README

Note

The default branch is now main. If your clone contains the old master branch, please rename and repoint your master remote to main.

This is an experimental project for exploring the analysis of the complexity of Python project source code in terms of cyclomatic complexity via directed graph representations of the associated CPython bytecode instructions. These directed graph representations have structural properties that can describe the complexity of the source code, which can be useful in a variety of applications, including unit testing.

Method

Computing the measures is made possible using the following method:

• From the given source code object - which could be a source fragment (string), or a code object, or a function or callable - the (CPython) bytecode object is obtained using a modifed version of the dis library (an old version from Python 3.7), and the bytecode object is then disassembled into an instruction map of individual CPython bytecode instructions.
• Each instruction is classified as follows: an entry point if the instruction is the very first bytecode step of the callable, a decision point if the instruction involves a comparison (e.g. COMPARE_OP), a branch point if the instruction is a branching instruction to another instruction (e.g. JUMP), or an exit point if the instruction stops or interrupts execution of the callable and returns control flow back to the caller (e.g. RETURN_VALUE, RAISE_VARARGS).
• Using the networkx library the bytecode instruction map is represented as a strongly connected directed graph, called the bytecode graph, with nodes representing individual bytecode instructions, edges representing (explicit or implicit) transitions between instructions.

Note: often instructions may not be entry points, decision points, branch points or exit points, but simply transitional instructions that follow and are succeeded by other instructions in order of execution.

The bytecode graph will have all of the structural information about the number of nodes, edges, connected components and the like, in order to compute the CCMs. It can also be used to create a (connected) digraph of the source lines (as the quotient graph of the bytecode graph under a simple equivalence relation on the instructions), as described later - this source code graph will be similar to the bytecode graph, but not necessarily the same and may produce slightly different values for the CCMs compared to the bytecode graph.

There are several CCMs that can be calculated with this approach:

1. McCabe complexity: #{edges} - #{nodes} + 2)
2. Generalised McCabe complexity: #{edges} - #{nodes} + 2 * #{connected components})
3. Henderson-Sellers complexity: #{edges} - #{nodes} + #{connected components} + 1)
4. Henderson-Sellers & Tegarden complexity: #{edges} - #{nodes} + #{connected components})
5. Generalised Henderson-Sellers & Tegarden complexity: #{edges} - #{nodes} + #{exit points per connected component} + 2)
6. Harrison complexity: #{decision points} - #{exit points} + 2)

A related measure, useful for software testing, is the number of linearly independent paths (so-called basis paths) in the directed acyclic graph (DAG) representation of a function's control flow. This number serves as an upper bound on the number of test cases required to cover all possible execution paths that the function could take, as used in test-driven development (TDD), and also a lower bound on the number of all test cases needed to cover all possible behaviours of the function, as used in behaviour-driven development (BDD). This will be described in more detail later.

Note: As noted here the standard McCabe complexity measure (1) only applies to individual functions or methods, which can be represented as a connected digraph with a single component. A class with more than one method, or a module with multiple functions and/or classes, is usually representable as a connected graph which is a disjoint union of components, and in these cases the generalised McCabe complexity (2), the Henderson-Sellers complexity (3), or related generalised measures, should be used instead.

Examples

Some examples are given below, to illustrate the concept and measures of cyclomatic complexity, as described above, for Python source code, and the data structures and methods used to compute the measures.

First, consider the following simple Python implementation of the sign function, for determining whether a given number (integer or float) is negative, zero, or positive:

def sign(x: int | float) -> typing.Literal[-1, 0, 1]:
    if x < 0:
        return -1
    if x == 0:
        return 0
    return 1

Using dis.dis this function can be disassembled into the following (CPython) bytecode:

2           0 LOAD_FAST                0 (x)
            2 LOAD_CONST               1 (0)
            4 COMPARE_OP               0 (<)
            6 POP_JUMP_IF_FALSE       12

3           8 LOAD_CONST               2 (-1)
           10 RETURN_VALUE

4     >>   12 LOAD_FAST                0 (x)
           14 LOAD_CONST               1 (0)
           16 COMPARE_OP               2 (==)
           18 POP_JUMP_IF_FALSE       24

5          20 LOAD_CONST               1 (0)
           22 RETURN_VALUE

6     >>   24 LOAD_CONST               3 (1)
           26 RETURN_VALUE

For more information on the details of the bytecode instructions, as displayed to the console, refer to the dis documentation (Python 3.7), but a brief summary is given below of the values contained in the lines in the bytecode printout:

• The first value is an integer representing the (unique) number of the source line associated with the bytecode instruction (block).
• The second value is an integer, called the instruction offset, representing the (unique) index of the bytecode instruction relative to the starting point of the complete sequence of bytecode instructions, which are in ascending order of offset.
• The third value, called the opname is the human readable name of the associated bytecode operation.
• The (possibly null) fourth value is an argument (or parameter) to the bytecode operation (if any).
• The (possibly null) fifth value, in parentheses if not null, is a human readable description of the operational argument.

