n) 
@dub(b,n) 
??? 
plus(p,n) 
p+n 
addition 
plus(n,p) 
n+p 
addition 
plus(m,n) 
m+n 
addition 
gtesubtb(b,p,q) 
@gtesubtb(b,p,q) 
substraction with borrow 
times(m,n) 
m*n 
multiplication 
div(m,p) 
m div p 
integer division 
mod(m,p) 
m mod p 
modulus 
exp(p,n) 
p^n 
exponentiation 
exp(m,n) 
m^n 
exponentiation 
even(n) 
@even(n) 
predicate to indicate 
monus(m,n) 
@monus(m,n) 

swap_zero(m,n) 
@swap_zero(m,n) 
??? 
swap_zero_add(m,n,u,v) 
@swap_zero_add(m,n,u,v) 
??? 
swap_zero_min(m,n,u,v) 
@swap_zero_min(m,n,u,v) 
??? 
swap_zero_monus(m,n,u,v) 
@swap_zero_monus(m,n,u,v) 
??? 
swap_zero_lte(m,n,u,v) 
@swap_zero_lte(m,n,u,v) 
??? 
To facilitate efficient rewriting, also a sort @NatPair
is available. Code
for this is also present in nat.h
, in namespace data::sort_nat
.
Let m
, n
be expressions of sort Nat.
Expression 
Syntax 
Meaning 

cpair(m,n) 
@cPair(m,n) 
tuple (m,n) 
Also functions for these pairs are available.
Let b
be a Boolean expression, p
, q
be positive numbers,
and n
, m
, u
, v
be natural numbers.
Expression 
Syntax 
Meaning 

first(cpair(m,n)) 
@first(@cPair(m,n)) 
projection of first argument 
second(cpair(m,n)) 
@second(@cPair(m,n)) 
projection of second argument 
divmod(p,q) 
@divmod(p,q) 
simultaneous division and modulus 
gdivmod(pair(m,n), b, p) 
@gdivmod(@pair(m,n), b, p) 
generalised simultaneous division and modulus 
ggdivmod(m, n, p) 
@ggdivmod(m ,n, p) 
doubly generalised simultaneous division and modulus 
Integers
All standard operations for integers are available in int.h, and can
be found in the namespace data::sort_int
. The integers have two
constructors, one interpreting a natural number as integer, and one
interpreting a positive number as a negative integer.
Let p
be a positive expression, and n
be a natural number.
Expression 
Syntax 
Meaning 

cint(n) 
@cInt(n) 

cneg(p) 
@cNeg(p) 

Furthermore the standard operations are available on Natural numbers.
Let b
be a Boolean expression, p
, q
be positive numbers,
n
, m
be natural numbers, and x
, y
be integers.
Expression 
Syntax 
Meaning 

Nat2Int(n) 
Nat2Int(n) 
explicit conversion of 
Int2Nat(x) 
Int2Nat(x) 
explicit conversion of 
Pos2Int(p) 
Pos2Int(p) 
explicit conversion of 
Int2Pos(x) 
Int2Pos(x) 
explicit conversion of 
max(p,x) 
max(p,x) 
maximum 
max(x,p) 
max(x,p) 
maximum 
max(n,x) 
max(n,x) 
maximum 
max(x,n) 
max(x,n) 
maximum 
max(x,y) 
max(x,y) 
maximum 
min(x,y) 
min(x,y) 
minimum 
abs(x) 
abs(x) 
absolute value 
negate(p) 
p 
unary minus 
negate(n) 
n 
unary minus 
negate(x) 
x 
unary minus 
succ(x) 
succ(x) 
successor 
pred(n) 
pred(n) 
predecessor 
pred(x) 
pred(x) 
predecessor 
dub(b,x) 
@dub(b,x) 
??? 
plus(x,y) 
x+y 
addition 
minus(p,q) 
pq 
subtraction 
minus(n,m) 
nm 
subtraction 
minus(x,y) 
xy 
subtraction 
times(x,y) 
x*y 
multiplication 
div(x,p) 
x div p 
integer division 
mod(x,p) 
x mod p 
modulus 
exp(x,n) 
x^n 
exponentiation 
Real numbers
All standard operations for real numbers are available in real.h, and can
be found in the namespace data::sort_real
. The real numbers do not have
any constructors, because they cannot be finitely enumerated.
Standard functions for real are available however.
Let p
, q
be positive numbers,``n``, m
be natural numbers, x
, y
be integers,
and r
, s
be real numbers.
Expression 
Syntax 
Meaning 

