• Basic Concepts
• Converting between ASTs and Text Strings
• The String Formula Syntax and Differences with MathML
• Methods for working with libSBML's Abstract Syntax Trees
• Reading and Writing MathML from/to ASTs

This section describes libSBML's facilities for working with SBML representations of mathematical expressions. Unless otherwise noted, all classes are in the Java package `org.sbml.libsbml`.

## 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 2.0). In libSBML, an AST is a collection of one or more objects of class `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 Java `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 `getNumerator()` and `getDenominator()` methods provided by the `ASTNode` class.
- The children of an `ASTNode` object 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 libSBML's text-string based translation functions such as `libsbml.formulaToString(ASTNode)` and `libsbml.parseFormula(java.lang.String)`. 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 represent 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.

    import org.sbml.libsbml.ASTNode;
    import org.sbml.libsbml.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   = libsbml.readMathMLFromString(input_mathml);
          String ast_as_string = libsbml.formulaToString(ast_result);

          if (ast_as_string.equals(expected))
          {
              System.out.println("Got expected result.");
          }
          else
          {
              System.out.println("Mismatch after readMathMLFromString().");
              System.exit(1);
          }

          ASTNode new_mathml = libsbml.parseFormula(ast_as_string);
          String new_string  = libsbml.writeMathMLToString(new_mathml);

          System.out.println("Result of writing AST to string:");
          System.out.print(new_string);
          System.out.println();
      }

      static
      {
        try
        {
          System.loadLibrary("sbmlj");
        }
        catch (Exception e)
        {
          System.err.println("Could not load libSBML library:" + e.getMessage());
        }
      }
    }

The text-string form of mathematical formulas produced by `libsbml.formulaToString(ASTNode)` and read by `libsbml.parseFormula(java.lang.String)` are simple C-inspired infix notation taken from SBML Level 1. 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 Level 1 mathematical expressions, or used as part of a translation system. In summary, the functions available are the following:

- `static java.lang.String libsbml.formulaToString(ASTNode)` 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.
- `static ASTNode libsbml.parseFormula(java.lang.String)` reads a text-string containing a mathematical expression in SBML Level 1 syntax, 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`.)

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 table of the expression operators and their precedence in the text-string format for mathematical expressions used by `libsbml.parseFormula(java.lang.String)`.

A program parsing a formula in an SBML model should assume that names appearing in the formula are the identifiers of `Species`, `Compartment`, `Parameter`, `FunctionDefinition`, (in Level 2) `Reaction`, or (in Level 3) `SpeciesReference` 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 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(10, x)`
`pow(x, y)` | `power(x, y)`
`sqr(x)` | `power(x, 2)`
`sqrt(x)` | `root(2, x)`
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 `libsbml.parseFormula(java.lang.String)`; the right column shows their equivalent function names in MathML 2.0, used in SBML Levels 2 and 3.

## Methods for working with libSBML's Abstract Syntax Trees

Every `ASTNode` in a libSBML abstract syntax tree has an associated type, which is a value taken from a set of constants having names beginning with `AST_` and defined in `org.sbml.libsbml.libsbmlConstants`. The list of possible AST types in libSBML is quite long, because it covers all the mathematical functions that are permitted in SBML. 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_NAME_AVOGADRO` (Level 3 only)
`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 defined by the `ASTNode` class are the following:

