GAP (openmath) - Chapter 2: OpenMath functionality in GAP

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2 OpenMath functionality in GAP
2.1 Viewing OpenMath representation of an object
2.1-1 OMPrint
2.1-2 OMString
2.2 Reading OpenMath code from streams and strings
2.2-1 OMGetObject
2.2-2 EvalOMString
2.3 Writing OpenMath code to streams
2.3-1 IsOpenMathWriter
2.3-2 OpenMathXMLWriter
2.3-3 OpenMathBinaryWriter
2.3-4 OMPutObject
2.3-5 OMPlainString
2.4 Utilities
2.4-1 OMTestXML
2.4-2 OMTestBinary

### 2 OpenMath functionality in GAP

#### 2.1 Viewing OpenMath representation of an object

##### 2.1-1 OMPrint

`‣ OMPrint`( obj ) | ( function )

OMPrint writes the default XML OpenMath encoding of GAP object obj to the standard output.

One can try it with different GAP objects to see if they can be converted to OpenMath and learn how their OpenMath representation looks like. Here we show the encoding for lists of integers and rationals:

    gap> OMPrint( [ 1, 1/2 ] );
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="list1" name="list"/>
    		<OMI>1</OMI>
    		<OMA>
    			<OMS cd="nums1" name="rational"/>
    			<OMI>1</OMI>
    			<OMI>2</OMI>
    		</OMA>
    	</OMA>
    </OMOBJ>

Strings are encoded using `<OMSTR>` tags:

    gap> OMPrint( "This is a string" );
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMSTR>This is a string</OMSTR>
    </OMOBJ>

Cyclotomics may be encoded in different ways dependently on their properties:

    gap> OMPrint( 1-2*E(4) );
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="complex1" name="complex_cartesian"/>
    		<OMI>1</OMI>
    		<OMI>-2</OMI>
    	</OMA>
    </OMOBJ>
    gap> OMPrint(E(3));
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="arith1" name="plus"/>
    		<OMA>
    			<OMS cd="arith1" name="times"/>
    			<OMI>1</OMI>
    			<OMA>
    				<OMS cd="algnums" name="NthRootOfUnity"/>
    				<OMI>3</OMI>
    				<OMI>1</OMI>
    			</OMA>
    		</OMA>
    	</OMA>
    </OMOBJ>

Various encodings may be used for various types of groups:

    gap> OMPrint( Group( (1,2) ) );
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="permgp1" name="group"/>
    		<OMS cd="permutation1" name="right_compose"/>
    		<OMA>
    			<OMS cd="permut1" name="permutation"/>
    			<OMI>2</OMI>
    			<OMI>1</OMI>
    		</OMA>
    	</OMA>
    </OMOBJ>
    gap> OMPrint( Group( [ [ [ 1, 2 ],[ 0, 1 ] ] ] ) );
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="group1" name="group_by_generators"/>
    		<OMA>
    			<OMS cd="linalg2" name="matrix"/>
    			<OMA>
    				<OMS cd="linalg2" name="matrixrow"/>
    				<OMI>1</OMI>
    				<OMI>2</OMI>
    			</OMA>
    			<OMA>
    				<OMS cd="linalg2" name="matrixrow"/>
    				<OMI>0</OMI>
    				<OMI>1</OMI>
    			</OMA>
    		</OMA>
    	</OMA>
    </OMOBJ>
    gap> OMPrint( FreeGroup( 2 ) );
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="fpgroup1" name="free_groupn"/>
    		<OMI>2</OMI>
    	</OMA>
    </OMOBJ>

