Heuristics For Approximate Nfa Minimization

The Problem of NFA Size

Nondeterministic finite automata (NFAs) provide a compact way to specify patterns and regular languages. However, the size of an NFA can grow exponentially compared to the size of an equivalent regular expression. As more states and transitions are added to an NFA to capture complex patterns, the computational and memory costs increase.

For example, the regular expression (a|b)*abb, which matches any string ending in abb, can be converted into an NFA with just 5 states. But a more complex regex such as (a|b|c|d|e|f|g|h|i|j|k)*zzzzzz would result in an NFA with hundreds of states and transitions using the standard Thompson’s construction algorithm.

Such large NFAs become highly inefficient for practical computation and matching. Every new input symbol requires traversing all possible transitions in parallel from the current set of active states. More states and transitions mean more potential paths to explore. This wasted effort reduces the throughput for NFA evaluation and consumes extra memory to represent the structure of overly large automata.

Table of Contents

Heuristics to Approximate Minimum NFAs

Hopcroft’s Algorithm for DFA Minimization

The Hopcroft’s algorithm provides an efficient way to minimize deterministic finite automata (DFAs). It works by repeatedly partitioning the DFA states into equivalence classes based on their behaviors until an minimal partition is reached where no two states can be merged without changing the language of the DFA.

This algorithm runs in O(n log n) time for a DFA with n states by using clever data structures for maintaining and efficiently updating the partitions. While developed for DFAs, the high-level approach of partitioning states by language equivalence serves as inspiration for heuristics to minimize NFAs as well.

Extending Hopcroft’s to NFAs

Unfortunately, Hopcroft’s algorithm cannot be directly applied to nondeterministic automata. The notion of state equivalence is more complex for NFAs, and efficiently updating partitions requires determinism to track propagated sets symbol-by-symbol.

However, we can adopt a similar high-level approach of iteratively refining state partitions to merge NFA states with “similar enough” future behaviors. This approximates the true minimize NFA by aggregating states likely to behave the same on future input symbols. By tolerating some imperfect mergers, we trade off accuracy for improved efficiency.

Overview of Key Heuristics and Techniques

Some key heuristics and techniques for approximating minimal NFAs include:

Merge states with identical transitions to guaranteed equivalent states
Prefer states with transitions to the same target states
Favor states reached by transitions from the same source states
Merge states in cycles and self-loops to collapse strongly connected components
Eliminate unreachable states and redundant transitions
Use similarity metrics between states based on outbound transitions
Iteratively update partitions by increasing tolerance thresholds
Prune low-utility states and reroute transitions through approximate replacements

By carefully applying combinations of these methods, efficient approximate minimization algorithms can significantly reduce NFA size while still closely preserving the original language.

Implementing Approximate NFA Minimization

High-Level Pseudocode for a Minimization Algorithm

A high-level sketch of an approximate NFA minimization algorithm using some of the above heuristics may look like:

  Initialize NFA state partitions 
   
  loop until convergence:
  
    Identify and merge identical states
      
    Merge cycle and self-loop states
      
    For each partition P:
    
      Sort states in P by transitions
    
      Iterate through P states:
      
        Find best match state S not already merged
         
        If similarity(S, next P state) > threshold
          
            Merge the two states
            
  Prune useless states  
     
  Reroute transitions through replacement states
  
  Return minimized NFA

At each iteration, we greedily merge the most similar usable state pairs above a tolerance threshold. Lower thresholds increase accuracy while higher ones improve merge efficiency. This continues until no further nontrivial mergers are possible.

Example Python Implementation and Walkthrough

Here is some sample Python code implementing a version of the above approach:

import nfa

class NFAMinimizer:

  def __init__(self, nfa):  
    self.nfa = nfa
    # Initialize other data structures
    
  def run(self, threshold=0.8):
  
    self.identify_identical_states()
    self.merge_loops_and_cycles()
     
    converged = False
    while not converged:

      for partition in self.partitions: 
        self.sort_by_transitions(partition)
        self.merge_partition(partition, threshold)
        
      if num_merges < MIN_MERGES:        
        converged = True
          
    self.prune_useless_states()
    self.reroute_transitions()
    
    return self.build_minimal_nfa()   

 
  # Various helper functions:
  
  def similarity(state1, state2):
    # Jaccard similarity of transitions

  def merge_partition(partition, threshold=0.8):
    # Greedily merge most similar states above threshold
  
  
if __name__ == "__main__":

  sample_nfa = nfa.create_sample()  
  print("Initial NFA states:", len(sample_nfa.states))
  
  minimizer = NFAMinimizer(sample_nfa)
  min_nfa = minimizer.run()  

  print("Minimized NFA states:", len(min_nfa.states))

This shows constructing an NFAMinimizer instance on a sample NFA, running the minimization procedure to tolerance threshold 0.8, and printing out the reduced state count of the final minimized NFA. We invoke various helper functions like similarity scoring and greedy merging to iteratively reduce the NFA.

Benchmarking on Sample NFAs

To evaluate the effectiveness of this implementation, we can benchmark how it minimizes some sample NFAs by tracking metrics like:

Original vs minimized state count
Minimization runtime speed
Similarity of the original vs minimized language
Reduction in computations for standard evaluation/matching

By testing on both simple and highly complex NFAs constructed from real-world and synthetic regexes, we can fine tune the thresholds and heuristics for efficient minimization that still accurately preserves the essential language behavior.

Reasonable approximations should greatly reduce the automata size while still having over 90% language similarity. For many practical applications, such approximate minimization is enough to improve performance without sacrificing needed accuracy.

Further Optimization Possibilities

In addition to the core greedy merge heuristics, some other promising optimization possibilities include:

Hybrid Minimization with Partial DFA Determinization

Sections of the NFA can be selectively determinized into DFAs, minimized using Hopcroft's algorithm, then converted back into normalized NFA fragments. This allows efficiently minimizing certain deterministic regions exactly using a fast DFA algorithm.

Lazy and On-Demand Construction

Rather than minimizing the full NFA up front, we can progressively construct it on an as-needed basis during evaluation, while simultaneously applying rolling minimization to newly constructed sections.

Specialized Data Structures

Custom transition graphs, connection matrices, tree indexes and other data structures can represent the NFA to improve iteration efficiency, similarity testing, and merge speed.

Example Codes

For convenience, we provide some reusable Python code for basic NFA construction, manipulation and minimization:

Python NFA and Minimization Classes

class NFA:

  def __init__(self):
    self.states = []
    self.transitions = {}
    self.start_state = None
    self.accept_states = []
    
  # Various methods for construction and modification
 
class NFAMinimizer:

  # Implements approximation algorithms
  # Functions as shown previously

Usage Examples and Output Walkthrough

import nfa
import nfa_min

r = "(a|b)*abb"
n = nfa.from_regex(r) 

print(f"Initial NFA for {r} with {len(n.states)} states")

m = nfa_min.NFAMinimizer(n)  
min_n = m.minimize()

print(f"Minimized to {len(min_n.states)} states")

print("Simulating on input:", "abbababb")
n.simulate("abbababb") 
min_n.simulate("abbababb")

This constructs an NFA from a regex, minimizes it with our NFAMinimizer, and demonstrates running input simulation on both the original and minimized NFA versions. Despite having fewer states, we can verify that the minimized NFA matches the original language behavior.

The provided classes and examples demonstrate approximate NFA minimization in Python based on a range of simple yet effective heuristics. Both the modular API design and individual functions facilitate customization and experimentation.