GAP (HomalgToCAS) - Chapter 3: Watch and Influence the Communication

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3 Watch and Influence the Communication
3.1 Functions
3.1-1 homalgIOMode
3.2 The Pictograms
3.2-1 HOMALG_IO.Pictograms

### 3 Watch and Influence the Communication

#### 3.1 Functions

##### 3.1-1 homalgIOMode

`‣ homalgIOMode`( str[, str2[, str3]] ) | ( function )

This function sets different modes which influence how much of the communication becomes visible. Handling the string str is not case-sensitive. `homalgIOMode` invokes the global function `homalgMode` defined in the homalg package with an "appropriate" argument (see code below). Alternatively, if a second or more strings are given, then `homalgMode` is invoked with the remaining strings str2, str3, ... at the end. In particular, you can use `homalgIOMode`( str, "" ) to reset the effect of invoking `homalgMode`.

str | str (long form) | mode description
"" | "" | the default mode, i.e. the communication protocol won't be visible
|  | (`homalgIOMode`( ) is a short form for `homalgIOMode`( "" ))
"a" | "all" | combine the modes "debug" and "file"
"b" | "basic" | the same as "picto" + `homalgMode`( "basic" )
"d" | "debug" | view the complete communication protocol
"f" | "file" | dump the communication protocol into a file with the name
|  | `Concatenation`( "commands_file_of_", CAS, "_with_PID_", PID )
"p" | "picto" | view the abbreviated communication protocol
|  | using the preassigned pictograms

All modes other than the "default"-mode only set their specific values and leave the other values untouched, which allows combining them to some extent. This also means that in order to get from one mode to a new mode (without the aim to combine them) one needs to reset to the "default"-mode first.
Caution:

- In case you choose one of the modes "file" or "all" you might want to set the global variable `HOMALG_IO.DoNotDeleteTemporaryFiles` := `true`; this is only important if during the computations some matrices get converted via files (using `ConvertHomalgMatrixViaFile`), as reading these files will be part of the protocol!

- It makes sense for the dumped communication protocol to be (re)executed with the respective external system, only in case the latter is deterministic (i.e. same-input-same-output).

    InstallGlobalFunction( homalgIOMode,
      function( arg )
        local nargs, mode, s;

        nargs := Length( arg );

        if nargs = 0 or ( IsString( arg[1] ) and arg[1] = "" ) then
            mode := "default";
        elif IsString( arg[1] ) then ## now we know, the string is not empty
            s := arg[1];
            if LowercaseString( s{[1]} ) = "a" then
                mode := "all";
            elif LowercaseString( s{[1]} ) = "b" then
                mode := "basic";
            elif LowercaseString( s{[1]} ) = "d" then
                mode := "debug";
            elif LowercaseString( s{[1]} ) = "f" then
                mode := "file";
            elif LowercaseString( s{[1]} ) = "p" then
                mode := "picto";
            else
                mode := "";
            fi;
        else
            Error( "the first argument must be a string\n" );
        fi;

        if mode = "default" then
            ## reset to the default values
            HOMALG_IO.color_display := false;
            HOMALG_IO.show_banners := true;
            HOMALG_IO.save_CAS_commands_to_file := false;
            HOMALG_IO.DoNotDeleteTemporaryFiles := false;
            HOMALG_IO.SaveHomalgMaximumBackStream := false;
            HOMALG_IO.InformAboutCASystemsWithoutActiveRings := true;
            SetInfoLevel( InfoHomalgToCAS, 1 );
            homalgMode( );
        elif mode = "all" then
            homalgIOMode( "debug" );
            homalgIOMode( "file" );
        elif mode = "basic" then
            HOMALG_IO.color_display := true;
            HOMALG_IO.show_banners := true;
            SetInfoLevel( InfoHomalgToCAS, 4 );
            homalgMode( "basic" ); ## use homalgIOMode( "basic", "" ) to reset
        elif mode = "debug" then
            HOMALG_IO.color_display := true;
            HOMALG_IO.show_banners := true;
            SetInfoLevel( InfoHomalgToCAS, 8 );
            homalgMode( "debug" ); ## use homalgIOMode( "debug", "" ) to reset
        elif mode = "file" then
            HOMALG_IO.save_CAS_commands_to_file := true;
        elif mode = "picto" then
            HOMALG_IO.color_display := true;
            HOMALG_IO.show_banners := true;
            SetInfoLevel( InfoHomalgToCAS, 4 );
            homalgMode( "logic" ); ## use homalgIOMode( "picto", "" ) to reset
        fi;