Producing OpenMath representation of polynomials, one may get a warning:

    gap> x:=Indeterminate(Rationals,"x");; y:=Indeterminate(Rationals,"y");;
    gap> OMPrint(x^2+y);
    #I  Warning : polynomial will be printed using its default ring
    #I  because the default OpenMath polynomial ring is not specified
    #I  or it is not contained in the default OpenMath polynomial ring.
    #I  You may ignore this or call SetOpenMathDefaultPolynomialRing to fix it.
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="polyd1" name="DMP"/>
    		<OMA id="polyring9qiY2oOaiITWUORb" >
    			<OMS cd="polyd1" name="poly_ring_d"/>
    			<OMS cd="setname1" name="Q"/>
    			<OMI>2</OMI>
    		</OMA>
    		<OMA>
    			<OMS cd="polyd1" name="SDMP"/>
    			<OMA>
    				<OMS cd="polyd1" name="term"/>
    				<OMI>1</OMI>
    				<OMI>0</OMI>
    				<OMI>1</OMI>
    			</OMA>
    			<OMA>
    				<OMS cd="polyd1" name="term"/>
    				<OMI>1</OMI>
    				<OMI>2</OMI>
    				<OMI>0</OMI>
    			</OMA>
    		</OMA>
    	</OMA>
    </OMOBJ>

Indeed, now when another polynomial will be printed, it will belong to a ring with a different identifier (despite GAP will be able to perform arithmetical operations on these polynomials like when they belong to the same ground ring):

    gap> OMPrint(x+1);
    #I  Warning : polynomial will be printed using its default ring
    #I  because the default OpenMath polynomial ring is not specified
    #I  or it is not contained in the default OpenMath polynomial ring.
    #I  You may ignore this or call SetOpenMathDefaultPolynomialRing to fix it.
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="polyd1" name="DMP"/>
    		<OMA id="polyring0LqlkhnCyLldcoBl" >
    			<OMS cd="polyd1" name="poly_ring_d_named"/>
    			<OMS cd="setname1" name="Q"/>
    			<OMV name="x"/>
    		</OMA>
    		<OMA>
    			<OMS cd="polyd1" name="SDMP"/>
    			<OMA>
    				<OMS cd="polyd1" name="term"/>
    				<OMI>1</OMI>
    				<OMI>1</OMI>
    			</OMA>
    			<OMA>
    				<OMS cd="polyd1" name="term"/>
    				<OMI>1</OMI>
    				<OMI>0</OMI>
    			</OMA>
    		</OMA>
    	</OMA>
    </OMOBJ>

Thus, the warning means that it is not guaranteed that the polynomial ring will have the same identifier `<OMA id="polyring..." >` when another or same polynomial from this ring will be printed next time. If this may constitute a problem, for example, if a list of polynomial is being exchanged with another system and it is crucial that all of them will belong to the same ring, then such ring must be created explicitly and then `SetOpenMathDefaultPolynomialRing` must be called:

    gap> x:=Indeterminate(Rationals,"x");; y:=Indeterminate(Rationals,"y");;
    gap> R:=PolynomialRing(Rationals,[x,y]);;
    gap> SetOpenMathDefaultPolynomialRing(R);
    gap> OMPrint(x^2+y);
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="polyd1" name="DMP"/>
    		<OMA id="polyring9eNcBGFHXkjl2kWh" >
    			<OMS cd="polyd1" name="poly_ring_d"/>
    			<OMS cd="setname1" name="Q"/>
    			<OMI>2</OMI>
    		</OMA>
    		<OMA>
    			<OMS cd="polyd1" name="SDMP"/>
    			<OMA>
    				<OMS cd="polyd1" name="term"/>
    				<OMI>1</OMI>
    				<OMI>0</OMI>
    				<OMI>0</OMI>
    			</OMA>
    			<OMA>
    				<OMS cd="polyd1" name="term"/>
    				<OMI>1</OMI>
    				<OMI>0</OMI>
    				<OMI>0</OMI>
    			</OMA>
    		</OMA>
    	</OMA>
    </OMOBJ>

Now we can see that both polynomials belong to the ring with the same identifier, and the OpenMath representation of the 2nd polynomial properly reflects that it belongs to a polynomial ring with two variables.

    gap> OMPrint(x+1);
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="polyd1" name="DMP"/>
    		<OMR href="#polyring9eNcBGFHXkjl2kWh" />
    		<OMA>
    			<OMS cd="polyd1" name="SDMP"/>
    			<OMA>
    				<OMS cd="polyd1" name="term"/>
    				<OMI>1</OMI>
    				<OMI>0</OMI>
    				<OMI>0</OMI>
    			</OMA>
    			<OMA>
    				<OMS cd="polyd1" name="term"/>
    				<OMI>1</OMI>
    				<OMI>0</OMI>
    				<OMI>0</OMI>
    			</OMA>
    		</OMA>
    	</OMA>
    </OMOBJ>

