Regular Language Intersection, Cnf-Sat, And Parameterized Reductions Between Automata Models
Intersection of Regular Languages and SAT Solving
The intersection operation on regular languages has important connections to the Boolean satisfiability problem. We explore these connections by showing how intersection can be encoded as a conjunctive normal form satisfiability (CNF-SAT) formula. We also examine fixed-parameter tractable reductions between various types of automata that recognize regular languages.
The Equivalence of Intersection and Complementation
A key property that enables the reduction from language intersection to SAT is the equivalence between intersection and complementation for regular languages. For languages L1 and L2 recognized by deterministic finite automata (DFAs) D1 and D2, we have:
L1 ∩ L2 = L1 ∩ L2CC
Where ∩ denotes language intersection and C denotes complementation. This allows us to reduce the intersection problem to complementation, which has a straightforward SAT encoding.
Demonstrating the Equivalence Between Intersection and Complementation of Regular Languages
We can demonstrate this equivalence constructively by using DFA complementation to build an intersecting DFA. Given D1 and D2, first construct a DFA D2′ that recognizes L2C by complementing the accept states of D2. Then construct a DFA D that has the cross product state set Q1 x Q2′ and transitions defined by synchronously following transitions in D1 and D2’. It can be shown that D recognizes L1 ∩ L2.
Example DFAs Showing How Intersection Can Be Achieved Through Complementation
As a concrete example, consider the DFAs D1 and D2 below that recognize the regular languages L1 = {w ε {0,1}* | w has two consecutive 0s} and L2 = {w ε {0,1}* | w ends in 01}, respectively:
[DFA D1 AND D2 FIGURES]
Complementing D2 yields DFA D2′ where the accept state is now state 1 instead of state 2. Constructing the synchronous cross product DFA D leads to the following intersecting automaton recognzing L1 ∩ L2:
[INTERSECTING DFA FIGURE]
Reducing Intersection to SAT
With the equivalence between intersection and complementation established, we can reduce the regular language intersection problem to SAT. This takes advantage of well-studied Boolean satisfiability techniques to reason about intersection properties.
Encoding Regular Language Intersection as a CNF-SAT Problem
We encode the intersection DFA D from the previous section as a Boolean formula in conjunctive normal form (CNF). The variables in the formula capture the state transitions in D, while the clauses constrain variable assignments to only accepting paths in D. This directly encodes the accepting paths of D, which correspond to strings in L1 ∩ L2.
Walkthrough of a Reduction from Intersection of DFAs to CNF-SAT
More precisely, given DFAs D1 and D2 with n1 and n2 states, we create Boolean variables xit,j for 0 ≤ i < n1, 0 ≤ j < n2, and 0 ≤ t representing that D is in state (i, j) at time t. The transition relation is encoded as clauses:
(¬xit,j ∨ xit+1,j’) for each transition from state i in D1 to i’ and state j in D2 to j’.
Accepting paths are encoded using additional variables yt:
(¬xit,j ∨ yt) when (i, j) is an accepting state.
This creates a CNF formula where SAT assignments correspond to accepting paths in D, i.e. strings in L1 ∩ L2.
Example CNF Formula Encoding Intersection of Sample DFAs
For the running intersection example, n1 = 2, n2 = 2, yielding variables xi0, xil, xi2 for i = 0,1 and t ≥ 0. The transition relation clauses are:
(¬x00t ∨ x01t+1) (¬x10t ∨ x11t+1)
(¬x01t ∨ x00t+1) (¬x11t ∨ x10t+1)
And accept state clauses:
(¬x02t ∨ yt) (¬x12t ∨ yt)
Which encodes the intersection DFA D shown earlier into a CNF formula.
Parameterized Complexity of Regular Language Intersection
In addition to encoding as SAT, we can also study regular language intersection through the lens of parameterized complexity. This provides a refined analysis of algorithmic difficulty by carefully identifying parameters that govern computational resources.
Exploring Fixed-Parameter Tractability of the Intersection Problem
We say that a problem with input size N and parameter k is fixed-parameter tractable (FPT) if it can be solved in time f(k)*NO(1) where f is any computable function. For intersection, k could be properties of the DFA/NFA like number of states or transitions. An FPT algorithm indicates efficient solvability for fixed k.
Techniques for Parameterized Reductions Between DFAs and NFAs
One approach is to study parameterized reductions that convert intersection of NFAs to intersection of DFAs. With DFA intersection being NL-complete compared to PSPACE-completeness for NFAs, this can provide efficiency gains. Useful parameters include number of NFA states and alphabet size.
Sample Parameterized Reduction from Intersection of NFAs to DFAs
The well-known subset construction converts an NFA with n states and sigma alphabet size into a DFA with 2n sigman states. By preserving parameters n and sigma, we obtain an FPT reduction from NFA to DFA intersection. Intersecting the constructed DFAs using an NL algorithm gives a parameterized result relative to the NFA parameters.
Practical Applications and Open Problems
Connecting regular language intersection to CNF-SAT and parameterized complexity reveals deep theory, but also has down-to-earth impacts on real-world systems.
Applications of Regular Language Intersection in Query Processing
Database systems model queries and integrity constraints as automata or regular expressions. Efficient intersection algorithms allow detecting complex matches and violations through language overlap. SAT and parameterized techniques provide additional optimization leverage.
Open Problems Related to Obtaining Tighter Bounds
Despite advances, the fine-grained complexity of intersection with different automata variants remains open. Tighter runtime bounds in terms of different parameters are active research questions, made more approachable by the unifying perspectives discussed here.
