@center Version 2.1, February 1999

@display

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@end display

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@heading END OF TERMS AND CONDITIONS

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@page

@subheading How to Apply These Terms to Your New Libraries

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That's all there is to it!

@menu

* General choices::             

* Functionalities of the interface at level 0::  

* Functionalities of the interface at level 1::  

@end menu

@c ===================================================================

@node General choices, Functionalities of the interface at level 0, APRON Rationale and Functionalities, APRON Rationale and Functionalities

@section General choices

@c ===================================================================

@menu

* Interface levels::            

* Programming language::        

* Compatibility with threads::  

* Interruptions::               

* Memory management::           

* Programming style::           

* Number representation::       

@end menu

@c -------------------------------------------------------------------

@node Interface levels, Programming language, General choices, General choices

@subsubheading Interface levels

There are two main goals for the APRON interface: efficiency of the

implementations, and ease of use for the user. In addition, code

duplication between libraries should be avoided. As a consequence, two

levels were identified:

@table @emph

@item Level 0 

Choices are guided by the efficiency and the precision of the operations;

@item Level 1

Choices are guided by ease of use, and code factorization.

@end table

The level 0 is directly connected to the underlying (existing)

library. It includes all the operations that are specific to an

abstract domain and whose code cannot be shared. The interface should

be minimal, @emph{unless} there is a strong algorithmical advantage to

include a combination of more basic operations.

The higher levels offers additional functionalities that are shared by

all the library connected to the level 0. For instance:

@itemize

@item

managing correspondance between numerical dimensions and names

(characters strings or more generally references);

@item

abstraction of non linear expressions in interval linear expressions;

@item

automatic call to redimensioning and permutation operations for

computing

@iftex 

@tex

$P(x,y)\sqcap Q(y,z)$

@end tex 

@end iftex

@ifnottex

P(x,y)/\Q(y,z).

@end ifnottex

@end itemize

Combination of abstract domain is possible at the level 0. One can

implement for instance the cartesian or reduced product of two

different abstract domains, the decomposition of abstract values into

a product of values of smaller dimensionality, ...

@c -------------------------------------------------------------------

@node Programming language, Compatibility with threads, Interface levels, General choices

@subsubheading Programming language

The reference version of the interface is the C version of the interface:

@itemize

@item

C can be easily interfaced with most programming languages;

@item

Most of the existing libraries implementing abstract domains for

numerical variables are programmed in C or C++.

@end itemize

An @sc{OCaml} version is already available. The interface between

OCaml and C is even generic and any libraries can benefit from it by

providing the glue for just one function (see XX).

@c -------------------------------------------------------------------

@node Compatibility with threads, Interruptions, Programming language, General choices

@subsubheading Compatibility with threads

In order to ensure compatibility with multithreading programming, a

context is explicitly passed to functions in order to ensure the

following points:

@itemize

@item

the transmission of data specific to each library (non-standard

options, workspace, ...);

@item

the transmission of standard options (selection of algorithms and their

precision inside a library);

@item

the management of exceptions (implemented as error codes in the C

interface) (@code{not_implemented}, @code{invalid_argument},

@code{overflow}, @code{timeout}, @code{out_of_space}).

@end itemize

@c -------------------------------------------------------------------

@node Interruptions, Memory management, Compatibility with threads, General choices

@subsubheading Interruptions

Interruptions mechanism is have possible for different cases:

@table @code

@item timeout

if the execution time for an operation exceeds some bound;

@item out_of_space

if the space consumption for an operation exceeds some bound;

@item overflow

if the magnitude or the space usage of manipulated numbers exceeds some bound;

@item not_implemented

if the operation is actually not implemented by the underlying library;

@item invalid_argument

if the arguments do not follow the requirements of an operation.

@end table

@quotation 

For instance, in a convex polyhedra library, the @code{out_of_space}

exception allows to abort an operation is the result appears to have

too many constraints and/or generators. If this happens, one can redo

the operation with another (less precise) algorithm. The

@code{overflow} may be useful when effective overflows are encountered

with machine integers or when multiprecision rational numbers have too

large numerators and denominators. The @code{not_implemented}

exception allows for a library to be linked to the interface even if

it does not provide some operation of the interface.

