This section describes libSBML's facilities for working with SBML representations of mathematical expressions.

Basic concepts

LibSBML uses Abstract Syntax Trees (ASTs) to provide a canonical, in-memory representation for all mathematical formulas regardless of their original format (i.e., C-like infix strings or MathML). In libSBML, an AST is a collection of one or more objects of type ASTNode. An AST node in libSBML is a recursive structure containing a pointer to the node's value (which might be, for example, a number or a symbol) and a list of children nodes. Each ASTNode node may have none, one, two, or more child depending on its type. The following diagram illustrates an example of how the mathematical expression "1 + 2" is represented as an AST with one plus node having two integer children nodes for the numbers 1 and 2. The figure also shows the corresponding MathML 2.0 representation:

Example AST representation of a mathematical expression.
Infix	AST	MathML
`1 + 2`		`<math xmlns="http://www.w3.org/1998/Math/MathML">` `<apply>` `<plus/>` `<cn type="integer"> 1 </cn>` `<cn type="integer"> 2 </cn>` `</apply>` `</math>`

The following are noteworthy about the AST representation in libSBML:

A numerical value represented in MathML 2.0 as a real number with an exponent is preserved as such in the AST node representation, even if the number could be stored in a double data type. This is done so that when an SBML model is read in and then written out again, the amount of change introduced by libSBML to the SBML during the round-trip activity is minimized.
Rational numbers are represented in an AST node using separate numerator and denominator values. These can be retrieved using the methods libsbmlcs.ASTNode.getNumerator() and libsbmlcs.ASTNode.getDenominator().
The children of an ASTNode are other ASTNode objects. The list of children is empty for nodes that are leaf elements, such as numbers. For nodes that are actually roots of expression subtrees, the list of children points to the parsed objects that make up the rest of the expression.

For many applications, the details of ASTs are irrelevant because the applications can use the text-string based translation functions such as libsbmlcs.formulaToString(), libsbmlcs.parseL3Formula(), and libsbmlcs.parseFormula(). If you find the complexity of using the AST representation of expressions too high for your purposes, perhaps the string-based functions will be more suitable.

Finally, it is worth noting that the AST and MathML handling code in libSBML remains written in C, not C++, as all of libSBML was originally written in C. Readers may occasionally wonder why some aspects are more C-like than following a C++ style, and that's the reason.

Converting between ASTs and Text Strings

SBML Levels 2 and 3 represents mathematical expressions using MathML 2.0 (more specifically, a subset of the content portion of MathML 2.0), but most software applications using libSBML do not use MathML directly. Instead, applications generally either interact with mathematics in text-string form, or else they use the API for working with Abstract Syntax Trees (described below). LibSBML provides support for both approaches. The libSBML formula parser has been carefully engineered so that transformations from MathML to infix string notation and back is possible with a minimum of disruption to the structure of the mathematical expression.

The example below shows a simple program that, when run, takes a MathML string compiled into the program, converts it to an AST, converts that to an infix representation of the formula, compares it to the expected form of that formula, and finally translates that formula back to MathML and displays it. The output displayed on the terminal should have the same structure as the MathML it started with. The program is a simple example of using the various MathML and AST reading and writing methods, and shows that libSBML preserves the ordering and structure of the mathematical expressions.

using System;
using libsbml;
public class Example
{
    public static void Main(String[] args)
    {
        String expected = "1 + f(x)";
        String input_mathml = "<?xml version='1.0' encoding='UTF-8'?>"
          + "<math xmlns='http://www.w3.org/1998/Math/MathML'>"
          + "  <apply> <plus/> <cn> 1 </cn>"
          + "                  <apply> <ci> f </ci> <ci> x </ci> </apply>"
          + "  </apply>"
          + "</math>";
        ASTNode ast_result   = libsbmlcs.libsbml.readMathMLFromString(input_mathml);
        String ast_as_string = libsbmlcs.libsbml.formulaToString(ast_result);
        if (ast_as_string == expected)
        {
            Console.WriteLine("Got expected result.");
        }
        else
        {
            Console.WriteLine("Mismatch after readMathMLFromString().");
            Environment.Exit(1);
        }
        ASTNode new_mathml = libsbmlcs.libsbml.parseFormula(ast_as_string);
        String new_string = libsbmlcs.libsbml.writeMathMLToString(new_mathml);
        Console.WriteLine("Result of writing AST to string:");
        Console.WriteLine(new_string);
    }
}