Pos2Real(p) 
Pos2Real(p) 
explicit conversion of 
Nat2Real(n) 
Nat2Real(n) 
explicit conversion of 
Int2Real(x) 
Int2Real(x) 
explicit conversion of 
Real2Pos(r) 
Real2Pos(r) 
explicit conversion of 
Real2Nat(r) 
Real2Nat(r) 
explicit conversion of 
Real2Int(r) 
Real2Int(r) 
explicit conversion of 
max(r,s) 
max(r,s) 
maximum 
min(r,s) 
min(r,s) 
minimum 
abs(r) 
abs(r) 
absolute value 
negate(r) 
r 
unary minus 
succ(r) 
succ(r) 
successor 
pred(r) 
pred(r) 
predecessor 
plus(r,s) 
r+s 
addition 
minus(r,s) 
rs 
subtraction 
times(r,s) 
r*s 
multiplication 
divides(p,q) 
p / q 
division 
divides(m,n) 
m / n 
division 
divides(x,y) 
x / y 
division 
divides(r,s) 
r / s 
division 
floor(r) 
floor(r) 
floor 
ceil(r) 
ceil(r) 
ceil 
round(r) 
round(r) 
round 
redfrac(x,y) 
@redfrac(x,y) 
reduce fraction x/y w.r.t. lowest common multiple 
redfracwhr(p,x,n) 
@redfracwhr(p,x,n) 
??? 
redfrachlp(r,x) 
@redfrachlp(r,x) 
??? 
Important
The sorts that are allowed as arguments to the functions for numeric sorts
are exactly the ones that correspond to the sorts of the variables in the tables
with functions. Note that e.g. sort_real::max(p,q) is also allowed, and the
correct result sort of Pos
will automatically be inferred.
Lists
All standard operations for lists are available in list.h, and can be found in the namespace data::sort_list. The lists have two constructors, the empty list ([]), and inserting an element into a list (>).
Let x be an element of sort S, and l of sort List(S).
Expression 
Syntax 
Meaning 

nil(S) 
[] 
The empty list of sort S 
cons(S,x,l) 
x > l 
The list l prefixed with x 
Also, the following functions operating on lists are available. Again, let x be an element of sort S, l of sort List(S), and n of sort Nat.
Expression 
Syntax 
Meaning 

in(S,x,l) 
x in l 
Test whether 
count(S,l) 
#l 
The size of 
snoc(S,l,x) 
l < x 
The list 
element_at(S,l,n) 
l.n 
The element at position 
head(S,l) 
head(l) 
The first element of 
tail(S,l) 
tail(l) 

rhead(S,l) 
rhead(l) 
The last element of 
rtail(S,l) 
rtail(l) 

Finite sets
The finite sets quite closely resemble lists. For sort FSet(S)
the following
constructors are available, assuming a sort S
, an element x
of sort S
, and
t
being of sort FSet(S)
.
Expression 
Syntax 
Meaning 

fset_empty(S) 
@fset_empty() 
The empty finite set of sort 
fset_cons(S,x,t) 
@fset_cons 
The finite set 
Let b
be a Boolean, x
an element of sort S
, f
, g
be functions of sort
S > Bool
, and s
, t
be of sort FSet(S)
. The operations of finite
sets are defined as follows.
Expression 
Syntax 
Meaning 

insert(S,x,s) 
@fsetinsert(x,s) 
Insert 
cinsert(S,x,b,s) 
@fsetinsert(x,b,s) 
??? 
in(S,x,s) 
@fsetin(x,s) 
Test whether 
lte(S,f,s,t) 
@fsetlte(f,s,t) 

union(S,f,g,s,t) 
@fsetunion(f,g,s,t) 
Union of 
intersection(S,f,g,s,t) 
@fsetinter(f,g,s,t) 
Intersection of 
Finite bags
Finite bags are defined in a similar vein as finite sets.
For sort FBag(S)
the following
constructors are available, assuming a sort S
, an element x
of sort S
,
p
being a positive number, and
b
being of sort FBag(S)
.
Expression 
Syntax 
Meaning 