Note: instructions which are jump targets have offsets prefixed by >> (refer here).

The bytecode can be represented as the following DAG:

The unique entry point, and the decision points, branch points and exit points are clear from this representation, but are also stored at the level of bytecode instruction as attributes.

To compute the cyclomatic complexity measures, as defined above, for a given Python source code object the bytecode graph must be connected, i.e. there must be a path between any two nodes, in any direction. This is done in the bytecode graph via derived edges that link all exit points back to the entry point.

Here's an iPython session using the sign function as an example.

In [1]: from ccm.complexity import *

In [2]: def sign(x) :
   ...:     if x < 0:
   ...:         return -1
   ...:     if x == 0:
   ...:         return 0
   ...:     return 1
   ...:

In [3]: mccabe_complexity(sign)
Out[3]: 4

In [4]: mccabe_generalised_complexity(sign)
Out[5]: 4

In [5]: henderson_sellers_complexity(sign)
Out[5]: 4

In [6]: henderson_sellers_tegarden_complexity(sign)
Out[6]: 3

In [7]: henderson_sellers_tegarden_generalised_complexity(sign)
Out[7]: 7

In [8]: harrison_complexity(sign)
Out[8]: 1

All the complexity functions are based on calculations involving numeric properties of the bytecode graphs (ccm.graphs.XBytecodeGraph, a custom subclass of nx.DiGraph) obtained from the source code, such as the number of edges, nodes, connectivity features etc. As described above, the bytecode graph of a source code object is a connected digraph where nodes represent bytecode instructions and edges represent transitions between the instructions. Additionally, each instruction stores a number of attributes which classify the instruction as an entry point, branch point, decision point, or exit point, or none of these, depending on the instruction.

The bytecode graph for sign can be obtained quite simply as follows:

In [9]: from ccm.graphs import XBytecodeGraph

In [10]: G = XBytecodeGraph(code=sign)

In [11]: G.nodes
Out[11]: NodeView((0, 2, 4, 6, 12, 8, 10, 14, 16, 18, 24, 20, 22, 26))

In [12]: G.edges
Out[12]: OutEdgeView([(0, 2), (2, 4), (4, 6), (6, 12), (6, 8), (12, 14), (8, 10), (10, 0), (14, 16), (16, 18), (18, 24), (18, 20), (24, 26), (20, 22), (22, 0), (26, 0)])

Note: each node label is an instruction offset (the relative index of the instruction with respect to the complete, ordered sequence of all the bytecode instructions). The instruction map is an ordered dictionary keyed by tuples consisting of source line numbers and instruction offets, and values are ccm.xdis.XInstruction objects. The instruction map can be obtained from the bytecode graph using the xbytecode.instructions attribute, e.g. for sign:

In [13]: G.instructions
Out[13]:
OrderedDict([((2, 0),
              XInstruction(opname='LOAD_FAST', opcode=124, arg=0, argval='x', argrepr='x', offset=0, starts_line=2, is_entry_point=True, is_jump_target=False, is_decision_point=False, is_branch_point=False, is_exit_point=False)),
             ((2, 2),
              XInstruction(opname='LOAD_CONST', opcode=100, arg=1, argval=0, argrepr='0', offset=2, starts_line=2, is_entry_point=False, is_jump_target=False, is_decision_point=False, is_branch_point=False, is_exit_point=False)),
             ...
             ...
             ((6, 26),
              XInstruction(opname='RETURN_VALUE', opcode=83, arg=None, argval=None, argrepr='', offset=26, starts_line=6, is_entry_point=False, is_jump_target=False, is_decision_point=False, is_branch_point=False, is_exit_point=True))])

Note: the last edge (6, 26), connecting the last instruction, which is the only exit point, with the first instruction, which is the entry point, makes the digraph strongly connected.

Each instruction object (ccm.xdis.XInstruction) has a property named dis_line which stores the ccm.xdis.dis console representation of the instruction, e.g.:

In [14]: G.instructions[(2,0)].dis_line
Out[14]: '  2           0 LOAD_FAST                0 (x)'

The console printed versions of the instructions are more human readable, and so it is possible to quickly identify classified instructions such as branch points, decision points and exit points in this way, e.g. for sign:

In [15]: for instr in G.xbytecode.instr_map.values():
     ...:     if instr.is_entry_point:
     ...:         print(instr.dis_line)
     ...:
     ...:
  2           0 LOAD_FAST                0 (x)

In [16]: for instr in G.xbytecode.instr_map.values():
     ...:     if instr.is_decision_point:
     ...:         print(instr.dis_line)
     ...:
  2           4 COMPARE_OP               0 (<)
  4          16 COMPARE_OP               2 (==)