@end quotation

When an interruption occurs, the function should still return a

correct result, in the abstract interpretation sense: it should be a

correct approximation, usable for next operations in the program. The

top value is always a correct approximation.

@c -------------------------------------------------------------------

@node Memory management, Programming style, Interruptions, General choices

@subsubheading Memory management

Memory is managed differently depending on the programming language. Currently:

@itemize

@item

No automatic garbage collection in the C interface

@item

Use of the @sc{OCaml} runtime garbage collector in the @sc{OCaml} interface

@end itemize

@c -------------------------------------------------------------------

@node Programming style, Number representation, Memory management, General choices

@subsubheading Programming style

Both functional and imperative (i.e., side-effect) signatures are

supported for operations. This allows to optimize the memory

allocation and to use whichever version is more convenient for an user

and the used programming language.

@c -------------------------------------------------------------------

@node Number representation,  , Programming style, General choices

@subsubheading Number representation

Inside a specific library, any number representation may be used

(floating-point numbers, machine integers, multiprecision

integers/rationals, ...). Existing libraries often offers the

possibility to select different representations.

However, in the interface, this representation should be normalized

and independent of underlying libraries, without being restrictive

either. As a consequence, the interface offers the choiced between

@itemize

@item GMP multiprecision rationals (which implements exact arithmetic);

@item and machine floating-point numbers (@code{double}).

@end itemize

@c ===================================================================

@node Functionalities of the interface at level 0, Functionalities of the interface at level 1, General choices, APRON Rationale and Functionalities

@section Functionalities of the interface at level 0

@c ===================================================================

@menu

* Representation of an abstract value::  

* Semantics of an abstract value::  

* Dimensions::                  

* Other datatypes::             

* Control of internal representation::  

* Printing::                    

* Serializaton/Deserialization::  

* Constructors::                

* Tests::                       

* Property extraction::         

* Lattice operations::          

* Assignement and Substitutions::  

* Operations on dimensions::    

* Other operations::            

@end menu

@c -------------------------------------------------------------------

@node Representation of an abstract value, Semantics of an abstract value, Functionalities of the interface at level 0, Functionalities of the interface at level 0

@subsubheading Representation of an abstract value

At the level 0 of the interface, an abstract value is a structure

@verbatim

struct ap_abstract0_t {

  ap_manager_t *manager; /* Explicit context */

  void         *value;   /* Abstract value representation

			    (only known by the underlying library) */

}

@end verbatim

The context is allocated by the underlying library, and contains an

array of function pointers pointing to the function of the underlying

library. Hence, it indicates the effective type of an abstract value.

The validity of the arguments of the functions called through the

interface is checked before the call to effective functions. In case

of problem, an @code{invalid_argument} exception is raised.

@c -------------------------------------------------------------------

@node Semantics of an abstract value, Dimensions, Representation of an abstract value, Functionalities of the interface at level 0

@subsubheading Semantics of an abstract value

The semantics of an abstract value is a subset

@iftex 

@tex

$$X\subseteq {\cal N}^p\times{\cal R}^q$$

@end tex 

@end iftex

@ifnottex

@quotation

X of N^p x R^q

@end quotation

@end ifnottex

@noindent Abstract values are typed according to their dimensionality

(p,q).

@c -------------------------------------------------------------------

@node Dimensions, Other datatypes, Semantics of an abstract value, Functionalities of the interface at level 0

@subsubheading Dimensions

Dimensions are numbered from 0 to p+q-1 and are typed either as

integer or real, depending on their rank w.r.t. the dimensionality of

the abstract value.

@quotation Note

Taking into account or not the fact that some dimensions are integers

is left to underlying libraries. Treating them as real is still a

correct approximation. The behaviour of the libraries in this regard

may also depend on some options.

@end quotation

@c -------------------------------------------------------------------

@node Other datatypes, Control of internal representation, Dimensions, Functionalities of the interface at level 0

@subsubheading Other datatypes

In addition to abstract values, the interface also manipulates the

following main datatypes:

@table @emph

@item scalar (number)

Either GMP multiprecision rationals or C @code{double}.

@item interval

composed of 2 scalar numbers. With rationals, plus (resp minus) infinity is represented by 1/0 (resp -1/0). With @code{double}, the IEEE754 is assumed and the corresponding standard representation is used.

@item coefficient

which is either a scalar or an interval.

