doc/isl.tex

   1 \section{\protect\isl/ interface}
   2
   3 \subsection{Library}
   4
   5 The \barvinok/ library currently supports just two
   6 functions that interface with the \isl/ library.
   7 In time, this interface will grow and is set to replace
   8 the \PolyLib/ interface.
   9 For more information on the \isl/ data structures, see
  10 the \isl/ user manual.
  11
  12 \begin{verbatim}
  13 __isl_give isl_pw_qpolynomial *isl_set_card(__isl_take isl_set *set);
  14 \end{verbatim}
  15 Compute the number of elements in an \ai[\tt]{isl\_set}.
  16 The resulting \ai[\tt]{isl\_pw\_qpolynomial} has purely parametric cells.
  17
  18 \begin{verbatim}
  19 __isl_give isl_pw_qpolynomial *isl_map_card(__isl_take isl_map *map);
  20 \end{verbatim}
  21 Compute a closed form expression for the number of image elements
  22 associated to any element in the domain of the given \ai[\tt]{isl\_map}.
  23 The union of the cells in the resulting \ai[\tt]{isl\_pw\_qpolynomial}
  24 is equal to the domain of the input \ai[\tt]{isl\_map}.
  25
  26 \begin{verbatim}
  27 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_sum(
  28         __isl_take isl_pw_qpolynomial *pwqp);
  29 \end{verbatim}
  30 Compute the sum of the given piecewise quasipolynomial over
  31 all integer points in the domain.  The result is a piecewise
  32 quasipolynomial that only involves the parameters.
  33
  34 \subsection{Calculator}
  35
  36 The \ai[\tt]{iscc} calculator offers an interface to some
  37 of the functionality provided by the \isl/ and \barvinok/
  38 libraries.
  39 The supported operations are shown in \autoref{t:iscc}.
  40 Here are some examples:
  41 \begin{verbatim}
  42 P := [n, m] -> { [i,j] : 0 <= i <= n and i <= j <= m };
  43 card P;
  44
  45 f := [n,m] -> { [i,j] -> i*j + n*i*i*j : i,j >= 0 and 5i + 27j <= n+m };
  46 sum f;
  47 s := sum f;
  48 s @ [n,m] -> { [] : 0 <= n,m <= 20 };
  49
  50 f := [n] -> { [i] -> 2*n*i - n*n + 3*n - 1/2*i*i - 3/2*i-1 :
  51                 (exists j : 0 <= i < 4*n-1 and 0 <= j < n and
  52                             2*n-1 <= i+j <= 4*n-2 and i <= 2*n-1 ) };
  53 ub f;
  54 u := ub f;
  55 u @ [n] -> { [] : 0 <= n <= 10 };
  56
  57 m := [n] -> { [i,j] -> [i+1,j+1] : 1 <= i,j < n;
  58               [i,j] -> [i+1,j-1] : 1 <= i < n and 2 <= j <= n };
  59 m^+;
  60 (m^+)[0];
  61 \end{verbatim}
  62
  63 \bottomcaption{{\tt iscc} operations.  The variables
  64 have the following types,
  65 $s$: set,
  66 $m$: map,
  67 $q$: piecewise quasipolynomial,
  68 $f$: piecewise quasipolynomial fold,
  69 $l$: list,
  70 $i$: integer,
  71 $b$: boolean
  72 }
  73 \label{t:iscc}
  74 \tablehead{
