Simulation vs. Equivalence

arXiv:1004.2426v1 [cs.FL] 14 Apr 2010 Simulation vs. Equivalence Zoltán Ésik1,∗and Andreas Maletti2,† 1Dept. of Computer Science, University of Szeged Árpád tér 2, 6720 Szeged, Hungary ze@inf.u-szeged.hu 2Dept. de Filologies Romàniques, Universitat Rovira i Virgili Avinguda de Catalunya 35, 43002 Tarragona, Spain andreas.maletti@urv.cat Abstract For several semirings S, two weighted finite automata with multiplic- ities in S are equivalent if and only if they can be connected by a chain of simulations. Such a semiring S is called “proper”. It is known that the Boolean semiring, the semiring of natural numbers, the ring of inte- gers, all finite commutative positively ordered semirings and all fields are proper. The semiring S is Noetherian if every subsemimodule of a finitely generated S-semimodule is finitely generated. First, it is shown that all Noetherian semirings and thus all commutative rings and all finite semi- rings are proper. Second, the tropical semiring is shown not to be proper. So far there has not been any example of a semiring that is not proper. Keywords: Weighted automaton, semiring, rational series, simulation, equivalence. 1 Introduction In this paper, we consider weighted (finite) automata [2, 5] with multiplicities (or weights) in a semiring S. A weighted automaton over a finite alphabet Σ with multiplicities in S defines a rational series [2, 5, 9] in the semiring S⟨⟨Σ∗⟩⟩ of all formal series over Σ with coefficients in S. Two automata are termed equivalent if they define the same rational series. In [3, 4], a notion of morphism between automata was introduced in order to relate equivalent automata. These morphisms, called “simulations” preserve the equivalence of automata. It has been demonstrated that for many semirings, any two equivalent automata over any finite alphabet can be connected by a finite ∗Partially supported by grant no. K 75249 from the National Foundation of Hungary for Scientific Research and by the TÁMOP-4.2.2/08/1/2008-0008 program of the Hungarian National Development Agency. †Supported by the Ministerio de Educación y Ciencia (MEC) grant JDCI-2007-760. 1 chain of simulations. Semirings with this property for all finite alphabets include the Boolean semiring [3], any finite commutative positively ordered semiring [4], any field [1], the semiring N of natural numbers and the ring of integers [1]. Such semirings are called proper. Until now, there has not been any example of a semiring that is not proper. In this note, our aim is two-fold. First, we point out additional classes of proper semirings. We call a semiring S Noetherian if every subsemimodule of a finitely generated S-semimodule is finitely generated. In Theorem 4.2 we show that any Noetherian semiring and thus any commutative ring and any finite semiring is proper. Then in Theorem 5.4 we prove that the tropical semiring [6, 7] used in many combinatorial optimization problems is not proper. 2 Semirings and semimodules We recall from [6, 7] that a semiring S = (S, +, ·, 0, 1) consists of a commutative monoid (S, +, 0) and a monoid (S, ·, 1) such that multiplication (or product) · distributes over addition (or sum) +, and moreover, s·0 = 0 = 0·s for all s ∈S. A semiring S is called commutative if ss′ = s′s for all s, s′ ∈S. (When writ- ing expressions, we will follow the standard convention that multiplication has higher precedence than addition.) Examples of semirings include all fields and rings, all bounded distributive lattices including the 2-element lattice B = {0, 1}, called the Boolean semiring, the semiring N of natural numbers, and the trop- ical semiring defined in Section 5. In order to avoid trivial situations, we will only consider nontrivial semirings in which 0 ̸= 1. When S is a semiring, so is the collection Sn×n of all n × n matrices over S with the usual operations and constants. We will identify any matrix in S1×n with the corresponding row vector, and any matrix in Sn×1 with the corresponding column vector. If S is a semiring, an S-semimodule is a commutative monoid V = (V, +, 0) that is equipped with a (left) S-action S × V →V with (s, v) 7→sv subject to the usual laws: (s + s′)v = sv + s′v s(v + v′) = sv + sv′ (ss′)v = s(s′v) 1v = v s0 = 0 0v = 0 for all s, s′ ∈S and v, v′ ∈V . Note that for any m, n ≥1, the set Sm×n of m×n matrices equipped with the pointwise sum operation is an S-semimodule. Suppose that S is a semiring and Σ is a finite alphabet. Let Σ∗denote the free monoid of all words over Σ including the empty word ǫ. Recall from [2, 9] that a formal series over Σ with multiplicities in S is a function s: Σ∗→S written as a formal sum P w∈Σ∗(s, w)w, where (s, w) = s(w) for each w ∈Σ∗. 2 The support supp(s) of a series s is {w | (s, w) ̸= 0}. We let S⟨⟨Σ∗⟩⟩denote the collection of all such series. Each s ∈S may be identified with the series mapping ǫ to s and all nonempty words to 0. This defines the series 0 and 1. Also, each letter a ∈Σ may be identified with the series mapping a to 1 and all other w

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