fbag_empty(S) 
@fbag_empty() 
The empty finite bag of sort 
fbag_cons(S,x,p,b) 
@bag_cons(x,p,b) 
The finite bag 
Let x
an element of sort S
, f
,``g`` be functions of sort
S > Nat
, t
of sort FSet(S)
, and b
,``c`` be elements of sort FBag(S)
.
The operations on finite bags are defined as follows.
Expression 
Syntax 
Meaning 

fbaginsert(S,x,p,b) 
@fbag_insert(x,p,b) 
Insert 
fsetcinsert(S,x,n,b) 
@fbag_cinsert(x,n,b) 
??? 
fbagcount(S,x,b) 
@fbag_count(x,b) 
Test count the number of occurrences of 
fbagin(S,x,b) 
@fbag_in(x,b) 
Test whether 
fbaglte(S,f,b,c) 
@fbag_lte(f,b,c) 

fbagjoin(S,f,g,b,c) 
@fbag_join(f,g,b,c) 
Join of 
fbagintersect(S,f,g,b,c) 
@fbag_inter(f,g,b,c) 
Intersection of 
fbagdifference(S,f,g,b,c) 
@fbag_diff(f,g,b,c) 
Difference of 
fset2fbag(S,t) 
@fset2fbag(t) 
Convert 
Sets
Like the Real numbers, sets and bags do not have constructors. This means that elements of these sorts are built using functions, as well as their more simple counterparts, the finite sets and bags.
For sets the following functions are available. Let d
, e
be of sort Set(S),
x
be of sort S
, s
be of sort FSet(S)
, and f
and g
be function of
sort S > Bool
.
Expression 
Syntax 
Meaning 

setconstructor(S, f, s) 
@set(f,s) 
Construct a set from a function and a finite set 
emptyset(S) 
{} 
Empty set of sort S 
setfset(S, s) 
@setfset(s) 
Interpret finite set s as a set 
setcomprehension(S, f) 
@setcomp(f) 
The set of all elements of sort 
in(S,x,d) 
x in d 
Test whether 
setcomplement(S,d) 
!d 
Set complement of 
setunion_(S,d,e) 
d + e 
Union of 
setintersection(S,d,e) 
d * e 
Intersection of 
setdifference(S,d,e) 
d  e 
Difference of 
false_function(S) 
@false_ 
The constant function returning false 
true_function(S) 
@true_ 
The constant function returning true 
false_function(S) 
@false_ 
The constant function returning false 
not_function(S,f) 
@not_(f) 
The function returning 
and_function(S,f,g) 
@and_(f,g) 
The function returning 
or_function(S,f,g) 
@or_(f,g) 
The function returning 
Note that the *_function operations are used as implementation details for representing sets.
Bags
For bags the following functions are available. Let b
, c
be of sort FBag(S)
,
e
of sort S
, f
,``g``, of sort S > Nat
, h
of sort S > Bool
, s
of sort
FSet(S)
, and x
,``y`` of sort Bag(S)
.
Expression 
Syntax 
Meaning 