@item (interval) linear expression

The term linear is used even if the proper term should rather be

affine.  A linear expression is a linear expression in the common

sense, using only scalar numbers. A quasi-linear expression is a

linear expression where the constant coefficient is an interval. An

interval linear expression is a linear expression where any

coefficient may be an interval. In order to have a unique datatype for

these variations, we introduced the notion of coefficient described

above.

@item ``linear'' constraints 

``Linear'' constraints includes proper linear constraints, linear

constraints in which the expression can be possibly an interval linear

expression, linear equalities modulo a number, and linear disequalities.

@item generators

A generator system for a subset of @math{X\subseteq R^n} is a finite

set of vectors, among which one distinguishes @emph{points}

@math{p_0,\ldots,p_m} and @emph{rays} @math{r_0,\ldots,r_n}, that

generates @math{X}:

@iftex 

@tex

$$X=\{ \lambda_0 \vec{p_0} + \ldots \lambda_m \vec{p_m} + \mu_0 \vec{r_0} +\ldots + \mu_n \vec{r_n} ~|~ \sum_i \lambda_i = 1 ~\wedge~ \forall j: \mu_j\geq 0 \}$$

@end tex 

@end iftex

@ifnottex

@quotation

X = @{ lambda0 p0 +...+ lambdaM pM + mu0 r0 +...+ muN rN | lambda0 +...+ lambdaN = 1 and forall J : muJ >= 0 @}

@end quotation

@end ifnottex

The APRON datatype for generators distinguishes points (sum of

coefficients equal to one), rays (positive coefficients), lines (or

bidirectional rays, with unconstrainted coefficients), integer rays

(integer positive coefficients) and integer lines (integer

coefficients).

@end table

@c -------------------------------------------------------------------

@node Control of internal representation, Printing, Other datatypes, Functionalities of the interface at level 0

@subsubheading Control of internal representation

We identified several notions:

@itemize

@item

Canonical form

@item

Minimal form (in term of space)

@item

Approximation notion left to the underlying library (taking into

account integers or not, ...).

@end itemize

@c -------------------------------------------------------------------

@node Printing, Serializaton/Deserialization, Control of internal representation, Functionalities of the interface at level 0

@subsubheading Printing

There are two printing operations:

@itemize

@item

Printing of an abstract value;

@item

Printing the difference between two abstract values.

@end itemize

@noindent The printing format is library dependent. However, the conversion of

abstract values to constraints (see below) allows a form of

standardized printing for abstract values.

@c -------------------------------------------------------------------

@node Serializaton/Deserialization, Constructors, Printing, Functionalities of the interface at level 0

@subsubheading Serializaton/Deserialization

Serialization and deserialization of abstract values to a memory

buffer is offered. It is entirely managed by the underlying

library. In particular, it is up to it to check that a value read from

the memory buffer has the right format and has not been written by a

different library.

Serialization is done to a memory buffer instead of to a file

descriptor because this mechanism is more general and is needed for

interfacing with languages like @sc{OCaml}.

@c -------------------------------------------------------------------

@node Constructors, Tests, Serializaton/Deserialization, Functionalities of the interface at level 0

@subsubheading Constructors

Four basic constructors are offered:

@itemize

@item

bottom (empty) and top (universe) values (with a specified dimensionality);

@item

abstraction of a bounding box;

@item

abstraction of conjunction of linear constraints (in the broad sense).

@end itemize

@c -------------------------------------------------------------------

@node Tests, Property extraction, Constructors, Functionalities of the interface at level 0

@subsubheading Tests

Predicates are offered for testing

@itemize

@item

emptiness and universality of an abstract value:

@item

inclusion and equality of two abstract values;

@item

inclusion of a dimension into an interval given an abstract value;

@iftex

@tex

$${\it abs}(\vec{x}) \models x_i \in I ~~ ?$$

@end tex

@end iftex

@item

satisfaction of a linear constraint by the abstract value.