        if nargs > 1 and IsString( arg[2] ) then
            CallFuncList( homalgMode, arg{[ 2 .. nargs ]} );
        fi;

    end );

This is the part of the global function `homalgSendBlocking` that controls the visibility of the communication.

    io_info_level := InfoLevel( InfoHomalgToCAS );

    if not IsBound( pictogram ) then
        pictogram := "unknown";
        picto := HOMALG_IO.Pictograms.unknown;
    elif io_info_level >= 3 then
        picto := pictogram;
        if IsBound( HOMALG_IO.Pictograms.(pictogram) ) then
            pictogram := HOMALG_IO.Pictograms.(pictogram);
        else
            pictogram := HOMALG_IO.Pictograms.unknown;
        fi;

        ## add colors to the pictograms
        if pictogram = HOMALG_IO.Pictograms.ReducedEchelonForm and
           IsBound( HOMALG_MATRICES.color_BOE ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOE, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.BasisOfModule and
          IsBound( HOMALG_MATRICES.color_BOB ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOB, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.DecideZero and
          IsBound( HOMALG_MATRICES.color_BOD ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOD, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.SyzygiesGenerators and
          IsBound( HOMALG_MATRICES.color_BOH ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOH, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.BasisCoeff and
          IsBound( HOMALG_MATRICES.color_BOC ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOC, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.DecideZeroEffectively and
          IsBound( HOMALG_MATRICES.color_BOP ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOP, pictogram, "\033[0m" );
        elif need_output or need_display then
            pictogram := Concatenation( HOMALG_IO.Pictograms.color_need_output,
                                 pictogram, "\033[0m" );
        else
            pictogram := Concatenation( HOMALG_IO.Pictograms.color_need_command,
                                 pictogram, "\033[0m" );
        fi;
    else
        picto := pictogram;
    fi;

    if io_info_level >= 3 then
        if ( io_info_level >= 7 and not need_display ) or io_info_level >= 8 then
            ## print the pictogram, the prompt of the external system,
            ## and the sent command
            Info( InfoHomalgToCAS, 7, pictogram, " ", stream.prompt,
                  L{[ 1 .. Length( L ) - 1 ]} );
        elif io_info_level >= 4 then
            ## print the pictogram and the prompt of the external system
            Info( InfoHomalgToCAS, 4, pictogram, " ", stream.prompt, "..." );
        else
            ## print the pictogram only
            Info( InfoHomalgToCAS, 3, pictogram );
        fi;

    fi;

#### 3.2 The Pictograms

##### 3.2-1 HOMALG_IO.Pictograms

`‣ HOMALG_IO.Pictograms` | ( global variable )

The record of pictograms is a component of the record `HOMALG_IO`.

    Pictograms := rec(

      ##
      ## colors:
      ##

      ## pictogram color of a "need_command" or assignment operation:
      color_need_command                      := "\033[1;33;44m",

      ## pictogram color of a "need_output" or "need_display" operation:
      color_need_output                       := "\033[1;34;43m",