bagconstructor(S, f, b) 
@bag(f,b) 
Construct a bag from a function and a finite bag 
emptybag(S) 
{} 
Empty bag of sort S 
bagfbag(S, b) 
@bagfset(b) 
Interpret finite bag 
bagcomprehension(S, f) 
@bagcomp(f) 
The bag of all elements of sort 
bagcount(S,e,x) 
count(e,x) 
The number of occurrences of 
bagin(S,e,x) 
in(e,x) 
Determine whether 
bagjoin(S,x,y) 
x + y 
Join of 
bagintersect(S,x,y) 
x * y 
Intersection of 
bagdifference(S,x,y) 
x  y 
Difference of 
bag2set(S,x) 
Bag2Set(x) 
Convert bag 
set2bag(S,t) 
Set2Bag(t) 
Convert set 
zero_function(S) 
@zero_ 
The constant function returning 
one_function(S) 
@one_ 
The constant function returning 
add_function(S,f,g) 
@add_(f,g) 
The function returning 
min_function(S,f,g) 
@min_(f,g) 
The function returning 
monus_function(S,f,g) 
@monus_(f,g) 
The function returning 
nat2bool_function(S,f) 
@Nat2Bool_(f) 
The function returning false if 
bool2nat_function(S,h) 
@Bool2Nat_(h) 
The function returning 
Note that, like for sets, the *_function operations are used as implementation details for representing bags.
Creating data expressions
Data expressions can be created in two ways: directly using constructors, or using a parser.
Constructing data expressions directly can be quite tedious:
basic_sort X("X");
basic_sort Y("Y");
basic_sort Z("Z");
sort_expression XYZ = function_sort(function_sort(X, Y), Z);
variable x("x", XYZ);
variable three("3", sort_pos::pos());
variable zero("0", sort_nat::nat());
For convenience a function parse_data_expression is available. This function takes a variable declaration as optional second argument, that can be used to specify unbound variables that appear in the expression. An example of this is:
#include "mcrl2/data/parse.h"
#include "mcrl2/data/pos.h"
#include "mcrl2/data/nat.h"
int main(int argc, char* argv[])
{
// two ways to create the expression m + n
std::string var_decl = "m, n: Pos;\n";
data_expression d1 = parse_data_expression("m+n", var_decl);
variable m = parse_data_expression("m", var_decl);
variable n = parse_data_expression("n", var_decl);
data_expression d2 = sort_pos::plus(m, n);
return 0;
}
Sort aliases and sort normalisation
Sort aliases are used to give an alternative name to a sort or a sort expression. Typical examples are (expressed in MCRL2):
sort Time=Nat;
L=List(List(Bool));
Tree=struct leaf(Nat)  node(Tree,Tree);
F=Nat>List(Nat);
Sort aliases are used to give alternative names or shorthands for existing sorts. Moreover, they allow to define recursive structured sorts. Note that at the left of a sort alias there is a basic_sort, and at the right there is a sort_expression.
An alias is declared as follows:
#include "mcrl2/data/alias.h"
#include "mcrl2/data/container_sort.h"
using namespace mcrl2::data;
void alias_demo()
{
basic_sort b("sort_id");
container_sort c(bag,sort_nat::sort_nat());
alias a(b,c);
std::cout << "Alias name: " << a.name() << " Alias rhs: " << pp(a.reference()) << "\n";
}
An important consequence of the use of aliases is that different sort expressions can denote the same sort. In the example above, Time and Nat denote the same sort. So, the variables x:Time and x:Nat are the same object. It is time consuming to continuously calculate whether sorts are the same, which is undesirable if it comes to verification. Therefore, we require that all sorts in expressions that are equal modulo sort aliases are represented by a unique sort expression. This process is called sort normalisation. Note that sort normalisation is dependent on a particular specification; the sorts in one expression can be normalised differently for two different data specifications.
More concretely, for a sort alias
sort A=B;
sort A and B are equal. Sort normalisation will rewrite each sort B to A, except if B is a Bool, Pos, Nat, Int or Real. In case there are more aliases referring to the same sort, as in the example below there are more options for the unique sort.
sort A1=List(B);
A2=List(B);
In this case either A1, or A2 is chosen as the representation for List(B).
Sort normalisation takes place automatically inside a data specification. Functions, sorts, equations, etc. that are added using for instance add_equation are automatically normalised. Aliases that are added are also automatically applied to all elements in the data type. If the elements of a data type are requested, e.g. the sorts, constructors, mappings and equations of a data type, then these are provided with normalised sorts. The functions user_defined_aliases, user_defined_sorts, etc. are provided to extract the aliases, sorts, mappings, constructors and equations in exactly the form they were added to the specification using the add_…. functions.
However, objects outside the data specification are not automatically normalised. These must be normalized explicitly. Normalisation functions, normalise_sort, exist for all types that derive from terms, such as data_expressions, assignments, sort_expressions, data_equations, etc., as well as for lists of these types.
