[Paper Review] Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages

Janusz Brzozowski, David Liu|arXiv (Cornell University)|Mar 13, 2012

semigroups and automata theory6 references3 citations

TL;DR

This paper investigates the syntactic complexity of finite/cofinite, definite, and reverse definite languages by analyzing the size of their syntactic semigroups as a function of state complexity $n$. It establishes tight upper bounds of $(n-1)!$ for finite/cofinite and reverse definite languages, and conjectures that $\lfloor e\cdot(n-1)!\rfloor$ is the tight upper bound for definite languages, verified for $n \leq 4$, with corresponding alphabet size requirements.

ABSTRACT

We study the syntactic complexity of finite/cofinite, definite and reverse definite languages. The syntactic complexity of a class of languages is defined as the maximal size of syntactic semigroups of languages from the class, taken as a function of the state complexity n of the languages. We prove that (n-1)! is a tight upper bound for finite/cofinite languages and that it can be reached only if the alphabet size is greater than or equal to (n-1)!-(n-2)!. We prove that the bound is also (n-1)! for reverse definite languages, but the minimal alphabet size is (n-1)!-2(n-2)!. We show that \lfloor e\cdot (n-1)! floor is a lower bound on the syntactic complexity of definite languages, and conjecture that this is also an upper bound, and that the alphabet size required to meet this bound is \floor{e \cdot (n-1)!} - \floor{e \cdot (n-2)!}. We prove the conjecture for n\le 4.

Motivation & Objective

• To determine the maximal syntactic complexity (size of syntactic semigroup) for finite/cofinite, definite, and reverse definite languages as a function of state complexity $n$.
• To identify the minimal alphabet size required to achieve the maximal syntactic complexity in each class.
• To establish tight upper bounds for finite/cofinite and reverse definite languages, and to conjecture and verify bounds for definite languages.
• To explore the asymmetry in syntactic complexity bounds between definite and reverse definite languages despite their duality.
• To extend the understanding of syntactic complexity in subclasses of star-free languages within the dot-depth hierarchy.

Proposed method

• Characterize the transformation semigroups of minimal DFAs for each language class using syntactic congruence and Myhill-Nerode equivalence.
• Define and analyze the semigroup $B_n$ of transformations corresponding to definite languages, identifying its structure and generators.
• Use combinatorial counting techniques to compute the size of the minimal generating set $H_n$ of $B_n$, yielding $|H_n| = \lfloor e(n-1)!\rfloor - \lfloor e(n-2)!\rfloor$.
• Prove that the syntactic complexity of reverse definite and finite/cofinite languages is exactly $(n-1)!$, using structural properties of their transformation semigroups.
• Construct minimal DFAs with $n$ states and alphabet size matching the derived bounds, proving minimality and correctness via reachability and distinguishability.
• Apply the exponential generating function and factorial-based bounds to analyze the growth of syntactic complexity, particularly for definite languages.

Experimental results

Research questions

• RQ1 What is the tight upper bound on the syntactic complexity of finite/cofinite languages with state complexity $n$?
• RQ2 What is the tight upper bound on the syntactic complexity of reverse definite languages with state complexity $n$?
• RQ3 What is the maximal syntactic complexity achievable for definite languages with state complexity $n$, and what alphabet size is required to reach it?
• RQ4 Can the conjectured upper bound $\lfloor e(n-1)!\rfloor$ for definite languages be proven for all $n$?
• RQ5 Why does a lack of left-right symmetry persist in syntactic complexity bounds, even though the syntactic congruence is symmetric?

Key findings

• The syntactic complexity of finite/cofinite languages with state complexity $n$ is exactly $(n-1)!$, and this bound is tight only if the alphabet size is at least $(n-1)! - (n-2)!$.
• The syntactic complexity of reverse definite languages with state complexity $n$ is also $(n-1)!$, but requires an alphabet of size $(n-1)! - 2(n-2)!$ to achieve the bound.
• For definite languages, $\lfloor e(n-1)!\rfloor$ is a lower bound on syntactic complexity, and this bound is achievable with an alphabet of size $\lfloor e(n-1)!\rfloor - \lfloor e(n-2)!\rfloor$.
• The conjecture that $\lfloor e(n-1)!\rfloor$ is the tight upper bound for definite languages is proven true for $n \leq 4$.
• The minimal generating set $H_n$ of the transformation semigroup $B_n$ for definite languages has size $\lfloor e(n-1)!\rfloor - \lfloor e(n-2)!\rfloor$, which matches the required alphabet size.
• The construction of a minimal DFA with $n$ states and transformation semigroup $B_n$ confirms that the language accepted is definite, with syntactic complexity $\lfloor e(n-1)!\rfloor$.