Lab 2: Parsing Maths

Language and AST Overview

You will parse and build the AST for standard maths notation over double-precision numbers. The constraints on the input language are:

• Operators to be supported are +, -, *, /, and ^ (exponentiation).

• Operators follow standard mathematical precedence.

• Operators follow standard mathematical associativity.

• Parentheses ( and ) can be used to override operator associativity and precedence.

• Functions to be supported are sqrt, exp, and log. Functions have one parameter, which must be surrounded by parentheses. exp and log are both natural (base e).

• Numbers are expressed as decimal numbers. These must start with an integer digit, may have a fractional part, and may be negative.

• Variables are sequences of lower-case alphabetic characters.

• Apart from operators, numbers, variables, and functions, the only other characters will be white-space (spaces, tabs, linebreaks).

The AST is defined as a class hierarchy, spread across the following header files.

• Expression: Provides the base class for the AST.

• Primitive: Sub-classes for handling variables and numbers.

• Operator: Sub-classes for representing binary operators.

• Function : Sub-classes for representing unary functions.

Some properties of the AST are:

• The AST lives in the heap, not on the stack. So in order to allocate a node representing 5.0, you would do:

  ExpressionPtr expr = new Number(5.0);
  

• The AST is immutable, which means that once it is constructed it cannot be modified. Every function is const, and there are no exposed member variables, so there is no way of modifying the internals of a node after construction. This makes the semantics simpler, but then we can't modify the tree; we can only build a new one.

• Expressions are owned by the holder of the pointer to that expression. So in Operator::Operator, the constructor takes ownership of the two sub-expressions, and will delete them when it is itself deleted in Operator::~Operator. This makes managing expression lifetimes easy, but means we can't share parts of the tree. (For example, when rewriting a tree, we can't have the new tree point to parts of the old tree that are the same).

Your task

You have two tasks for this lab:

• complete the parser for building the AST, and

• complete the evaluation functions for evaluating the expression with a given set of inputs.

Task 1: Parsing

Your parser should complete the implemention of the function: const Expression *parseAST(); which is declared in ast.hpp, and defined in parser.y. This will parse the input from stdin (the default source for Flex, which is providing tokens to Bison), then return the AST.

There is a program called bin/print_canonical, which will use the function parseAST to parse input, then print it as a canonical representation. You can build it using make bin/print_canonical.

The idea of a canonical form is that if two expressions are the same, they should print to the same string. Here that is true except for the floating-point values, which are not printed to full precision. So for example, we might find that two expressions print to ( 4.1 + x ) and ( 4.1 + x ), but one expression might be (4.0999999999 + x), and the other could be (4.1000000001 + x).

Examples

Given the input:

7+8

the code should do the equivalent of:

const Expression *expr = new AddOperator(
    new Number(7),
    new Number(8)
);

and the canonical printer will produce:

$ bin/print_canonical
7+8

( 7 + 8 )

Note that the input 7+8 can be terminated by using CTRL+d to indicate the end of the input stream. This will cause the pretty-printer (or any program) to conclude that no more input is going to arrive over stdin, and it should conclude its work. Until the parser knows that there is no more input, it can't return the AST.

When manually typing input, the whitespace between input and output could vary while still being correct. So it could look like this:

$ bin/print_canonical
7+8


( 7 + 8 )

if you hit return twice before the stream ends, or like this:

$ bin/print_canonical
7+8( 7 + 8 )

if you end the stream without pressing return. Generally you should use typed input as a debugging and exploration tool, and rely more on automated and repeatable tests from files for proper testing.

Given the input:

7*x +5*
 -5 * y

the code should do the equivalent of:

const Expression *expr  = new AddOperator(
    new MulOperator(
        new Number(7),
        new Variable("x")
    ),
    new MulOperator(
        new MulOperator(
            new Number(5),
            new Number(-5)
        ),
        new Variable("y")
    )
);

and the canonical printer will produce:

$ bin/print_canonical
7*x +5*
 -5 * y
 
( ( 7 * x ) + ( ( 5 * -5 ) * y ) )

Testing

There is a script called test_parser.sh, which applies the tests used during assessment. During testing it will perform two tasks:

• Check that the expressions in test/valid_expressions.input.txt are parsed into the correct canonical string. The file consists of an input string and an output string, separated by a comma; and

• Check that the expressions in test/invalid_expressions.input.txt are rejected by the parser.

Task 2: Evaluation

There is a virtual function called Expression::evaluate, which takes a mapping from variable names to numeric values as input, and produces the numeric value of the expression as output. The function is implemented for the primitive types (Number and Variable), and for addition (AddOperator). You need to implement it for the remaining nodes.

There is a driver program called src/eval_expr.cpp, which is built using make bin/eval_expr. This takes a list of variable bindings on the command line, where each binding is a variable and a number separated by spaces, then evaluates an expression read from stdin.

Example

Assume we have the expression x+6, and want to evaluate it when x is 7. The usage is:

$ make bin/eval_expr
$ bin/eval_expr x 7
x+6
13.000000
$

Another example:

$ bin/eval_expr a 4 b 7 c -3
(c+7)*b+c
25.000000

A set of test vectors is supplied, which checks that both the parsing and functionality is correct. These are also run using:

./test_parser.sh

Submission

Passing all given tests is worth 50% of the marks, with the remaining marks coming from unseen test-cases.

As before, ensure you have committed your code, then do a full test. Then push your code to your private GitHub repo, and submit the commit hash to Blackboard.