#include mcrl2/data/data_specification.h
#include "mcrl2/data/alias.h"
#include "mcrl2/data/function_symbol.h"
using namespace mcrl2::data;
void normalise_sort_demo()
{
data_specification spec;
const basic_sort a("A");
const basic_sort b("B");
spec.add_alias(alias(a,b));
const function_symbol f("f",b));
// An example of normalising a function symbol explicitly.
std::cout << "Not normalised: " << pp(f) << " has sort " << pp(f.sort()) << "\n";
const function_symbol normalised_f=spec.normalise(f);
std::cout << "Normalised: " << pp(normalised_f) << " has sort " << pp(normalised_f.sort()) << "\n";
spec.add_mapping(function_symbol);
// Get the mapping and the sorts, which are normalised. So, f:A is replaced by f:B.
mappings_const_range m=mappings();
for(function_symbol::const_iterator i=m.begin(); i!=m.end(); ++i)
{ std::cout << "Function symbol " << pp(*i) << " has sort " << pp(i>sort()) << "\n";
}
}
Given a particular sort, it is sometimes useful to find the sort it represents. E.g. suppose that sort F is defined by F=A>B. Then from sort F alone it cannot be seen that F is actually a function sort. The function unalias yields the structure of an alias. So, in this particular case data_spec.unalias(F) yields A>B. As aliases may be recursive, as in sort E=struct nil  insert(Nat,E), unalias will only unfold an alias until it cannot be unfolded further, or until a type with a type constructing operator occurs.
Data rewriters
A rewriter is a function that rewrites terms using a number of rewrite rules.
In the mCRL2 toolset a class data::rewriter
is available that operates on data expressions,
and that is initialized using a data specification. The equations of the data specification
are interpreted as rewrite rules from left to right. An example is given below.
// rewrite two data expressions, and check if they are the same
rewriter r;
data_expression d1 = parse_data_expression("2+7");
data_expression d2 = parse_data_expression("4+5");
assert(d1 != d2);
assert(r(d1) == r(d2));
Tip
Rewriters can be used to determine equivalence between data expressions. In general this
problem is undecidable. Only if [^r(d1) == r(d2)] one can conclude that the expressions d1
and d2
are equivalent, otherwise the answer is unknown.
For efficiency reasons a rewriter can be invoked with an optional substitution function
__sigma as a second argument, where __sigma maps data variables to data expressions. The
function __sigma must satisfy the property that for all data variables v
__sigma(v) == r(__sigma(v)).
Under this condition the following property holds:
r(d,__sigma) == r(__sigma(d)).
Note that in general the computation of [^r(d,__sigma)] can be done more efficiently than the computation of [^r(__sigma(d))]. In the mCRL2 toolset substition functions are used that take constant time.
An example of rewriting with a substitution function is given below.
rewriter r;
// Create a substitution sequence sigma with two substitutions: [m:=3, n:=4]
std::string var_decl = "m, n: Pos;\n";
mutable_map_substitution sigma;
sigma[parse_data_expression("m", var_decl)] = r(parse_data_expression("3"));
sigma[parse_data_expression("n", var_decl)] = r(parse_data_expression("4"));
data::data_expression d1 = parse_data_expression("m+n", var_decl);
data::data_expression d2 = parse_data_expression("7");
assert(r(d1, sigma) == r(d2));
Caution
The current implementation of rewriting with substitutions to data variables is
inefficient. The interface of the underlying Rewriter
class needs to be adapted
to get rid of these inefficiencies.
Rewriter Concept
In the mCRL2 toolset a Rewriter
is a concept with the following requirements:
Expression 
Meaning 

Rewriter::term_type 
the type of the terms on which the rewriter operates 
Rewriter::variable_type 
the type of the variables 
A SubstitutionFunction
is a function that maps variables to terms. Let sigma
be a
substitution function, and let v
be an expression of type Rewriter::variable_type
.
Expression 
Result 

sigma(v) 
Returns an expression of type 
Let r
be a Rewriter
, let d
be an expression of type Rewriter::term_type
and let sigma
be a SubstitutionFunction
.
Expression 
Result 

r(d) 
Returns an expression of type Rewriter::term_type that is the result of applying the rewriter r to term d 
r(d, sigma) 
Returns an expression of type Rewriter::term_type that is the result of applying the rewriter r to term , while on the fly applying the substitution function sigma to all data variables in d. 
Algorithms using a rewriter
Algorithms that use a rewriter are typically implemented with a template parameter for the rewriter. An example of this is the constelm algorithm of the LPS library:
template <typename Rewriter>
specification constelm(const specification& spec, Rewriter r, bool verbose = false)
{
...
}
The algorithm may only assume that the requirements of the Rewriter Concept hold, with proper choices for the nested variable and term types.
Data enumerator
Documentation not yet available