@iftex

@tex

$${\it abs}(\vec{x}) \models {\it cons}(\vec{x}) ~~ {\tt or} ~~ {\it abs}(\vec{x}) \Rightarrow {\it cons}(\vec{x}) ~~ ?$$

@end tex

@end iftex

@end itemize

@c -------------------------------------------------------------------

@node Property extraction, Lattice operations, Tests, Functionalities of the interface at level 0

@subsubheading Property extraction

Some properties may be inferred given an abstract values:

@itemize

@item Interval of variation of a dimension in an abstract value;

@iftex

@tex

$$\bigcap\{ I ~|~ {\it abs}(\vec{x}) \models x_i\in I\}$$

@end tex

@end iftex

@item Interval of variation of a linear expression in an abstract value;

@iftex

@tex

$$\bigcap\{ I ~|~ {\it abs}(\vec{x}) \models {\it expr}(\vec{x})\in I\}$$

@end tex

@end iftex

@item Conversion to a bounding box

@iftex

@tex

$$\bigcap\{ B ~|~ {\it abs}(\vec{x}) \subseteq B \}$$

@end tex

@end iftex

@item Conversion to a set of linear constraints (in the broad sense).

@end itemize

@noindent Notice that the second operation implements linear programming if it

is exact. The third operation is not minimal, as it can be implemented

using the first one, but it was convenient to include it. But the

fourth operation is minimal and cannot be implemented using the second

one, as the number of linear expression is infinite.

@c -------------------------------------------------------------------

@node Lattice operations, Assignement and Substitutions, Property extraction, Functionalities of the interface at level 0

@subsubheading Lattice operations

@itemize

@item

Least upper bound and greatest lower bound of two abstract values, and of arrays of abstract values;

@item

Intersection with one or several linear constraints;

@iftex

@tex

$$\alpha\left(\gamma({\it abs}(\vec{x})) \cap \bigcap_i {\it cons}_i(\vec{x})\right)$$

@end tex

@end iftex

@item

Addition of rays (for instance for implement generalized time elapse

operator in linear hybrid systems).

@iftex

@tex

$$\alpha\left(\left\{ \vec{x} + \sum_i \lambda_i \vec{r}_i ~|~ \vec{x}\in\gamma({\it abs}), \lambda_i\geq 0\right\}\right)$$

@end tex

@end iftex

@end itemize

@c -------------------------------------------------------------------

@node Assignement and Substitutions, Operations on dimensions, Lattice operations, Functionalities of the interface at level 0

@subsubheading Assignement and Substitutions

@itemize

@item

of a dimension by a (interval) linear expression

@iftex

@tex

Assignement:

$$\alpha\left(\biggl(

\exists x_i: \Bigl(\gamma({\it abs}(\vec{x})) \cap x_i'={\it expr}(\vec{x})\Bigr)\biggr)[x_i\leftarrow x_i']\right)$$

Substitution:

$$\alpha\biggl(

\exists x_i': \Bigl(\gamma({\it abs}(\vec{x}))[x_i'\leftarrow x_i] \cap x_i'={\it expr}(\vec{x})\Bigr)\biggr)$$

@end tex

@end iftex

@item

in parallel of several dimensions by several (interval) linear expressions

@iftex

@tex

Assignement:

$$\alpha\left(\biggl(

\exists \vec{x}: \Bigl(\gamma({\it abs}(\vec{x})) \cap \bigcap_i x_i'={\it expr}_i(\vec{x})\Bigr)\biggr)[\vec{x}\leftarrow \vec{x'}]\right)$$

Substitution:

$$\alpha\biggl(

\exists \vec{x'}: \Bigl(\gamma({\it abs}(\vec{x'})) \cap \bigcap_i x_i'={\it expr}(\vec{x})\Bigr)\biggr)$$

@end tex

@end iftex

@end itemize

@noindent Parallel assignement and substitution ar enot minimal operations, but

for some abstract domains implementing them directly results in more

efficient or more precise operations.

@c -------------------------------------------------------------------

@node Operations on dimensions, Other operations, Assignement and Substitutions, Functionalities of the interface at level 0

@subsubheading Operations on dimensions

@itemize

@item

Projection/Elimination of one or several dimensions with constant

dimensionality;

@iftex

@tex

Elimination: $$\exists x_i:{\it abs}(\vec{x})$$

Projection: $$(\exists x_i:{\it abs}(\vec{x}))\cap x_i=0$$

@end tex

@end iftex

@item

Addition/Removal/Permutation of dimensions with corresponding change

of dimensionality (with the exception of permutation). These

operations allows to resize abstract values, and reorganize

dimensions.

