[Paper Review] State Complexity and the Monoid of Transformations of a Finite Set

Bryan Krawetz, John Lawrence+1 more|ArXiv.org|Jun 29, 2003

semigroups and automata theory8 references4 citations

TL;DR

This paper investigates the state complexity of the root operation on regular languages, defined as root(L) = {w | w^n ∈ L for some n ≥ 1}. It establishes tight upper and lower bounds on the blow-up in state complexity when transforming a DFA of size n into a DFA recognizing root(L), showing that the worst-case complexity is n^n − binom(n,2) for alphabets of size ≥3, and significantly lower for smaller alphabets, using algebraic structure of transformation monoids on finite sets.

ABSTRACT

In this paper we consider the state complexity of an operation on formal languages, root(L). This naturally entails the discussion of the monoid of transformations of a finite set. We obtain good upper and lower bounds on the state complexity of root(L) over alphabets of all sizes.

Motivation & Objective

• To determine the worst-case state complexity of the root operation on regular languages, defined as root(L) = {w | w^n ∈ L for some n ≥ 1}.
• To analyze how the structure of transformation monoids on finite sets influences the state complexity of root(L).
• To derive tight upper and lower bounds on the state complexity of root(L) for alphabets of all sizes, particularly distinguishing between unary, binary, and larger alphabets.
• To characterize the maximal blow-up in state complexity when transforming a minimal DFA for L into a minimal DFA for root(L), using monoid-theoretic techniques.

Proposed method

• Constructs a new DFA A' over the monoid of all transformations of the state set Q, with states Q^Q, to recognize root(L).
• Defines the transition function δ' on A' via δ'(f,a) = f ∘ δ_a, where δ_a is the transformation induced by symbol a.
• Uses the fact that δ'_q0(w) = δ_w, enabling the characterization of w ∈ root(L) iff δ_w^n(q0) ∈ F for some n ≥ 1.
• Applies algebraic results on transformation monoids, particularly the structure of two-generated submonoids U_{k,l} for coprime k ≥ 2, l ≥ 3 with k + l = n.
• Employs combinatorial bounds on monoid size, especially the fact that any monoid M ⊆ T_n with |M| > n^n − binom(n,2) must be the full transformation monoid T_n.
• Leverages computational verification and known results (e.g., from Holzer and König) to establish tightness of bounds for n ≥ 7.

Experimental results

Research questions

• RQ1 What is the maximal state complexity blow-up for the root operation on regular languages over a finite alphabet?
• RQ2 How does the structure of the transformation monoid of a minimal DFA influence the state complexity of root(L)?
• RQ3 Can tight upper and lower bounds on state complexity of root(L) be established for alphabets of size 1, 2, and ≥3?
• RQ4 Is the bound n^n − binom(n,2) tight for alphabets of size ≥3, and what monoid structures achieve this maximum?
• RQ5 What is the role of two-generated submonoids U_{k,l} in determining the worst-case state complexity of root(L)?

Key findings

• For alphabets of size ≥3, the worst-case state complexity of root(L) is n^n − binom(n,2), and this bound is tight.
• When the transformation monoid of the minimal DFA for L is the full transformation monoid T_n, the state complexity of root(L) is exactly n^n − binom(n,2).
• For binary alphabets, the maximal state complexity of root(L) is bounded by |U_{k,l}| − binom(n,2), where U_{k,l} is the largest two-generated submonoid of T_n with coprime k ≥ 2, l ≥ 3, k + l = n.
• For unary languages, the state complexity of root(L) is bounded by the original state complexity, i.e., sc(root(L)) ≤ sc(L), and this bound is tight.
• The construction of a language achieving the bound n^n − binom(n,2) exists for any n ≥ 1 when the alphabet size is at least 3, by using a generating set X of size min(n,3) that generates T_n.
• For n ≥ 7, the bound |U_{k,l}| − binom(n,2) is tight for binary alphabets, assuming the conjecture that U_{k,l} is the largest such submonoid.