  75 Syntax & Meaning
  76 \\
  77 \hline
  78 }
  79 \tabletail{
  80 \multicolumn{2}{r}{\small\sl continued on next page}
  81 \\
  82 }
  83 \tablelasttail{}
  84 \begin{supertabular}{lp{0.7\textwidth}}
  85 $s_2$ := \ai[\tt]{aff} $s_1$ & affine hull of $s_1$
  86 \\
  87 $m_2$ := \ai[\tt]{aff} $m_1$ & affine hull of $m_1$
  88 \\
  89 $q$ := \ai[\tt]{card} $s$ &
  90 number of elements in the set $s$
  91 \\
  92 $q$ := \ai[\tt]{card} $m$ &
  93 number of elements in the image of a domain element
  94 \\
  95 $s_2$ := \ai[\tt]{coalesce} $s_1$ &
  96 simplify the representation of set $s_1$ by trying
  97 to combine pairs of basic sets into a single
  98 basic set
  99 \\
 100 $m_2$ := \ai[\tt]{coalesce} $m_1$ &
 101 simplify the representation of map $m_1$ by trying
 102 to combine pairs of basic maps into a single
 103 basic map
 104 \\
 105 $q_2$ := \ai[\tt]{coalesce} $q_1$ &
 106 simplify the representation of $q_1$ by trying
 107 to combine pairs of basic sets in the domain
 108 of $q_1$ into a single basic set
 109 \\
 110 $f_2$ := \ai[\tt]{coalesce} $f_1$ &
 111 simplify the representation of $f_1$ by trying
 112 to combine pairs of basic sets in the domain
 113 of $f_1$ into a single basic set
 114 \\
 115 $s_3$ := $s_1$ \ai[\tt]{cross} $s_2$ &
 116 Cartesian product of $s_1$ and $s_2$
 117 \\
 118 $m_3$ := $m_1$ \ai[\tt]{cross} $m_2$ &
 119 Cartesian product of $m_1$ and $m_2$
 120 \\
 121 $s$ := \ai[\tt]{deltas} $m$ &
 122 the set $\{\, y - x \mid x \to y \in m \,\}$
 123 \\
 124 $s$ := \ai[\tt]{dom} $m$ &
 125 domain of map $m$
 126 \\
 127 $s$ := \ai[\tt]{dom} $q$ &
 128 domain of piecewise quasipolynomial $q$
 129 \\
 130 $s$ := \ai[\tt]{dom} $f$ &
 131 domain of piecewise quasipolynomial fold $f$
 132 \\
 133 $s$ := \ai[\tt]{ran} $m$ &
 134 range of map $m$
 135 \\
 136 $s_2$ := \ai[\tt]{lexmin} $s_1$ &
 137 lexicographically minimal element of $s_1$
 138 \\
 139 $m_2$ := \ai[\tt]{lexmin} $m_1$ &
 140 lexicographically minimal image element
 141 \\
 142 $s_2$ := \ai[\tt]{lexmax} $s_1$ &
 143 lexicographically maximal element of $s_1$
 144 \\
 145 $m_2$ := \ai[\tt]{lexmax} $m_1$ &
 146 lexicographically maximal image element
 147 \\
 148 $s_2$ := \ai[\tt]{sample} $s_1$ &
 149 a sample element of the set $s_1$
 150 \\
 151 $m_2$ := \ai[\tt]{sample} $m_1$ &
 152 a sample element of the map $m_1$
 153 \\
 154 $q_2$ := \ai[\tt]{sum} $q_1$ &
 155 sum $q_1$ over all integer points in the domain of $q_1$
 156 \\
 157 $f$ := \ai[\tt]{ub} $q$ &
 158 upper bound on the piecewise quasipolynomial $q$ over
 159 all integer points in the domain of $q$.
 160 This operation is only available if
 161 \ai[\tt]{GiNaC} support was compiled in.