In [17]: for instr in G.xbytecode.instr_map.values():
     ...:    if instr.is_branch_point:
     ...:        print(instr.dis_line)

  2           6 POP_JUMP_IF_FALSE       12
  4          18 POP_JUMP_IF_FALSE       24

In [18]: for instr in G.xbytecode.instr_map.values():
     ...:     if instr.is_exit_point:
     ...:         print(instr.dis_line)
     ...:
  3          10 RETURN_VALUE
  5          22 RETURN_VALUE
  6          26 RETURN_VALUE

If we define a simple equivalence relation on the bytecode instructions that two instructions are related if their associated source lines are identical, then the blocks of this relation correspond to the source lines. This means there is a connected digraph of the source lines, which we call the source code graph, which is the quotient graph of the bytecode graph, under this equivalence relation, with one condition: in the special case of a source code object with just a single source line a looped edge is added between the single source line and itself.

The bytecode graph stores the associated source code graph in the source_code_graph attribute, e.g. for the sign function:

In [19]: G.source_code_graph
Out[19]: <networkx.classes.digraph.DiGraph at 0x12105db90>

In [20]: G.source_code_graph.nodes
Out[20]: NodeView((4, 6, 2, 3, 5))

In [21]: G.source_code_graph.edges
Out[21]: OutEdgeView([(4, 6), (4, 5), (6, 2), (2, 4), (2, 3), (3, 2), (5, 2)])

Note: as with the bytecode graph, the source code graph has edges between any source line representing an exit point (if the associated bytecode instruction block contains an exit point) and the (unique) entry point, including the special case where we have just a single source line, with a looped edge on itself. This is what makes the source code graph (strongy) connected. This includes the special case of a source code object with just a single source line.

The bytecode graph and the source code graph for functions and class methods will be similar, but not necessarily the same. There are several points to note.

• As the source code graph is the quotient of the bytecode graph under the equivalence relation described above, its nodes correspond to blocks of bytecode instructions associated with a unique source line, and edges correspond to edges between instructions in different instruction blocks associated with different source lines.
• If n is the number of instruction blocks (same as the number of source lines), and for a given block B we have D(B) decision points and X(B) exit points, then the source code graph will have n nodes and at least Sum(D(B) + X(B)) edges, where this is a sum over all instruction blocks B.
• There is only one entry point in a bytecode graph, because it is defined as the first bytecode instruction (one with the unique offset 0). Thus there is only one source line in the source code graph associated with this entry point, namely, the first source line in the body of the source code object.
• Decision points, branch points and exit points in the bytecode graph are also associated with unique source lines in the source code graph.
• The bytecode graph is (strongly) connected with only one component, namely, itself, which means the source code graph is also (strongly) connected with only one component.

This means that for a given bytecode graph the CCMs, as defined above, will be an upper bound for the CCMs calculated using the associated source code graph. Here are three examples for McCabe complexity, using simple functions. First, the sign function, which has five source lines (excluding the signature):

In [22]: G = XBytecodeGraph(code=sign)

In [23]: G.nodes
Out[23]: NodeView((0, 2, 4, 6, 12, 8, 10, 14, 16, 18, 24, 20, 22, 26))

In [24]: G.edges
Out[24]: OutEdgeView([(0, 2), (2, 4), (4, 6), (6, 12), (6, 8), (12, 14), (8, 10), (10, 0), (14, 16), (16, 18), (18, 24), (18, 20), (24, 26), (20, 22), (22, 0), (26, 0)])

In [25]: G.number_of_edges() - G.number_of_nodes() + 2
Out[25]: 4

In [26]: G.source_code_graph.nodes
Out[26]: NodeView((2, 3, 4, 5, 6))

In [27]: G.source_code_graph.edges
Out[27]: OutEdgeView([(2, 4), (2, 3), (3, 2), (4, 6), (4, 5), (5, 2), (6, 2)])

In [28]: G.source_code_graph.number_of_edges() - G.source_code_graph.number_of_nodes() + 2
Out[28]: 4

The second example is an identity function for arbitrary arguments, with just a single source line:

In [29]: from ccm.xdis import dis as xdis

In [30]: def identity(x):
     ...:    return x

In [30]: xdis(identity)
 2           0 LOAD_FAST                0 (x)
             2 RETURN_VALUE

In [32]: H = XBytecodeGraph(code=identity)

In [33]: H.nodes
Out[33]: NodeView((0, 2))

In [34]: H.edges
Out[34]: OutEdgeView([(0, 2), (2, 0)])

In [35]: H.number_of_edges() - H.number_of_nodes() + 2
Out[35]: 2

In [36]: H.source_code_graph.nodes
Out[36]: NodeView((2,))

In [37]: H.source_code_graph.edges
Out[37]: OutEdgeView([(2, 2)])

In [38]: H.source_code_graph.number_of_edges() - H.source_code_graph.number_of_nodes() + 2
Out[38]: 2

In both these examples, the CCMs computed using the bytecode graph and source code graph were identical - this is because the decision points in both represent simple conditions involving a comparison of two values, and do not consist of a compound condition composed of two or more comparisons. With a decision point involving a simple condition, both branches of the associated branching instruction will lead to instructions in other blocks. This is not the case where a decision point involves a compound condition.