@item

Expansion and folding of dimensions. This is useful for the

abstraction of arrays, where a dimension may represent several

variables.

@iftex

@tex

Expansion of $i$ into $i$, $j_1$, $j_2$ assuming $x_{j_1}$, $x_{j_2}$ are new dimensions:

$${\it abs}(\vec{x}) \sqcap {\it abs}(\vec{x})[x_{j_1}\leftarrow x_i] \sqcap {\it abs}(\vec{x})[x_{j_2}\leftarrow x_i] ...

$$

Folding of $j_0$ and $j_1$ into $j_0$:

$$(\exists x_{j_1}:{\it abs}(\vec{x})) \sqcup (\exists x_{j_0}:{\it abs}(\vec{x})[x_{j_0}\leftarrow x_{j_1}] 

$$

@end tex

@end iftex

@end itemize

@c -------------------------------------------------------------------

@node Other operations,  , Operations on dimensions, Functionalities of the interface at level 0

@subsubheading Other operations

Widening, either simple or with threshold, is offered. A generic

widening with threshold function is offered in the interface.

Topological closure (i.e., relaxation of strict inequalities) is

offered.

@c ===================================================================

@node Functionalities of the interface at level 1,  , Functionalities of the interface at level 0, APRON Rationale and Functionalities

@section Functionalities of the interface at level 1

@c ===================================================================

We focus on the changes brought by the level 1 w.r.t. the level 0.

@menu

* Variables and Environments::  

* Semantics and Representation of an abstract value::  

* Operations on environments::  

* Dynamic typing w.r.t. environments::  

* Operations on variables in abstract values::  

@end menu

@c -------------------------------------------------------------------

@node Variables and Environments, Semantics and Representation of an abstract value, Functionalities of the interface at level 1, Functionalities of the interface at level 1

@subsubheading Variables

Dimensions are replaced by @emph{variables}.

In the C interface, variables are defined by a generic type

(@code{char*}, structured type, ...), equipped with the operations

@code{compare}, @code{copy}, @code{free}, @code{to_string}. In the

@sc{OCaml}, for technical reasons, the type is just the @code{string}

type.

@emph{Environments} manages the correspondance between the numerical

dimensions of level 0 and the variables of level 1.

@c -------------------------------------------------------------------

@node Semantics and Representation of an abstract value, Operations on environments, Variables and Environments, Functionalities of the interface at level 1

@subsubheading Semantics and Representation of an abstract value

The semantics of an abstract value is a subset

@iftex 

@tex

$$X\subseteq V\rightarrow ({\cal N}\cup{\cal R})$$

@end tex 

@end iftex

@ifnottex

@quotation

X -> (N+R).

@end quotation

@end ifnottex

where @math{X} is a set of variables.

@noindent

Abstract values are typed according to their environment.

It is represented by a structure

@verbatim

struct ap_abstract1_t {

  ap_abstract0_t    *abstract0;

  ap_environment_t  *env;

};

@end verbatim

Other datatypes of level 0 are extend in the same way. For instance,

@verbatim

struct ap_linexpr1_t {

  ap_linexpr0_t    *linexpr0;

  ap_environment_t *env;

};

@end verbatim

@c -------------------------------------------------------------------

@node Operations on environments, Dynamic typing w.r.t. environments, Semantics and Representation of an abstract value, Functionalities of the interface at level 1

@subsubheading Operations on environments

@itemize

@item creation, merging, destruction

@item addition/removal/renaming of variables

@end itemize

@c -------------------------------------------------------------------

@node Dynamic typing w.r.t. environments, Operations on variables in abstract values, Operations on environments, Functionalities of the interface at level 1

@subsubheading Dynamic typing w.r.t. environments

For binary operations on abstract values, the environments should be

the same.

For operations involving an abstract value and an other datatype

(expression, constraint, ...), one checks that the environment of

the expression is a subenvironment of the environment of the abstract

value, and one resize if necessary.

@c -------------------------------------------------------------------

@node Operations on variables in abstract values,  , Dynamic typing w.r.t. environments, Functionalities of the interface at level 1

@subsubheading Operations on variables in abstract values

Operations on dimensions are lifted to operations on variables:

@itemize

@item

Projection/Elimination of one or several variables with constant

environment;

@item

Addition/Removal/Renaming of variables with corresponding change

of environment;

@item

Change of environment (possibly combining removal and addition of variables);

@item

Expansion and folding of variables. 

