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<H2> Functionalities of the interface at level 1 </H2>
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<P>
We focus on the changes brought by the level 1 w.r.t. the level 0.
</P><P>
<BLOCKQUOTE><TABLE BORDER=0 CELLSPACING=0>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="apron_6.html#SEC33">Variables and Environments</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="apron_6.html#SEC34">Semantics and Representation of an abstract value</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="apron_6.html#SEC35">Operations on environments</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="apron_6.html#SEC36">Dynamic typing w.r.t. environments</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="apron_6.html#SEC37">Operations on variables in abstract values</A></TD><TD> </TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
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<A NAME="Variables and Environments"></A>
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<H4> Variables </H4>
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<P>
Dimensions are replaced by <EM>variables</EM>.
</P><P>
In the C interface, variables are defined by a generic type
(<CODE>char*</CODE>, structured type, ...), equipped with the operations
<CODE>compare</CODE>, <CODE>copy</CODE>, <CODE>free</CODE>, <CODE>to_string</CODE>. In the
OCAML, for technical reasons, the type is just the <CODE>string</CODE>
type.
</P><P>
<EM>Environments</EM> manages the correspondance between the numerical
dimensions of level 0 and the variables of level 1.
</P><P>
<A NAME="Semantics and Representation of an abstract value"></A>
<A NAME="SEC34"></A>
<H4> Semantics and Representation of an abstract value </H4>
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<P>
The semantics of an abstract value is a subset
<BLOCKQUOTE>
X -> (N+R).
</BLOCKQUOTE>
where <EM>X</EM> is a set of variables.
Abstract values are typed according to their environment.
<P>
It is represented by a structure
@verbatim
struct ap_abstract1_t {
ap_abstract0_t *abstract0;
ap_environment_t *env;
};
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;
};
</P><P>
<A NAME="Operations on environments"></A>
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<H4> Operations on environments </H4>
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<P>
<UL>
<LI>creation, merging, destruction
<LI>addition/removal/renaming of variables
</UL>
<P>
<A NAME="Dynamic typing w.r.t. environments"></A>
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<H4> Dynamic typing w.r.t. environments </H4>
<!--docid::SEC36::-->
<P>
For binary operations on abstract values, the environments should be
the same.
</P><P>
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.
</P><P>
<A NAME="Operations on variables in abstract values"></A>
<A NAME="SEC37"></A>
<H4> Operations on variables in abstract values </H4>
<!--docid::SEC37::-->
<P>
Operations on dimensions are lifted to operations on variables:
</P><P>
<UL>
<LI>
Projection/Elimination of one or several variables with constant
environment;
<LI>
Addition/Removal/Renaming of variables with corresponding change
of environment;
<LI>
Change of environment (possibly combining removal and addition of variables);
<LI>
Expansion and folding of variables.
</UL>
<P>
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