The text-string form of mathematical formulas produced by libsbmlcs.libsbml.formulaToString() and read by libsbmlcs.libsbml.parseL3Formula() and libsbmlcs.libsbml.parseFormula() are in a simple C-inspired infix notation. It is summarized in the next section below. A formula in this text-string form therefore can be handed to a program that understands SBML mathematical expressions, or used as part of a translation system. In summary, the functions available are the following:

libsbmlcs.libsbml.formulaToString(ASTNode) $\rightarrow$ string reads an AST, converts it to a text string in SBML Level 1 formula syntax, and returns it. The caller owns the character string returned and should free it after it is no longer needed.
libsbmlcs.libsbml.parseFormula(string) $\rightarrow$ ASTNode reads a text-string containing a mathematical expression in SBML Level 1 syntax, and returns an AST corresponding to the expression.
libsbmlcs.libsbml.parseL3Formula(string) $\rightarrow$ ASTNode reads a text-string containing a mathematical expression in an expanded syntax more compatible with SBML Levels 2 and 3, and returns an AST corresponding to the expression.

The String Formula Syntax and Differences with MathML

The text-string formula syntax is an infix notation essentially derived from the syntax of the C programming language and was originally used in SBML Level 1. The formula strings may contain operators, function calls, symbols, and white space characters. The allowable white space characters are tab and space. The following are illustrative examples of formulas expressed in the syntax:

0.10 * k4^2

(vm * s1)/(km + s1)

The following table shows the precedence rules in this syntax. In the Class column, operand implies the construct is an operand, prefix implies the operation is applied to the following arguments, unary implies there is one argument, and binary implies there are two arguments. The values in the Precedence column show how the order of different types of operation are determined. For example, the expression a * b + c is evaluated as (a * b) + c because the * operator has higher precedence. The Associates column shows how the order of similar precedence operations is determined; for example, a - b + c is evaluated as (a - b) + c because the + and - operators are left-associative. The precedence and associativity rules are taken from the C programming language, except for the symbol ^, which is used in C for a different purpose. (Exponentiation can be invoked using either ^ or the function power.)

A table of the expression operators and their precedence in the text-string format for mathematical expressions used by SBML_parseFormula().
Token	Operation	Class	Precedence	Associates
name	symbol reference	operand	6	n/a
`(`expression`)`	expression grouping	operand	6	n/a
`f(`...`)`	function call	prefix	6	left
`-`	negation	unary	5	right
`^`	power	binary	4	left
`*`	multiplication	binary	3	left
`/`	divison	binary	3	left
`+`	addition	binary	2	left
`-`	subtraction	binary	2	left
`,`	argument delimiter	binary	1	left

A program parsing a formula in an SBML model should assume that names appearing in the formula are the identifiers of Species, Parameter, Compartment, FunctionDefinition, or Reaction objects defined in a model. When a function call is involved, the syntax consists of a function identifier, followed by optional white space, followed by an opening parenthesis, followed by a sequence of zero or more arguments separated by commas (with each comma optionally preceded and/or followed by zero or more white space characters), followed by a closing parenthesis. There is an almost one-to-one mapping between the list of predefined functions available, and those defined in MathML. All of the MathML funcctions are recognized; this set is larger than the functions defined in SBML Level 1. In the subset of functions that overlap between MathML and SBML Level 1, there exist a few differences. The following table summarizes the differences between the predefined functions in SBML Level 1 and the MathML equivalents in SBML Level 2:

Table comparing the names of certain functions in the SBML text-string formula syntax and MathML. The left column shows the names of functions recognized by SBML_parseFormula(); the right column shows their equivalent function names in MathML 2.0, used in SBML Levels 2 and 3.
Text string formula functions	MathML equivalents in SBML Levels 2 and 3
`acos`	`arccos`
`asin`	`arcsin`
`atan`	`arctan`
`ceil`	`ceiling`
`log`	`ln`
`log10(x)`	`log(x)` or `log(10, x)`
`pow(x, y)`	`power(x, y)`
`sqr(x)`	`power(x, 2)`
`sqrt(x)`	`root(x)` or `root(2, x)`

