[Paper Review] State Complexity of Two Combined Operations: Reversal-Catenation and Star-Catenation

Bo Cui, Yuan Gao+2 more|arXiv (Cornell University)|Jun 23, 2010

Chemical Synthesis and Analysis25 references3 citations

TL;DR

This paper investigates the state complexity of two combined regular language operations: reversal followed by catenation (L₁ᴿL₂) and star followed by catenation (L₁*L₂). It shows that due to structural constraints in the resulting automata—such as final state distributions after reversal or the initial state being final after star—the state complexity of the combined operation is not simply the composition of individual operation complexities, but is instead bounded by tighter, often lower, values than expected, with exact upper bounds proven via witness automata constructions.

ABSTRACT

In this paper, we show that, due to the structural properties of the resulting automaton obtained from a prior operation, the state complexity of a combined operation may not be equal but close to the mathematical composition of the state complexities of its component operations. In particular, we provide two witness combined operations: reversal combined with catenation and star combined with catenation.

Motivation & Objective

• To analyze the state complexity of combined operations in regular language theory, particularly when individual operations are composed in sequence.
• To investigate why the state complexity of a combined operation may differ from the mathematical composition of its component operations' complexities.
• To identify and prove tight upper bounds on state complexity for two specific combined operations: reversal-catenation and star-catenation.
• To construct witness DFAs that achieve the derived upper bounds, demonstrating the bounds are tight.
• To highlight how structural properties of intermediate automata (e.g., final state distribution after reversal or star) affect the final complexity.

Proposed method

• Construct a product automaton for the combined operation using the standard NFA-to-DFA construction, tracking pairs of states from the input automata.
• Define the state set of the resulting DFA as pairs (P, T), where P is a subset of states from the first automaton and T is a subset of states from the second, with constraints based on operation semantics.
• Apply reachability and equivalence analysis to prove minimality of the constructed automaton, showing all states are reachable and no two are equivalent.
• Use specific witness automata with four-letter alphabets and carefully designed transitions to achieve the upper bounds, particularly for the star-catenation case.
• Prove that for reversal-catenation, the upper bound is 3/4·2^{m+n} when m,n ≥ 2, and lower values when m=1 or n=1.
• For star-catenation, derive two different bounds: 5·2^{m+n−3} − 2^{m−1} − 2^{n} + 1 when A has multiple final states, and m(2^n − 1) − 2^{n−1} + 1 when A is star-closed (L(A) = L(A)*).

Experimental results

Research questions

• RQ1 Why does the state complexity of a combined operation like reversal-catenation not equal the composition of the individual operation complexities?
• RQ2 How do structural properties of the intermediate automaton (e.g., final state distribution after reversal) affect the state complexity of the next operation?
• RQ3 What is the tightest possible upper bound on the state complexity of L₁ᴿL₂ for m-state and n-state DFAs?
• RQ4 What is the state complexity of L₁*L₂ when the first automaton A is not star-closed (i.e., L(A)* ≠ L(A))?
• RQ5 Can witness automata be constructed to achieve the derived upper bounds for both reversal-catenation and star-catenation?

Key findings

• The state complexity of reversal-catenation L₁ᴿL₂ is bounded above by 3/4·2^{m+n} for m,n ≥ 2, and this bound is tight and achievable via witness automata.
• When m=1 and n≥1, the state complexity of L₁ᴿL₂ drops to 2^{n−1}, and when m≥2 and n=1, it is 2^{m−1}+1, showing significant reduction due to structural constraints.
• For star-catenation L₁*L₂, if A is star-closed (i.e., its initial state is the only final state), the complexity is m(2^n − 1) − 2^{n−1} + 1, which is lower than the general catenation complexity m·2^n − 2^{n−1}.
• When A is not star-closed (i.e., contains final states other than the initial state), the state complexity of L₁*L₂ is 5·2^{m+n−3} − 2^{m−1} − 2^{n} + 1, which is strictly less than the naive composition 3/4·2^{m+n} − 2^{n−1}.
• The witness automata for both operations are constructed over a four-letter alphabet with transitions that allow full reachability and distinguishability of states, proving the bounds are tight.
• The results demonstrate that the structural properties of intermediate automata—such as the final state configuration after reversal or star—significantly reduce the state complexity of the final combined operation, preventing it from reaching the theoretical composition of individual complexities.