Note: This is based on the ddmin in the fuzzingbook.

### About Delta Debugging

Delta Debugging is a method to reduce failure inducing inputs to their smallest required size that still induces the same failure. It was first formally introduced in the paper Simplifying and Isolating Failure-Inducing Input by Zeller and Hildebrandt.

The idea of delta debugging is fairly simple. We start by partitioning the given input string, starting with two partitions – which have a given partition length. Then, we check if any of these parts can be removed without removing the observed failure. If any of these can be removed, we remove all such parts of the given length. Once no such parts of the given length can be removed, we reduce the partition length by two, and do the same process again. This obtains us the 1-minimal failure causing string where removal of even a single character will remove the observed failure.

Given a causal function as below,

def test(s):
v = re.match("<SELECT.*>", s)
print("%s  %s %d" % (('+' if v else '.'),  s, len(s)))
return v


Here is an example run:

$python ddmin.py '<SELECT NAME="priority" MULTIPLE SIZE=7>' . ty" MULTIPLE SIZE=7> 20 . <SELECT NAME="priori 20 . ME="priority" MULTIPLE SIZE=7> 30 + <SELECT NAty" MULTIPLE SIZE=7> 30 + <SELECT NALE SIZE=7> 20 . <SELECT NA 10 . CT NALE SIZE=7> 15 . <SELELE SIZE=7> 15 + <SELECT NAZE=7> 15 . <SELECT NA 10 . ELECT NAZE=7> 13 . <SECT NAZE=7> 13 . <SELT NAZE=7> 13 . <SELECNAZE=7> 13 + <SELECT ZE=7> 13 + <SELECT =7> 11 + <SELECT > 9 . <SELECT 8 . SELECT > 8 . <ELECT > 8 . <SLECT > 8 . <SEECT > 8 . <SELCT > 8 . <SELET > 8 . <SELEC > 8 + <SELECT> 8 . <SELECT 7 <SELECT>  ## Implementation How do we implement this? First, the prerequisites: import random import string  ### remove_check_each_fragment() Given a partition length, we want to split the string into that many partitions, remove each partition one at a time from the string, and check if for any of them, the causal() succeeds. If it succeeds for any, then return the succeeding string. def remove_check_each_fragment(instr, start, part_len, causal): for i in range(start, len(instr), part_len): stitched = instr[:i] + instr[i+part_len:] if causal(stitched): return i, stitched return -1, instr  ### ddmin() The main function. We start by the smallest number of partitions – 2. Then, we check by removing each fragment for success. If removing one fragment succeeds, we change the current string to the string without that fragment. Since we succeeded in removing one fragment at this partition length, we do not know if we can remove other parts of the string. So, we want to redo with the same fragment length, so we keep the current partition length. If none of the fragments could be removed, then it is time to decrease the partition length. We reduce the partition length by half. If the partition length is now single chars, then we break and return. def ddmin(cur_str, causal_fn): start, part_len = 0, len(cur_str) // 2 while part_len >= 1: start, cur_str = remove_check_each_fragment(cur_str, start, part_len, causal_fn) if start != -1: if not cur_str: return '' else: start, part_len = 0, part_len // 2 return cur_str  The driver. def test(s): print("%s %d" % (s, len(s))) return set('()') <= set(s) inputstring = ''.join(random.choices(string.digits + string.ascii_letters + string.punctuation, k=1024)) if __name__ == "__main__": assert test(inputstring) solution = ddmin(inputstring, test) print(solution)  Usage: $ python3 py.py
...
)(


The nice thing is that, if you invoke the driver, you can see the reduction in input length in action. Note that our driver is essentially a best case scenario. In the worst case, the complexity is $O(n^2)$

### Recursive

That was of course illuminating. However, is that the only way to implement this? delta-debug at its heart, is a divide and conquer algorithm. Can we implement it recursively?

The basic idea is that given a string, we can split it into parts, and check if either part reproduces the failure. If either one does, then call ddrmin() on the part that reproduced the failure.

If neither one did, then it means that there is some part in the first partition that is required for failure, and there is some part in the second partition too that is required for failure. All that we need to do now, is to isolate these parts. How should we do that?

Call ddrmin() but with an updated check. For example, for the first part, rather than checking if some portion of the first part alone produces the failure, check if some part of first, when combined with the second will cause the failure.

All we have left to do, is to define the base case. In our case, a character of length one can not be partitioned to strictly smaller parts. Further, we already know that any string passed into ddrmin() was required for reproducing the failure. So, we do not have to worry about empty string. Hence, we can return it as is.

Here is the implementation.

### ddrmin()

def ddrmin(cur_str, causal_fn, pre='', post=''):
if len(cur_str) == 1: return cur_str

part_i = len(cur_str) // 2
string1, string2 = cur_str[:part_i], cur_str[part_i:]
if causal_fn(pre + string1 + post):
return ddrmin(string1, causal_fn, pre, post)
elif causal_fn(pre + string2 + post):
return ddrmin(string2, causal_fn, pre, post)
s1 = ddrmin(string1, causal_fn, pre, string2 + post)
s2 = ddrmin(string2, causal_fn, pre + s1, post)
return s1 + s2

ddmin = ddrmin


Given that it is a recursive procedure, one may worry about stack exhaustion, especially in languages such as Python which allocates just the bare minimum stack by default. The nice thing here is that, since we split the string by half again and again, the maximum stack size required is $log(N)$ of the input size. So there is no danger of exhaustion.

The recursive algorithm is given in Yesterday, my program worked.Today, it does not. Why? by Zeller in 1999.