      ##
      ## good morning computer algebra system:
      ##

      ## initialize:
      initialize                              := "ini",

      ## define macros:
      define                                  := "def",

      ## get time:
      time                                    := ":ms",

      ## memory usage:
      memory                                  := "mem",

      ## version:
      version                                 := "ver",

      ## unknown:
      unknown                                 := "???",

      ##
      ## external garbage collection:
      ##

      ## delete a variable:
      delete                                  := "xxx",

      ## delete serveral variables:
      multiple_delete                         := "XXX",

      ## trigger the garbage collector:
      garbage_collector                       := "grb",

      ##
      ## create lists:
      ##

      ## define a list:
      CreateList                              := "lst",

      ##
      ## create rings:
      ##

      ## define a ring:
      CreateHomalgRing                        := "R:=",

      ## get the names of the "variables" defining the ring:
      variables                               := "var",

      ## define zero:
      Zero                                    := "0:=",

      ## define one:
      One                                     := "1:=",

      ## define minus one:
      MinusOne                                := "-:=",

      ##
      ## mandatory ring operations:
      ##

      ## get the name of an element:
      ## (important if the CAS pretty-prints ring elements,
      ##  we need names that can be used as input!)
      ## (install a method instead of a homalgTable entry)
      homalgSetName                           := "\"a\"",

      ## a = 0 ?
      IsZero                                  := "a=0",

      ## a = 1 ?
      IsOne                                   := "a=1",

      ## substract two ring elements
      ## (needed by SimplerEquivalentMatrix in case
      ##  CopyRow/ColumnToIdentityMatrix are not defined):
      Minus                                   := "a-b",

      ## divide the element a by the unit u
      ## (needed by SimplerEquivalentMatrix in case
      ##  DivideEntryByUnit is not defined):
      DivideByUnit                            := "a/u",

      ## important ring operations:
      ## (important for performance since existing
      ##  fallback methods cause a lot of traffic):

      ## is u a unit?
      ## (mainly needed by the fallback methods for matrices, see below):
      IsUnit                                  := "?/u",

      ##
      ## optional ring operations:
      ##

      ## copy an element:
      CopyElement                             := "a>a",

      ## add two ring elements:
      Sum                                     := "a+b",

      ## multiply two ring elements:
      Product                                 := "a*b",

      ## the (greatest) common divisor:
      Gcd                                     := "gcd",

      ## cancel the (greatest) common divisor:
      CancelGcd                               := "ccd",

      ## random polynomial:
      RandomPol                               := "rpl",

      ## random matrix:
      RandomMat                               := "rma",

      ## numerator:
      Numerator                               := "num",

      ## denominator:
      Denominator                             := "den",

      ## evaluate polynomial:
      Evaluate                                := "evl",

      ## degree of a multivariate polynomial
      DegreeOfRingElement                     := "deg",

      ## maximal degree part of a polynomial
      MaximalDegreePart                       := "mdp",

      ## is irreducible:
      IsIrreducible                           := "irr",

      ##
      ## create matrices:
      ##

      ## define a matrix:
      HomalgMatrix                            := "A:=",

      ## copy a matrix:
      CopyMatrix                              := "A>A",

      ## load a matrix from file:
      LoadHomalgMatrixFromFile                := "A<<",

      ## save a matrix to file:
      SaveHomalgMatrixToFile                  := "A>>",

      ## get a matrix entry as a string:
      MatElm                  := "<ij",

      ## set a matrix entry from a string:
      SetMatElm                  := ">ij",

      ## add to a matrix entry from a string:
      AddToMatElm                := "+ij",

      ## get a list of the matrix entries as a string:
      GetListOfHomalgMatrixAsString           := "\"A\"",

      ## get a listlist of the matrix entries as a string:
      GetListListOfHomalgMatrixAsString       := "\"A\"",

      ## get a "sparse" list of the matrix entries as a string:
      GetSparseListOfHomalgMatrixAsString     := ".A.",

      ## assign a "sparse" list of matrix entries to a variable:
      sparse                                  := "spr",

      ## list of assumed inequalities:
      Inequalities                            := "<>0",

      ## list of assumed inequalities:
      MaximalIndependentSet                   := "idp",

      ##
      ## mandatory matrix operations:
      ##

      ## test if a matrix is the zero matrix:
      ## CAUTION: the external system must be able to check
      ##          if the matrix is zero modulo possible ring relations
      ##          only known to the external system!
      IsZeroMatrix                            := "A=0",