- `int getType()` returns the type of this AST node.
- `boolean isConstant()` returns `true` if this AST node is a MathML constant (`true`, `false`, `pi`, `exponentiale`), `false` otherwise.
- `boolean 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`).
- `boolean 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` object in the enclosing `Model` object).
- `boolean isInfinity()` returns `true` if this AST node is the special IEEE 754 value infinity.
- `boolean isInteger()` returns `true` if this AST node is holding an integer value.
- `boolean isNumber()` returns `true` if this AST node is holding any number.
- `boolean isLambda()` returns `true` if this AST node is a MathML `lambda` construct.
- `boolean isLog10()` returns `true` if this AST node represents the `log10` function, specifically, that its type code is `AST_FUNCTION_LOG` and it has two children, the first of which is an integer equal to 10.
- `boolean isLogical()` returns `true` if this AST node is a logical operator (`and`, `or`, `not`, `xor`).
- `boolean isName()` returns `true` if this AST node is a user-defined name or (in SBML Levels 2 and 3) one of the two special `csymbol` constructs "delay" or "time".
- `boolean isNaN()` returns `true` if this AST node has the special IEEE 754 value "not a number" (NaN).
- `boolean isNegInfinity()` returns `true` if this AST node has the special IEEE 754 value of negative infinity.
- `boolean isOperator()` returns `true` if this AST node is an operator (e.g., `+`, `-`, etc.)
- `boolean isPiecewise()` returns `true` if this AST node is the MathML `piecewise` function.
- `boolean isRational()` returns `true` if this AST node is a rational number having a numerator and a denominator.
- `boolean isReal()` returns `true` if this AST node is a real number (specifically, `AST_REAL_E` or `AST_RATIONAL`).
- `boolean isRelational()` returns `true` if this AST node is a relational operator.
- `boolean isSqrt()` returns `true` if this AST node is the square-root operator
- `boolean isUMinus()` returns `true` if this AST node is a unary minus.
- `boolean isUnknown()` returns `true` if this AST node's type code is `AST_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 methods are available for returning values from nodes:

- `int getInteger()`
- `char getCharacter()`
- `java.lang.String getName()`
- `int getNumerator()`
- `int getDenominator()`
- `double getReal()`
- `double getMantissa()`
- `int getExponent()`

Of course, all of this would be of little use if libSBML didn't also provide methods for setting the values of AST node objects! And it does. The methods are the following:

- `void setCharacter(char value)` sets the value of this AST node to the given character. If character is one of `+`, `-`, `void *`, `/` or `^`, the node type will be to the appropriate operator type. For all other characters, the node type will be set to the type code `AST_UNKNOWN`.
- `void setName(java.lang.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(node) != 0`) or number (`isNumber(node) != 0`). This allows names to be set for `AST_FUNCTION` nodes and the like.
- `void setValue(int value)` sets the value of the node to the given integer, `value`.
- `void setValue(int numerator, int denominator)` sets the value of this AST node to the given rational in two parts: the `numerator` and `denominator`. The node type code is set to `AST_RATIONAL`.
- `void setValue(double value)` sets the value of this AST node to the given `double` `value` and sets the node type code to `AST_REAL`.
- `void setValue(double mantissa, int exponent)` sets the value of this AST node to the given real number in two parts: the `mantissa` and the `exponent`. The node type code is set to `AST_REAL_E`.

Finally, `ASTNode` also defines some miscellaneous methods for manipulating

- `ASTNode ASTNode(int type)` creates a new AST node object and returns a pointer to it. The returned node will have the given `type` code. The type value must be chosen from among the constants beginning with the characters `AST_` defined in `org.sbml.libsbml.libsbmlConstants`.
- `ASTNode ASTNode()` creates a new AST node object and returns a pointer to it. The returned node will have the type `AST_UNKNOWN`. Its type should be set by the caller as soon as possible using
- `long getNumChildren()` returns the number of children of this AST node or `0` is this node has no children.
- `addChild(ASTNode)` adds the given node as a child of this AST node. Child nodes are added in left-to-right order.
- `prependChild()` adds the given node as a child of this AST node. This method adds child nodes in right-to-left order.
- `ASTNode getChild(long n)` returns the `n`th child of this AST node or `NULL` if this node has no `n`th child (`n > (ASTNode.getNumChildren() - 1)`).
- `ASTNode getLeftChild()` returns the left child of this AST node. This is equivalent to `ASTNode.getChild(0)`.
- `ASTNode getRightChild()` returns the right child of this AST node or `NULL` if this node has no right child.
- `swapChildren(ASTNode that)` swaps the children of this AST node with the children of `that` AST node.
- `setType(int type)` sets the type of this AST node to the given `type` code. The value must be chosen from among the constants beginning with the characters `AST_` defined in `org.sbml.libsbml.libsbmlConstants`.

## 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(java.lang.String)` reads raw MathML from a text string, constructs an AST from it, then returns the root AST node of the resulting expression tree.
- `java.lang.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.

The example program given above demonstrate the use of these methods.