@end itemize

<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">

<html>

<head>

<link rel="stylesheet" href="style.css" type="text/css">

<meta content="text/html; charset=iso-8859-1" http-equiv="Content-Type">

<link rel="Start" href="index.html">

<link title="Index of types" rel=Appendix href="index_types.html">

<link title="Index of exceptions" rel=Appendix href="index_exceptions.html">

<link title="Index of values" rel=Appendix href="index_values.html">

<link title="Index of modules" rel=Appendix href="index_modules.html">

<link title="Scalar" rel="Chapter" href="Scalar.html">

<link title="Interval" rel="Chapter" href="Interval.html">

<link title="Coeff" rel="Chapter" href="Coeff.html">

<link title="Var" rel="Chapter" href="Var.html">

<link title="Environment" rel="Chapter" href="Environment.html">

<link title="Linexpr1" rel="Chapter" href="Linexpr1.html">

<link title="Lincons1" rel="Chapter" href="Lincons1.html">

<link title="Generator1" rel="Chapter" href="Generator1.html">

<link title="Texpr1" rel="Chapter" href="Texpr1.html">

<link title="Tcons1" rel="Chapter" href="Tcons1.html">

<link title="Abstract1" rel="Chapter" href="Abstract1.html">

<link title="Dim" rel="Chapter" href="Dim.html">

<link title="Linexpr0" rel="Chapter" href="Linexpr0.html">

<link title="Lincons0" rel="Chapter" href="Lincons0.html">

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<link title="Texpr0" rel="Chapter" href="Texpr0.html">

<link title="Tcons0" rel="Chapter" href="Tcons0.html">

<link title="Manager" rel="Chapter" href="Manager.html">

<link title="Abstract0" rel="Chapter" href="Abstract0.html">

<link title="Box" rel="Chapter" href="Box.html">

<link title="Oct" rel="Chapter" href="Oct.html">

<link title="Polka" rel="Chapter" href="Polka.html">

<link title="Ppl" rel="Chapter" href="Ppl.html">

<link title="PolkaGrid" rel="Chapter" href="PolkaGrid.html">

<link title="Mpz" rel="Chapter" href="Mpz.html">

<link title="Mpq" rel="Chapter" href="Mpq.html">

<link title="Mpf" rel="Chapter" href="Mpf.html">

<link title="Mpfr" rel="Chapter" href="Mpfr.html">

<link title="Gmp_random" rel="Chapter" href="Gmp_random.html"><title></title>

<link title="Mpzf" rel="Chapter" href="Mpzf.html">

<link title="Mpqf" rel="Chapter" href="Mpqf.html">

<link title="Mpftf" rel="Chapter" href="Mpftf.html">

</head>

<body>

<center><h1></h1></center>

<a href="index_types.html">Index of types</a><br>

<a href="index_exceptions.html">Index of exceptions</a><br>

<a href="index_values.html">Index of values</a><br>

<a href="index_modules.html">Index of modules</a><br>

<br/><br>

<h1>Managers</h1>

Managers are allocated by abstract domains and defines the

effective type of abstract values. Information about compilation

and linking is provided in modules implementing a specific

abstract domain.

<table class="indextable">

<tr><td><a href="Manager.html">Manager</a></td><td><div class="info">

APRON Managers

</div>

</td></tr>

<tr><td><a href="Box.html">Box</a></td><td><div class="info">

BOX: intervals abstract domain

</div>

</td></tr>

<tr><td><a href="Oct.html">Oct</a></td><td><div class="info">

OCT: octagon abstract domain.

</div>

</td></tr>

<tr><td><a href="Polka.html">Polka</a></td><td><div class="info">

POLKA: Convex Polyhedra and Linear Equalities abstract domain

</div>

</td></tr>

<tr><td><a href="Ppl.html">PPL</a></td><td><div class="info">

PPL: Convex Polyhedra and Linear Congruences abstract domain

</div>

</td></tr>

<tr><td><a href="PolkaGrid.html">PolkaGrid</a></td><td><div class="info">

PolkaGrid: reduced product of (NewPolka) Convex Polyhedra and (PPL) Linear Congruences abstract domain

</div>

</td></tr>

</table>

<h1>Coefficients</h1>

<table class="indextable">

<tr><td><a href="Scalar.html">Scalar</a></td><td><div class="info">

APRON Scalar numbers.