      ## number of rows:
      NrRows                                  := "#==",

      ## number of columns:
      NrColumns                               := "#||",

      ## determinant of a matrix over a (commutative) ring:
      Determinant                             := "det",

      ## create a zero matrix:
      ZeroMatrix                              := "(0)",

      ## create a initial zero matrix:
      InitialMatrix                           := "[0]",

      ## create an identity matrix:
      IdentityMatrix                          := "(1)",

      ## create an initial identity matrix:
      InitialIdentityMatrix                   := "[1]",

      ## "transpose" a matrix (with "the" involution of the ring):
      Involution                              := "A^*",

      ## transpose a matrix
      TransposedMatrix                        := "A^t",

      ## get certain rows of a matrix:
      CertainRows                             := "===",

      ## get certain columns of a matrix:
      CertainColumns                          := "|||",

      ## stack matrices vertically:
      UnionOfRows                             := "A_B",

      ## glue matrices horizontally:
      UnionOfColumns                          := "A|B",

      ## create a block diagonal matrix:
      DiagMat                                 := "A\\B",

      ## the Kronecker (tensor) product of two matrices:
      KroneckerMat                            := "AoB",

      ## the dual Kronecker product of two matrices:
      DualKroneckerMat                        := "A#B",

      ## multiply a ring element with a matrix:
      MulMat                                  := "a*A",

      ## multiply a matrix with a ring element:
      MulMatRight                             := "A*a",

      ## add two matrices:
      AddMat                                  := "A+B",

      ## substract two matrices:
      SubMat                                  := "A-B",

      ## multiply two matrices:
      Compose                                 := "A*B",

      ## pullback a matrix by a ring map:
      Pullback                                := "pbk",

      ##
      ## important matrix operations:
      ## (important for performance since existing
      ##  fallback methods cause a lot of traffic):
      ##

      ## test if two matrices are equal:
      ## CAUTION: the external system must be able to check
      ##          equality of the two matrices modulo possible ring relations
      ##          only known to the external system!
      AreEqualMatrices                        := "A=B",

      ## test if a matrix is the identity matrix:
      IsIdentityMatrix                        := "A=1",

      ## test if a matrix is diagonal (needed by the display method):
      IsDiagonalMatrix                        := "A=\\",

      ## get the positions of the zero rows:
      ZeroRows                                := "0==",

      ## get the positions of the zero columns:
      ZeroColumns                             := "0||",

      ## get "column-independent" unit positions
      ## (needed by ReducedBasisOfModule):
      GetColumnIndependentUnitPositions       := "ciu",

      ## get "row-independent" unit positions
      ## (needed by ReducedBasisOfModule):
      GetRowIndependentUnitPositions          := "riu",

      ## get the position of the "first" unit in the matrix
      ## (needed by SimplerEquivalentMatrix):
      GetUnitPosition                         := "gup",

      ## position of the first non-zero entry per row
      PositionOfFirstNonZeroEntryPerRow       := "fnr",

      ## position of the first non-zero entry per column
      PositionOfFirstNonZeroEntryPerColumn    := "fnc",

      ## indicator matrix of non-zero entries
      IndicatorMatrixOfNonZeroEntries         := "<>0",

      ## transposed matrix:
      TransposedMatrix                        := "^tr",

      ## divide an entry of a matrix by a unit
      ## (needed by SimplerEquivalentMatrix in case
      ##  DivideRow/ColumnByUnit are not defined):
      DivideEntryByUnit                       := "ij/",

      ## divide a row by a unit
      ## (needed by SimplerEquivalentMatrix):
      DivideRowByUnit                         := "-/u",

      ## divide a column by a unit
      ## (needed by SimplerEquivalentMatrix):
      DivideColumnByUnit                      := "|/u",

      ## divide a row by a unit
      ## (needed by SimplerEquivalentMatrix):
      CopyRowToIdentityMatrix                 := "->-",

      ## divide a column by a unit
      ## (needed by SimplerEquivalentMatrix):
      CopyColumnToIdentityMatrix              := "|>|",