Methods for working with libSBML's Abstract Syntax Trees

Every ASTNode in a libSBML AST has an associated type, a value taken from a set of constants with names beginning with AST_ and defined in the interface class libsbml. The list of possible types is quite long, because it covers all the mathematical functions that are permitted in SBML's subset of MathML. The values are shown in the following table; their names hopefully evoke the construct that they represent:


`AST_UNKNOWN`	`AST_FUNCTION_ARCCOTH`	`AST_FUNCTION_POWER`
`AST_PLUS`	`AST_FUNCTION_ARCCSC`	`AST_FUNCTION_ROOT`
`AST_MINUS`	`AST_FUNCTION_ARCCSCH`	`AST_FUNCTION_SEC`
`AST_TIMES`	`AST_FUNCTION_ARCSEC`	`AST_FUNCTION_SECH`
`AST_DIVIDE`	`AST_FUNCTION_ARCSECH`	`AST_FUNCTION_SIN`
`AST_POWER`	`AST_FUNCTION_ARCSIN`	`AST_FUNCTION_SINH`
`AST_INTEGER`	`AST_FUNCTION_ARCSINH`	`AST_FUNCTION_TAN`
`AST_REAL`	`AST_FUNCTION_ARCTAN`	`AST_FUNCTION_TANH`
`AST_REAL_E`	`AST_FUNCTION_ARCTANH`	`AST_LOGICAL_AND`
`AST_RATIONAL`	`AST_FUNCTION_CEILING`	`AST_LOGICAL_NOT`
`AST_NAME`	`AST_FUNCTION_COS`	`AST_LOGICAL_OR`
`AST_NAME_TIME`	`AST_FUNCTION_COSH`	`AST_LOGICAL_XOR`
`AST_CONSTANT_E`	`AST_FUNCTION_COT`	`AST_RELATIONAL_EQ`
`AST_CONSTANT_FALSE`	`AST_FUNCTION_COTH`	`AST_RELATIONAL_GEQ`
`AST_CONSTANT_PI`	`AST_FUNCTION_CSC`	`AST_RELATIONAL_GT`
`AST_CONSTANT_TRUE`	`AST_FUNCTION_CSCH`	`AST_RELATIONAL_LEQ`
`AST_LAMBDA`	`AST_FUNCTION_EXP`	`AST_RELATIONAL_LT`
`AST_FUNCTION`	`AST_FUNCTION_FACTORIAL`	`AST_RELATIONAL_NEQ`
`AST_FUNCTION_ABS`	`AST_FUNCTION_FLOOR`
`AST_FUNCTION_ARCCOS`	`AST_FUNCTION_LN`
`AST_FUNCTION_ARCCOSH`	`AST_FUNCTION_LOG`
`AST_FUNCTION_ARCCOT`	`AST_FUNCTION_PIECEWISE`

There are a number of methods for interrogating the type of an ASTNode and for testing whether a node belongs to a general category of constructs. The methods are the following:

int ASTNode.getType() returns the type of this AST node.
bool ASTNode.isConstant() returns true if this AST node is a MathML constant (true, false, pi, exponentiale), false otherwise.
bool ASTNode.isBoolean() returns true if this AST node returns a boolean value (by being either a logical operator, a relational operator, or the constant true or false).
bool ASTNode.isFunction() returns true if this AST node is a function (i.e., a MathML defined function such as exp or else a function defined by a FunctionDefinition in the Model).
bool ASTNode.isInfinity() returns true if this AST node is the special IEEE 754 value infinity.
bool ASTNode.isInteger() returns true if this AST node is holding an integer value.
bool ASTNode.isNumber() returns true if this AST node is holding any number.
bool ASTNode.isLambda() returns true if this AST node is a MathML lambda construct.
bool ASTNode.isLog10() returns true if this AST node represents the log10 function, specifically, that its type is AST_FUNCTION_LOG and it has two children, the first of which is an integer equal to 10.
bool ASTNode.isLogical() returns true if this AST node is a logical operator (and, or, not, xor).
bool ASTNode.isName() returns true if this AST node is a user-defined name or (in SBML Level 2) one of the two special csymbol constructs "delay" or "time".
bool ASTNode.isNaN() returns true if this AST node has the special IEEE 754 value "not a number" (NaN).
bool ASTNode.isNegInfinity() returns true if this AST node has the special IEEE 754 value of negative infinity.
bool ASTNode.isOperator() returns true if this AST node is an operator (e.g., +, -, etc.)
bool ASTNode.isPiecewise() returns true if this AST node is the MathML piecewise function.
bool ASTNode.isRational() returns true if this AST node is a rational number having a numerator and a denominator.
bool ASTNode.isReal() returns true if this AST node is a real number (specifically, AST_REAL_E or AST_RATIONAL).
bool ASTNode.isRelational() returns true if this AST node is a relational operator.
bool ASTNode.isSqrt() returns true if this AST node is the square-root operator
bool ASTNode.isUMinus() returns true if this AST node is a unary minus.
bool ASTNode.isUnknown() returns true if this AST node's type is unknown.

Programs manipulating AST node structures should check the type of a given node before calling methods that return a value from the node. The following meethods are available for returning values from nodes:

Finally (and rather predictably), libSBML provides methods for setting the values of AST nodes.

ASTNode.setCharacter(char) sets the value of this ASTNode to the given character. If character is one of +, -, *, / or ^, the node type will be to the appropriate operator type. For all other characters, the node type will be set to AST_UNKNOWN.
ASTNode.setName(string) sets the value of this AST node to the given name. The node type will be set (to AST_NAME) only if the AST node was previously an operator (isOperator() != 0) or number (isNumber() != 0). This allows names to be set for AST_FUNCTIONs and the like.
ASTNode.setValue(int) sets the value of the node to the given integer.
ASTNode.setValue(int, int) sets the value of this ASTNode to the given rational in two parts: the numerator and denominator. The node type is set to AST_RATIONAL.
ASTNode.setValue(double) sets the value of this ASTNode to the given floating-point number and sets the node type to AST_REAL.
ASTNode.setValue(double, int) sets the value of this ASTNode to the given floating-point number in two parts: the mantissa and the exponent. The node type is set to AST_REAL_E.

The following are some miscellaneous methods for manipulating ASTs:

ASTNode ASTNode.ASTNode(int) creates a new ASTNode object and returns a pointer to it. The returned node will have the type identified by the code passed as the argument, or a type of AST_UNKNOWN if no type is explicitly given or the type code is unrecognized.
unsigned int ASTNode.getNumChildren() returns the number of children of this AST node or 0 is this node has no children.
ASTNode.addChild(ASTNode) adds the given node as a child of this AST node. Child nodes are added in left-to-right order.
ASTNode.prependChild(ASTNode) adds the given node as a child of this AST node. This method adds child nodes in right-to-left order.
ASTNode (long) ASTNode.getChild (long) returns the nth child of this AST node or NULL if this node has no nth child (n > (ASTNode.getNumChildren() - 1)).
ASTNode ASTNode.getLeftChild() returns the left child of this AST node. This is equivalent to ASTNode.getChild(0);
ASTNode ASTNode.getRightChild() returns the right child of this AST node or NULL if this node has no right child.
ASTNode.swapChildren(ASTNode) swaps the children of this ASTNode with the children of that ASTNode.
ASTNode.setType(int) sets the type of this ASTNode to the type identified by the type code passed as argument, or to AST_UNKNOWN if the type is unrecognized.

Reading and Writing MathML from/to ASTs

As mentioned above, applications often can avoid working with raw MathML by using either libSBML's text-string interface or the AST API. However, when needed, reading MathML content directly and creating ASTs, as well as the converse task of writing MathML, is easily done using two methods designed for this purpose:

ASTNode readMathMLFromString(string) reads raw MathML from a text string, constructs an AST from it, then returns the root ASTNode of the resulting expression tree.
string writeMathMLToString(ASTNode) writes an AST to a string. The caller owns the character string returned and should free it after it is no longer needed.