##### 2.1-2 OMString

`‣ OMString`( obj ) | ( function )

OMString returns a string with the default XML OpenMath encoding of GAP object obj. If used with the `noomobj` option, then initial and final <OMOBJ> tags will be omitted.

    gap> OMString(42);
    "<OMOBJ xmlns=\"http://www.openmath.org/OpenMath\" version=\"2.0\"> <OMI>42</OMI> </OMOBJ>"
    gap> OMString([1,2]:noomobj);
    "<OMA> <OMS cd=\"list1\" name=\"list\"/> <OMI>1</OMI> <OMI>2</OMI> </OMA>"

#### 2.2 Reading OpenMath code from streams and strings

##### 2.2-1 OMGetObject

`‣ OMGetObject`( stream ) | ( function )

stream is an input stream (see `InputTextFile` (Reference: InputTextFile), `InputTextUser` (Reference: InputTextUser), `InputTextString` (Reference: InputTextString), `InputOutputLocalProcess` (Reference: InputOutputLocalProcess), `InputOutputTCPStream` (SCSCP: InputOutputTCPStream for client), `InputOutputTCPStream` (SCSCP: InputOutputTCPStream for server)) with an OpenMath object on it. `OMGetObject` takes precisely one object off stream and returns it as a GAP object. Both XML and binary OpenMath encoding are supported: autodetection is used.

This may be used to retrieve objects from a file. In the following example we demonsrate reading the same content in binary and XML formats using the test files supplied with the package (the package autodetects whether binary or XML encoding is used):

    gap> txml:=Filename(DirectoriesPackageLibrary("openmath","tst"),"test3.omt");;
    gap> tbin:=Filename(DirectoriesPackageLibrary("openmath","tst"),"test3.bin");;
    gap> xstream := InputTextFile( txml );; bstream := InputTextFile( tbin );;
    gap> x:=OMGetObject(xstream); y:=OMGetObject(bstream);
    912873912381273891
    912873912381273891
    gap> x:=OMGetObject(xstream); y:=OMGetObject(bstream);
    E(4)
    E(4)
    gap> CloseStream(xstream);CloseStream(bstream);

To paste an OpenMath object directly into standard input execute the following command in GAP:

    gap> s:= InputTextUser();; g := OMGetObject(s); CloseStream(s);
    gap>

For XML OpenMath, this function requires that the GAP package GAPDoc is available.

##### 2.2-2 EvalOMString

`‣ EvalOMString`( omstr ) | ( function )

This function is an analog of `EvalString` (Reference: EvalString). Its argument omstr must be a string containing a single OpenMath object. `EvalOMString` will return the GAP object represented by omstr.

If omstr contains more OpenMath objects, the rest will be ignored.

    gap> s:="<OMOBJ><OMS cd=\"setname1\" name=\"Z\"/></OMOBJ>";;
    gap> EvalOMString(s);
    Integers
    gap> G:=SL(2,5);; G=EvalOMString(OMString(G));
    true

#### 2.3 Writing OpenMath code to streams

While it is possible to read OpenMath code directly from a stream, writing OpenMath to streams uses a different setup. It requires special objects called OpenMath writers, which encapsulate streams and may be viewed as transducers accepting GAP objects and writing them to a stream in the XML or binary OpenMath

Such setup makes it possible to re-use the same stream for both binary and XML OpenMath communication, using different OpenMath writers in different calls. It also allows to re-use most of the high-level code for GAP to OpenMath conversion, having separate methods for generating binary and XML OpenMath only for low-level output (OpenMath tags and basic objects). This makes easier adding support to new mathematical objects and private content dictionaries as described in Chapter 3 since it does not require changing the low-level functionality.

##### 2.3-1 IsOpenMathWriter

`‣ IsOpenMathWriter` | ( category )

`‣ IsOpenMathXMLWriter` | ( category )

`‣ IsOpenMathBinaryWriter` | ( category )

`IsOpenMathWriter`is a category for OpenMath writers. It has two subcategories: `IsOpenMathXMLWriter` and `IsOpenMathBinaryWriter`.