Here is a third example involving a function with a decision point involving a compound condition, where the CCMs from the bytecode graph and source code graph differ.

In [39]: def nonzero(x):
     ...:     if x < 0 or x > 0 :
     ...:         return True
     ...:     return False

In [40]: xdis(nonzero)
 2           0 LOAD_FAST                0 (x)
             2 LOAD_CONST               1 (0)
             4 COMPARE_OP               0 (<)
             6 POP_JUMP_IF_TRUE        16
             8 LOAD_FAST                0 (x)
            10 LOAD_CONST               1 (0)
            12 COMPARE_OP               4 (>)
            14 POP_JUMP_IF_FALSE       20

 3     >>   16 LOAD_CONST               2 (True)
            18 RETURN_VALUE

 4     >>   20 LOAD_CONST               3 (False)
            22 RETURN_VALUE

In [41]: Z = XBytecodeGraph(code=nonzero)

In [42]: Z.number_of_edges() - Z.number_of_nodes() + 2
Out[42]: 4

In [43]: Z.source_code_graph.number_of_edges() - Z.source_code_graph.number_of_nodes() + 2
Out[43]: 3

From the bytecode graph the instructions which represent entry points, decision points, branch points and exit points can be easily accessed using dictionary attributes (the dicts are keyed by tuples consisting of the source line and instruction offset):

In [44]: Z.entry_points
Out[44]:
OrderedDict([((2, 0),
              XInstruction(opname='LOAD_FAST', opcode=124, arg=0, argval='x', argrepr='x', offset=0, starts_line=2, is_entry_point=True, is_jump_target=False, is_decision_point=False, is_branch_point=False, is_exit_point=False))])

In [45]: Z.decision_points
Out[45]:
OrderedDict([((2, 4),
              XInstruction(opname='COMPARE_OP', opcode=107, arg=0, argval='<', argrepr='<', offset=4, starts_line=2, is_entry_point=False, is_jump_target=False, is_decision_point=True, is_branch_point=False, is_exit_point=False)),
             ((2, 12),
              XInstruction(opname='COMPARE_OP', opcode=107, arg=4, argval='>', argrepr='>', offset=12, starts_line=2, is_entry_point=False, is_jump_target=False, is_decision_point=True, is_branch_point=False, is_exit_point=False))])

In [46]: Z.branch_points
Out[46]:
OrderedDict([((2, 6),
              XInstruction(opname='POP_JUMP_IF_TRUE', opcode=115, arg=16, argval=16, argrepr='', offset=6, starts_line=2, is_entry_point=False, is_jump_target=False, is_decision_point=False, is_branch_point=True, is_exit_point=False)),
             ((2, 14),
              XInstruction(opname='POP_JUMP_IF_FALSE', opcode=114, arg=20, argval=20, argrepr='', offset=14, starts_line=2, is_entry_point=False, is_jump_target=False, is_decision_point=False, is_branch_point=True, is_exit_point=False))])

In [47]: Z.exit_points
Out[47]:
OrderedDict([((3, 18),
              XInstruction(opname='RETURN_VALUE', opcode=83, arg=None, argval=None, argrepr='', offset=18, starts_line=3, is_entry_point=False, is_jump_target=False, is_decision_point=False, is_branch_point=False, is_exit_point=True)),
             ((4, 22),
              XInstruction(opname='RETURN_VALUE', opcode=83, arg=None, argval=None, argrepr='', offset=22, starts_line=4, is_entry_point=False, is_jump_target=False, is_decision_point=False, is_branch_point=False, is_exit_point=True))])

Limitations

Currently, the functionality of bytecode graphs does not extend to Python classes or modules. This will be addressed in future versions.

Implications for Software Testing

TODO

References

[1]

Henderson-Sellers, B., Tegarden, D. (1995). A Critical Re-examination of Cyclomatic Complexity Measures. In: Lee, M., Barta, BZ., Juliff, P. (eds) Software Quality and Productivity. IFIP Advances in Information and Communication Technology. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-34848-3_51

[2]	Harrison, W. A. (1984), Applying Mccabe's complexity measure to multiple-exit programs. Softw: Pract. Exper., 14: 1004-1007. https://doi.org/10.1002/spe.4380141009

[3]	dis - Disassembler for Python bytecode. https://docs.python.org/3.7/library/dis.html