</div>

</td></tr>

<tr><td><a href="Interval.html">Interval</a></td><td><div class="info">

APRON Intervals on scalars

</div>

</td></tr>

<tr><td><a href="Coeff.html">Coeff</a></td><td><div class="info">

APRON Coefficients (either scalars or intervals)

</div>

</td></tr>

</table>

<h1>Level 1 of APRON interface</h1>

<table class="indextable">

<tr><td><a href="Var.html">Var</a></td><td><div class="info">

APRON Variables

</div>

</td></tr>

<tr><td><a href="Environment.html">Environment</a></td><td><div class="info">

APRON Environments binding dimensions to names

</div>

</td></tr>

<tr><td><a href="Linexpr1.html">Linexpr1</a></td><td><div class="info">

APRON Linear Expressions of level 1

</div>

</td></tr>

<tr><td><a href="Lincons1.html">Lincons1</a></td><td><div class="info">

APRON Linear Constraints and array of constraints of level 1

</div>

</td></tr>

<tr><td><a href="Generator1.html">Generator1</a></td><td><div class="info">

APRON Generators and array of generators of level 1

</div>

</td></tr>

<tr><td><a href="Texpr1.html">Texpr1</a></td><td><div class="info">

APRON Non-Linear Expressions of level 1

</div>

</td></tr>

<tr><td><a href="Tcons1.html">Tcons1</a></td><td><div class="info">

APRON Non-Linear Constraints and array of constraints of level 1

</div>

</td></tr>

<tr><td><a href="Abstract1.html">Abstract1</a></td><td><div class="info">

APRON Abstract values of level 1

</div>

</td></tr>

<tr><td><a href="Parser.html">Parser</a></td><td><div class="info">

APRON Parser for expressions, constraints and generators

</div>

</td></tr>

</table>

<h1>Level 0 of APRON interface</h1>

Normally not needed for a user.

<table class="indextable">

<tr><td><a href="Dim.html">Dim</a></td><td><div class="info">

APRON Dimensions and related types

</div>

</td></tr>

<tr><td><a href="Linexpr0.html">Linexpr0</a></td><td><div class="info">

APRON Linear expressions of level 0

</div>

</td></tr>

<tr><td><a href="Lincons0.html">Lincons0</a></td><td><div class="info">

APRON Linear constraints of level 0

</div>

</td></tr>

<tr><td><a href="Generator0.html">Generator0</a></td><td><div class="info">

APRON Generators of level 0

</div>

</td></tr>

<tr><td><a href="Texpr0.html">Texpr0</a></td><td><div class="info">

APRON Non-Linear Expressions of level 0

</div>

</td></tr>

<tr><td><a href="Tcons0.html">Tcons0</a></td><td><div class="info">

APRON Non-Linear Constraints and array of constraints of level 0

</div>

</td></tr>

<tr><td><a href="Abstract0.html">Abstract0</a></td><td><div class="info">

APRON Abstract value of level 0

</div>

</td></tr>

</table>

<h1>Misc</h1>

<table class="indextable">

<tr><td><a href="Mpz.html">Mpz</a></td><td><div class="info">

GMP multi-precision integers

</div>

</td></tr>

<tr><td><a href="Mpq.html">Mpq</a></td><td><div class="info">

GMP multiprecision rationals

</div>

</td></tr>

<tr><td><a href="Mpf.html">Mpf</a></td><td><div class="info">

GMP multiprecision floating-point numbers

</div>

<tr><td><a href="Mpfr.html">Mpfr</a></td><td><div class="info">

MPFR Multiprecision floating-point numbers with guaranteed rounding

</div>

</td></tr>

<tr><td><a href="Gmp_random.html">Gmp_random</a></td><td><div class="info">

GMP random generation functions

</div>

</td></tr>

<tr><td><a href="Mpzf.html">Mpzf</a></td><td><div class="info">

GMP multi-precision integers, functional version

</div>

</td></tr>

<tr><td><a href="Mpqf.html">Mpqf</a></td><td><div class="info">

GMP multi-precision rationals, functional version

</div>

</td></tr>

<tr><td><a href="Mpfrf.html">Mpfrf</a></td><td><div class="info">

MPFR multi-precision floating-point numbers, functional version

</div>

</td></tr>

</table>

</body>

</html>

@c This file is part of the APRON Library, released under LGPL

@c license. Please read the COPYING file packaged in the distribution

@c to be included from apron.texi

The @sc{APRON PPL} library is an APRON wrapper around the

@uref{http://www.cs.unipr.it/ppl/, Parma Polyhedra Library (PPL)}. The

wrapper offers the convex polyhedra and linear congruences abstract

domains.