### Is this Enough?

One of the problems with the unstructured version of ddmin() above is that it assumes that parts of the inputs can be cut away, while still retaining validity in that it will actually reach the testing function. This, however, may not be a reasonable assumption especially in the case of structured inputs. The problem is that if you have a JSON [{"a": null}] that produces an error because the key value is null, ddmin() will try to partition it as [{"a": followed by null}] neither of which are valid. Further, any chunk removed from either parts are also likely to be invalid. Hence, ddmin() will not be able to proceed.

The solution here is to go for a variant called Hierarchical Delta Debugging described in HDD: hierarchical delta debugging by Misherghi et al. in 2006. The basic idea is to first parse the input using a grammar, and then try and apply ddmin() on each level of the derivation tree. Another notable improvement is Automatically Reducing Tree-Structured Test Inputs by Herfert et al. in 2017. In this paper, the authors describe a simple strategy: Try and replace a given node by one of its child nodes. This works reasonably well for inputs that are context sensitive such as programs that can trigger bugs in compilers. Another is Perses: Syntax-Guided Program Reduction by Sun et al. in 2018 which uses the program grammar directly to avoid creating invalid nodes.

The problem in the above approach is that, it assumes that there exist a child node that is of same type as the parent node. This need not be the case. Hence an even better idea might be to simply do a breadth first search of the first node that has the same symbol as that of the current node, and try replacing with that node, continuing with the next symbol of the same kind if it fails.

### Perses

Below is a variant of hierarchical ddmin() from “Perses: syntax-guided program reduction” by Sun et al. 2018

import sys
import re
import copy
import heapq
from fuzzingbook.Parser import IterativeEarleyParser as Parser
from fuzzingbook.Parser import canonical


#### A few helpers

Check if a given token is a noterminal

def is_nt(v):
return (v[0], v[-1]) == ('<', '>')

def tree_to_string(tree):
name, children, *rest = tree
if not is_nt(name): return name
else: return ''.join([tree_to_string(c) for c in children])


#### We prioritize the smallest trees.

def count_leaves(node):
name, children = node
if not children:
return 1
return sum(count_leaves(i) for i in children)


#### A priority queue

def add_to_queue(node, q):
heapq.heappush(q, (count_leaves(node), node))


#### A way to tempoararily replace a node in a tree.

class ReplaceNode:
def __init__(self, node, new_node = None):
self.node, self.new_node = node, ['', []] if new_node is None else new_node
self.node_copy = copy.copy(self.node)

def __enter__(self):
# we dont worry about legal grammar here as this is temporary
self.node.clear()
if self.new_node is not None:
self.node.extend(self.new_node)

def __exit__(self, *args):
self.node.clear()
self.node.extend(self.node_copy)


#### Find all subtrees with the given symbol

def subtrees_with_symbol(node, symbol, result=None, depth=0):
if result is None: result = []
name, children = node
if name == symbol:
result.append((depth, node))
for c in children:
subtrees_with_symbol(c, symbol, result, depth+1)
return result


#### The perses algorithm itself.

def perses_delta_debug(grammar, orig_tree, predicate):
tree = copy.deepcopy(orig_tree) # we minify the original tree
p_q = []
while p_q:
reprocess = None
_, biggest_node = heapq.heappop(p_q)
bsymbol = biggest_node[0]
_root, *subtrees = subtrees_with_symbol(biggest_node, bsymbol)
ssubtrees = sorted(subtrees, reverse=True)

if '' in grammar[bsymbol]: # empty expansion
ssubtrees.insert(0, (0, (bsymbol, [])))

for _depth,stree in subtrees:
with ReplaceNode(biggest_node, stree):
s = tree_to_string(tree)
if predicate(s):
reprocess = stree
break
if reprocess is not None:
biggest_node.clear()
biggest_node.extend(reprocess)
else:
for c in biggest_node[1]:
if is_nt(c[0]):
return patterns, tree


#### The driver

First, we define the predicate. Our predicate is simple. We check if the expression has doubled parenthesis.

def test_bb(inp):
if re.match(r'.*[(][(].*[)][)].*', inp):
return True
return False


We have the following grammar

EXPR_GRAMMAR = {'<start>': ['<expr>'],
'<expr>': ['<term> + <expr>', '<term> - <expr>', '<term>'],
'<term>': ['<factor> * <term>', '<factor> / <term>', '<factor>'],
'<factor>': ['+<factor>',
'-<factor>',
'(<expr>)',
'<integer>.<integer>',
'<integer>'],
'<integer>': ['<digit><integer>', '<digit>'],
'<digit>': ['0', '1', '2', '3', '4', '5', '6', '7', '8', '9']}


And the following input

my_input = '1 + ((2 * 3 / 4))'


We parse and generate a derivation tree as follows

d_tree, *_ = Parser(EXPR_GRAMMAR).parse(my_input)


The derivation tree looks like this

d_tree = ['<start>',
[['<expr>',
[['<term>', [['<factor>', [['<integer>', [['<digit>', [['1', []]]]]]]]]],
[' + ', []],
['<expr>',
[['<term>',
[['<factor>',
[['(', []],
['<expr>',
[['<term>',
[['<factor>',
[['(', []],
['<expr>',
[['<term>',
[['<factor>', [['<integer>', [['<digit>', [['2', []]]]]]]],
[' * ', []],
['<term>',
[['<factor>',
[['<integer>', [['<digit>', [['3', []]]]]]]],
[' / ', []],
['<term>',
[['<factor>',
[['<integer>', [['<digit>', [['4', []]]]]]]]]]]]]]]],
[')', []]]]]]]],
[')', []]]]]]]]]]]]


Now, we are ready to use the perses_delta_debug()

EXPR_GRAMMAR_C = canonical(EXPR_GRAMMAR)
patterns, tree = perses_delta_debug(EXPR_GRAMMAR_C, d_tree, test_bb)
print(tree_to_string(tree))


This prints

((4))