# Linear process specifications¶

The LPS library contains classes and algorithms for linear processes. The code in the LPS library is contained in the namespace =lps=. LPS is shorthand for linear process specification.

## Definitions¶

A linear process is defined as

$P(d:D)=\sum\limits_{i\in I}s_{i}(d)$

with

$s_{i}(d)=\sum_{e:E_{i}}c_{i}(d,e)\rightarrow a_{i}(d,e)^@t_{i}(d,e)\cdot P(g_{i}(d,e))$

and

$$a_{i}(d,e) = a_{i}^1(f_{i}^1(d,e)) \mid \ldots \mid a_{i}^n(f_{i}^n(d,e))$$ or $$\delta$$

where

• $$d$$ is a vector of data variables, called the process parameters. The corresponding vector of sorts $$D$$ models the states of the process.
• $$s_{i}$$ is a linear process term, called a summand
• $$E_{i}$$ is a sort, and the elements $$e$$ of $$E_{i}$$ are called summation variables.
• $$c_{i}$$ is a boolean term, called the condition
• $$a_{i}$$ is a term called the action
• $$t_{i}$$ is a real valued term, called the time
• $$g_{i}$$ is an assignment function to data variables, with the following interpretation. The expression $$g_{i}(d,e)$$ is a state that can be reached from state $$d$$ by performing the action $$a_{i}$$ at time $$t_{i}$$.

A linear process usually has an accompanying initial value $$d_0:D$$.

Note

The time $$t_{i}$$ and the condition $$c_{i}$$ are optional. The action $$a_{i}(d,e)$$ can have the value $$\delta$$, corresponding to deadlock.

## Specification¶

A process can be defined using a (textual) mCRL2 specification. A typical process specification is the following:

sort
D     = struct d1 | d2;
Error = struct e;

act
r1,s4: D;
s2,r2,c2: D # Bool;
s3,r3,c3: D # Bool;
s3,r3,c3: Error;
s5,r5,c5: Bool;
s6,r6,c6: Bool;
s6,r6,c6: Error;
i;

proc
S(b:Bool)     = sum d:D. r1(d).T(d,b);
T(d:D,b:Bool) = s2(d,b).(r6(b).S(!b)+(r6(!b)+r6(e)).T(d,b));

R(b:Bool)     = sum d:D. r3(d,b).s4(d).s5(b).R(!b)+
(sum d:D.r3(d,!b)+r3(e)).s5(!b).R(b);

K             = sum d:D,b:Bool. r2(d,b).(i.s3(d,b)+i.s3(e)).K;

L             = sum b:Bool. r5(b).(i.s6(b)+i.s6(e)).L;

init
allow({r1,s4,c2,c3,c5,c6,i},
comm({r2|s2->c2, r3|s3->c3, r5|s5->c5, r6|s6->c6},
S(true) || K || L || R(true)
)
);


This process can be stored in an instance of the class lps::specification after linearizing it using

std::string text = ... ;
specification spec = linearise(text);


The class lps::specification represents a linearized process specification. It consists of

• a linear process
• a data specification, i.e. a specification of the sorts, constructors, mappings and equations
• an action specification, i.e. a sequence of actions that may occur during execution of the process
• an initial state
• global variables

## Linear processes¶

An instance of lps::specification contains a linear process of type lps::linear_process. This linear process in turn contains a sequence of summands, and an initial state. Note that the implementation distinguishes between action summands and deadlock summands (i.e. those summands containing a $$\delta$$).

const linear_process& proc = spec.process();
const atermpp::vector<action_summand>& as = proc.action_summands();
const process_initializer& init = proc.init();


Both classes action_summand and deadlock_summand have an optional attribute time. It is necessary to check if the time is available, before using it:

action_summand s = proc.action_summands().front();
if (s.has_time())
{
std::cout << "time = " << s.time() << std::endl;
}


Action summands have an associated multi action, which consists of a sequence of actions. An illustration of it’s usage is

action_summand s;
for (const process::action& a: s.actions())
{
core::identifier_string name = a.label().name();
data::data_expression_list arguments = a.arguments();
}


Here name corresponds with $$a_{i}$$, and arguments with $$f_i(d,e)$$ in the earlier given formulas.

Note

There is a convention that a linear process without any summands represents the process delta @ 0.

# Classes in the LPS library¶

Several classes in the LPS library are just thin wrappers around an ATerm pointer (see also the Introduction to the atermpp library). This means that instances of these classes are immutable, and instances with the same value are shared in memory. The following table gives an overview of the ATerm based classes:

ATerm based classes
process::action_label
process::action_label_list
lps::process_initializer

## Correctness checks¶

For many classes there are restrictions to what expressions are considered valid, the so called well typedness constraints. These constraints are implemented in the class lps/detail/lps_well_typed_checker.h. For example, the following checks are done for linear processes:

• process parameters have unique names
• process parameters and summation variables have different names
• the left hand sides of the assignments of summands are contained in the process parameters

Such constraints are only checked in debug mode in the load and save functions of lps::specification. The descriptions of the well typedness constraints are found in the reference documentation.