@menu

* Use of APRON PPL::

* Allocating APRON PPL managers::

* APRON PPL standard options::

@end menu

@c ===================================================================

@node Use of APRON PPL, Allocating APRON PPL managers,,PPL

@subsection Use of APRON PPL

@c ===================================================================

To use APRON PPL in C, you need of course to install PPL, @emph{after

having patched it} following the recommendations of the @file{README}

file.  You need also to add

@example

#include "apron_ppl.h"

@end example

in your source file(s) and add @samp{-I$(APRON_PREFIX)/include} in the

command line in your Makefile.

You should also link your object files with the APRON PPL library to

produce an executable, @emph{using} @samp{g++} (instead of @samp{gcc},

because @file{libppl.a} is a C++ library), and adding something like

@samp{-L$(APRON_PREFIX)/lib -lapron_ppl -L$(PPL_PREFIX)/lib -lppl

-L$(GMP_PREFIX)/lib -lgmpxx} in the command line in your Makefile

(followed by the standard @samp{-lapron -litvmpq -litvdbl

-L$(MPFR_PREFIX)/lib -lmpfr -L$(GMP_PREFIX)/lib -lgmp}). The

@file{libgmpxx.a} library is the C++ wrapper on top of the GMP

library. Ensure that your GMP installation contains it, as it is not

always installed by default.

All scalars of type @code{double} are converted to scalars of type

@code{mpq_t} inside APRON PPL, as APRON PPL works internally with exact

rational arithmetics. So when possible it is better for the user (in

term of efficiency) to convert already @code{double} scalars to

@code{mpq_t} scalars.

The wrapper library is available in debug mode

(@samp{libapron_ppl_debug.a}).

@c ===================================================================

@node Allocating APRON PPL managers, APRON PPL standard options, Use of APRON PPL, PPL

@subsection Allocating APRON PPL managers

@c ===================================================================

@deftypefun ap_manager_t* ap_ppl_poly_manager_alloc (bool @var{strict})

Allocate a APRON manager for convex polyhedra, linked to the PPL

library.

The @var{strict} option, when true, enables strict constraints in polyhedra

(like @code{x>0}). Managers in strict mode or in loose mode

(strict constraints disabled) are not compatible, and so are

corresponding abstract values.

@end deftypefun

@deftypefun ap_manager_t* ap_ppl_grid_manager_alloc ()

Allocate an APRON manager for linear equalities, linked to the PPL

library.

@end deftypefun

@c ===================================================================

@node APRON PPL standard options,  , Allocating APRON PPL managers, PPL

@subsection APRON PPL standard options

@c ===================================================================

Currently, the only options available are related to the widening

operators. 

@multitable @columnfractions .2 .06 .74

@item Function              @tab algo      @tab Comments

@item

@item widening

@tab <=0 

@tab CH78 standard widening (Cousot & Halbwachs, POPL'1978).

@item

@tab >0

@tab BHRZ03 widening (Bagnara, Hill, Ricci & Zafanella, SAS'2003)

@item

@item widening_threshold

@tab <=0

@tab standard widening with threshold

@item

@tab =1

@tab standard widening with threshold, intersected by the bounding box of the convex hull pof the two arguments

@item

@tab <=0

@tab standard widening with threshold

@item

@tab =1

@tab standard widening with threshold, intersected by the bounding box of the convex hull of the second argument. This is actually an extrapolation rather than a widening (termination is not guaranteed)

@item

@tab =2

@tab BHRZ03 widening with threshold

@item

@tab =3

@tab BHRZ03 widening with threshold, intersected by the bounding box of the convex hull of the second argument. This is actually an extrapolation rather than a widening (termination is not guaranteed)

@end multitable