 162 \\
 163 $s_3$ := $s_1$ \ai{$+$} $s_2$ & union
 164 \\
 165 $m_3$ := $m_1$ \ai{$+$} $m_2$ & union
 166 \\
 167 $q_3$ := $q_1$ \ai{$+$} $q_2$ & sum
 168 \\
 169 $s_3$ := $s_1$ \ai{$-$} $s_2$ & set difference
 170 \\
 171 $m_3$ := $m_1$ \ai{$-$} $m_2$ & set difference
 172 \\
 173 $q_3$ := $q_1$ \ai{$-$} $q_2$ & difference
 174 \\
 175 $s_3$ := $s_1$ \ai{$*$} $s_2$ & intersection
 176 \\
 177 $m_3$ := $m_1$ \ai{$*$} $m_2$ & intersection
 178 \\
 179 $q_3$ := $q_1$ \ai{$*$} $q_2$ & product
 180 \\
 181 $m_2$ := $m_1$ \ai{$*$} $s$ & intersect domain of $m_1$ with $s$
 182 \\
 183 $q_2$ := $q_1$ \ai{$*$} $s$ & intersect domain of $q_1$ with $s$
 184 \\
 185 $f_2$ := $f_1$ \ai{$*$} $s$ & intersect domain of $f_1$ with $s$
 186 \\
 187 $m_3$ := $m_1$ \ai[\tt]{.} $m_2$ & join of $m_1$ and $m_2$
 188 \\
 189 $m$ := $s_1$ \ai[\tt]{->} $s_2$ & universal map with domain $s_1$
 190 and range $s_2$
 191 \\
 192 $q_2$ := $q_1$ \ai{@} $s$ &
 193 evaluate the piecewise quasipolynomial $q_1$ in each element
 194 of the set $s$ and return a piecewise quasipolynomial
 195 mapping each of the individual elements to the resulting
 196 constant
 197 \\
 198 $q$ := $f$ \ai{@} $s$ &
 199 evaluate the piecewise quasipolynomial fold $f$ in each element
 200 of the set $s$ and return a piecewise quasipolynomial
 201 mapping each of the individual elements to the resulting
 202 constant
 203 \\
 204 $s_3$ := $s_1$ \ai[\tt]{\%} $s_2$ &
 205 simplify $s_1$ in the context of $s_2$, i.e., compute
 206 the gist of $s_1$ given $s_2$
 207 \\
 208 $m_3$ := $m_1$ \ai[\tt]{\%} $m_2$ &
 209 simplify $m_1$ in the context of $m_2$, i.e., compute
 210 the gist of $m_1$ given $m_2$
 211 \\
 212 $q_2$ := $q_1$ \ai[\tt]{\%} $s$ &
 213 simplify $q_1$ in the context of the domain $s$, i.e., compute
 214 the gist of $q_1$ given $s$
 215 \\
 216 $f_2$ := $f_1$ \ai[\tt]{\%} $s$ &
 217 simplify $f_1$ in the context of the domain $s$, i.e., compute
 218 the gist of $f_1$ given $s$
 219 \\
 220 $m_2$ := $m_1$\ai[\tt]{\^{}-1} & inverse of $m_1$
 221 \\
 222 $l$ := $m$\ai[\tt]{\^{}+} &
 223 compute an overapproximation of the transitive closure
 224 of $m$ and return a list containing the overapproximation
 225 and a boolean that is true if the overapproximation
 226 is known to be exact
 227 \\
 228 $l$[$i$] &
 229 the element at position $i$ in the list $l$
 230 \\
 231 $b$ := $s_1$ \ai[\tt]{=} $s_2$ & is $s_1$ equal to $s_2$?
 232 \\
 233 $b$ := $m_1$ \ai[\tt]{=} $m_2$ & is $m_1$ equal to $m_2$?
 234 \\
 235 $b$ := $s_1$ \ai[\tt]{<=} $s_2$ & is $s_1$ a subset of $s_2$?
 236 \\
 237 $b$ := $m_1$ \ai[\tt]{<=} $m_2$ & is $m_1$ a subset of $m_2$?
 238 \\
 239 $b$ := $s_1$ \ai[\tt]{<} $s_2$ & is $s_1$ a proper subset of $s_2$?
 240 \\
 241 $b$ := $m_1$ \ai[\tt]{<} $m_2$ & is $m_1$ a proper subset of $m_2$?
 242 \\
 243 $b$ := $s_1$ \ai[\tt]{>=} $s_2$ & is $s_1$ a superset of $s_2$?
 244 \\
 245 $b$ := $m_1$ \ai[\tt]{>=} $m_2$ & is $m_1$ a superset of $m_2$?
 246 \\
 247 $b$ := $s_1$ \ai[\tt]{>} $s_2$ & is $s_1$ a proper superset of $s_2$?
 248 \\
 249 $b$ := $m_1$ \ai[\tt]{>} $m_2$ & is $m_1$ a proper superset of $m_2$?
 250 \\
 251 \end{supertabular}