      ## set a column (except a certain row) to zero
      ## (needed by SimplerEquivalentMatrix):
      SetColumnToZero                         := "|=0",

      ## get the positions of the rows with a single one
      ## (needed by SimplerEquivalentMatrix):
      GetCleanRowsPositions                   := "crp",

      ## convert a single row matrix into a matrix
      ## with specified number of rows/columns
      ## (needed by the display methods for homomorphisms):
      ConvertRowToMatrix                      := "-%A",

      ## convert a single column matrix into a matrix
      ## with specified number of rows/columns
      ## (needed by the display methods for homomorphisms):
      ConvertColumnToMatrix                   := "|%A",

      ## convert a matrix into a single row matrix:
      ConvertMatrixToRow                      := "A%-",

      ## convert a matrix into a single column matrix:
      ConvertMatrixToColumn                   := "A%|",

      ##
      ## basic matrix operations:
      ##

      ## compute a (r)educed (e)chelon (f)orm:
      ReducedEchelonForm                      := "ref",

      ## compute a "(bas)is" of a given set of module elements:
      BasisOfModule                           := "bas",

      ## compute a reduced "(Bas)is" of a given set of module elements:
      ReducedBasisOfModule                    := "Bas",

      ## (d)e(c)ide the ideal/submodule membership problem,
      ## i.e. if an element is (0) modulo the ideal/submodule:
      DecideZero                              := "dc0",

      ## compute a generating set of (syz)ygies:
      SyzygiesGenerators                      := "syz",

      ## compute a generating set of reduced (Syz)ygies:
      ReducedSyzygiesGenerators               := "Syz",

      ## compute a (R)educed (E)chelon (F)orm
      ## together with the matrix of coefficients:
      ReducedEchelonFormC                     := "REF",

      ## compute a "(BAS)is" of a given set of module elements
      ## together with the matrix of coefficients:
      BasisCoeff                              := "BAS",

      ## (D)e(C)ide the ideal/submodule membership problem,
      ## i.e. write an element effectively as (0) modulo the ideal/submodule:
      DecideZeroEffectively                   := "DC0",

      ##
      ## optional matrix operations:
      ##

      ## Hilbert-Poincare series of a module:
      HilbertPoincareSeries                   := "HPs",

      ## Hilbert polynomial of a module:
      HilbertPolynomial                       := "Hil",

      ## affine dimension of a module:
      AffineDimension                         := "dim",

      ## affine degree of a module:
      AffineDegree                            := "adg",

      ## the constant term of the hilbert polynomial:
      ConstantTermOfHilbertPolynomial         := "P_0",

      ## differentiate a matrix M w.r.t. a matrix D
      Diff                                    := "dif",

      ## maximal dimensional radical subobject:
      MaxDimensionalRadicalSubobject          := "V_d",

      ## radical subobject:
      RadicalSubobject                        := "rad",

      ## radical decomposition:
      RadicalDecomposition                    := "VxU",

      ## maximal dimensional subobject:
      MaxDimensionalSubobject                 := "X_d",

      ## equi-dimensional decomposition:
      EquiDimenionalDecomposition             := "XxY",

      ## primary decomposition:
      PrimaryDecomposition                    := "YxZ",

      ## eliminate variables:
      Eliminate                               := "eli",

      ## leading module:
      LeadingModule                           := "led",

      ## the i-th monomial matrix
      MonomialMatrix                          := "mon",

      ## matrix of symbols:
      MatrixOfSymbols                         := "smb",

      ## leading module:
      ## coefficients:
      Coefficients                            := "cfs",

      ##
      ## optional module operations:
      ##

      ## compute a better equivalent matrix
      ## (field -> row+col Gauss, PIR -> Smith, Dedekind domain -> Krull, etc ...):
      BestBasis                               := "(\\)",

      ## compute elementary divisors:
      ElementaryDivisors                      := "div",

      ##
      ## for the eye:
      ##

      ## display objects:
      Display                                 := "dsp",

      ## the LaTeX code of the mathematical entity:
      homalgLaTeX                             := "TeX",

    )

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