##### 2.3-2 OpenMathXMLWriter

`‣ OpenMathXMLWriter`( s ) | ( function )

for a stream s, returns an object in the category `IsOpenMathXMLWriter` (2.3-1).

##### 2.3-3 OpenMathBinaryWriter

`‣ OpenMathBinaryWriter`( s ) | ( function )

for a stream s, returns an object in the category `OpenMathBinaryWriter`.

##### 2.3-4 OMPutObject

`‣ OMPutObject`( stream, obj ) | ( function )

`‣ OMPutObjectNoOMOBJtags`( stream, obj ) | ( function )

`OMPutObject` writes (appends) the XML OpenMath encoding of the GAP object obj to output stream stream (see `InputTextFile` (Reference: InputTextFile), `OutputTextUser` (Reference: OutputTextUser), `OutputTextString` (Reference: OutputTextString), `InputOutputTCPStream` (SCSCP: InputOutputTCPStream for client), `InputOutputTCPStream` (SCSCP: InputOutputTCPStream for server)).

The second version does the same but without <OMOBJ> tags, what may be useful for assembling complex OpenMath objects.

    gap> g := [[1,2],[1,0]];;
    gap> t := "";
    ""
    gap> s := OutputTextString(t, true);;
    gap> w:=OpenMathXMLWriter( s );
    <OpenMath XML writer to OutputTextString(0)>
    gap> OMPutObject(w, g);
    gap> CloseStream(s);
    gap> Print(t);
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMA>
    		<OMS cd="linalg2" name="matrix"/>
    		<OMA>
    			<OMS cd="linalg2" name="matrixrow"/>
    			<OMI>1</OMI>
    			<OMI>2</OMI>
    		</OMA>
    		<OMA>
    			<OMS cd="linalg2" name="matrixrow"/>
    			<OMI>1</OMI>
    			<OMI>0</OMI>
    		</OMA>
    	</OMA>
    </OMOBJ>

##### 2.3-5 OMPlainString

`‣ OMPlainString`( string ) | ( function )

`OMPlainString` wraps the string into a GAP object of a special kind called an OpenMath plain string. Internally such object is represented as a string, but `OMPutObject` (2.3-4) threat it in a different way: instead of converting it into a <OMSTR> object, an OpenMath plain string will be plainly substituted into the output (this explains its name) without decorating it with <OMSTR> tags.

It is assumed that OpenMath plain string contains valid OpenMath code; no actual validation is performed during its creation. Such functionality may be useful to compose some OpenMath code at the GAP level to communicate it to the other system, in particular, to send there symbols which are not supported by GAP, for example:

    gap> s:=OMPlainString("<OMS cd=\"nums1\" name=\"pi\"/>");
    <OMS cd="nums1" name="pi"/>
    gap> OMPrint(s);
    <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    	<OMS cd="nums1" name="pi"/>
    </OMOBJ>

#### 2.4 Utilities

##### 2.4-1 OMTestXML

`‣ OMTestXML`( obj ) | ( function )

`‣ OMTest`( obj ) | ( function )

Converts obj to XML OpenMath and back. Returns true if and only if obj is unchanged (as a GAP object) by this operation. The OpenMath standard does not stipulate that converting to and from OpenMath should be the identity function so this is a useful diagnostic tool.

    gap> OMTestXML([[1..10],[1/2,2+E(4)],ZmodnZObj(2,6),(1,2),true,"string"]);
    true

`OMTest` is a synonym to `OMTestXML`

##### 2.4-2 OMTestBinary

`‣ OMTestBinary`( obj ) | ( function )

Converts obj to binary OpenMath and back. Returns true if and only if obj is unchanged (as a GAP object) by this operation. The OpenMath standard does not stipulate that converting to and from OpenMath should be the identity function so this is a useful diagnostic tool.

    gap> OMTestBinary([[1..10],[1/2,2+E(4)],ZmodnZObj(2,6),(1,2),true,"string"]);
    true

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