## Equational Theories of Tropical Semirings (2001)

Venue: | THEORETICAL COMPUTER SCIENCE |

Citations: | 3 - 0 self |

### BibTeX

@TECHREPORT{Aceto01equationaltheories,

author = {Luca Aceto and Zoltan Esik and Anna Ingolfsdottir},

title = {Equational Theories of Tropical Semirings},

institution = {THEORETICAL COMPUTER SCIENCE},

year = {2001}

}

### OpenURL

### Abstract

This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max, +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

### Citations

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Citation Context ...in a number of simple inequations that is exponential in the number of variables. However, the validity of a simple inequation can be tested in polynomial time by using linear programming (see, e.g., =-=[33]-=-). The interested reader is referred to [1] for more information. # Remark 3.13 The decidability of the equational theories of the structures Z# , N# and N# also follows from well-known results in log... |

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Citation Context ... not finitely based, and has no axiomatization in a bounded number of variables. 4 Throughout the paper, we shall use standard notions and notations from universal algebra that can be found, e.g., in =-=[8, 13]-=-. This paper collects, and improves upon, all of the results first announced without proof in [3, 4]. In addition, the material in Sects. 5 and 6 is new. 2 Background Definitions We begin by introduci... |

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Citation Context ...st-order theory of addition on the natural numbers. It is interesting to compare the above result on the complexity of the equational theory of N# and N# with the classic results by Fischer and Rabin =-=[11]-=- on the complexity of the first-order theory of the real numbers under addition, and of Presburger arithmetic. There is a fixed constant c > 0 such that for every (non-deterministic) decision procedur... |

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Citation Context ...operations, respectively.) Interest in idempotent semirings arose in the 1950s through the observation that some problems in discrete optimization could be linearized over such structures (see, e.g., =-=[10]-=- for some of the early references and [38] for a survey). Since then, the study of idempotent semirings has forged productive connections with such diverse fields as, e.g., performance evaluation of m... |

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(Show Context)
Citation Context ...atistical physics, large deviations), and automata and language theory (automata with multiplicities). The interested reader is referred to [14] for a survey of these more recent developments, and to =-=[12]-=- for further applications of the (max, +) semiring. Here we limit ourselves to mentioning some of the deep applications of variations on the tropical semiring in automata theory and the study of forma... |

43 |
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Citation Context ... finitely generated group must necessarily be finite if each element has finite order [7].) The tropical semiring was also used by Hashiguchi in his independent solution to the aforementioned problem =-=[16]-=-, and in his study of the star height of regular languages (see, e.g., [17, 18, 19]). (For a tutorial introduction on how the tropical semiring is used to solve the finite power property problem, we r... |

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Citation Context ...equatorial semirings are studied in [23]. Another classic question for the language of regular expressions, with or without multiplicities, is the study of complete axiom systems for them (see, e.g., =-=[9, 21, 32]-=-). Along this line of research, Bonnier-Rigny and Krob have o#ered a complete system of identities for one-letter regular expressions with multiplicities in the tropical semiring [6]. However, to the ... |

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Citation Context ...ation is idempotent---two prime examples are the (max, +) semiring (R # {-#},max,+,-#, 0) (see [5, Chapter 3] for a general reference), and the tropical semiring (N # {#}, min, +, #, 0) introduced in =-=[34]-=-. (Henceforth, we shall write # and # for the binary maximum and minimum operations, respectively.) Interest in idempotent semirings arose in the 1950s through the observation that some problems in di... |

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Citation Context ...equatorial semirings are studied in [23]. Another classic question for the language of regular expressions, with or without multiplicities, is the study of complete axiom systems for them (see, e.g., =-=[9, 21, 32]-=-). Along this line of research, Bonnier-Rigny and Krob have o#ered a complete system of identities for one-letter regular expressions with multiplicities in the tropical semiring [6]. However, to the ... |

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Citation Context ... semiring to reformulate the finite power property as a Burnside problem. (The original Burnside problem asks if a finitely generated group must necessarily be finite if each element has finite order =-=[7]-=-.) The tropical semiring was also used by Hashiguchi in his independent solution to the aforementioned problem [16], and in his study of the star height of regular languages (see, e.g., [17, 18, 19]).... |

24 | Tropical semirings
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Citation Context ...of the star height of regular languages (see, e.g., [17, 18, 19]). (For a tutorial introduction on how the tropical semiring is used to solve the finite power property problem, we refer the reader to =-=[31]-=-.) The tropical semiring also plays a key role in Simon's study of the nondeterministic complexity of a standard finite automaton [36]. In his thesis [26], Leung introduced topological ideas into the ... |

23 |
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Citation Context ...equatorial semirings are studied in [23]. Another classic question for the language of regular expressions, with or without multiplicities, is the study of complete axiom systems for them (see, e.g., =-=[9, 21, 32]-=-). Along this line of research, Bonnier-Rigny and Krob have o#ered a complete system of identities for one-letter regular expressions with multiplicities in the tropical semiring [6]. However, to the ... |

13 |
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(Show Context)
Citation Context ... number of variables, then V(A) also has such an axiomatization. # We now turn to positive ciw-semirings. Definition 4.8 An equation is regular if its two sides contain the same variables (cf., e.g., =-=[37]-=-). An inequation is regular if every variable contained in the left-hand side of the inequation also appears on the right-hand side. An inequation is strictly regular if its two sides contain the same... |

12 |
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(Show Context)
Citation Context ...ee, e.g., [9, 21, 32]). Along this line of research, Bonnier-Rigny and Krob have o#ered a complete system of identities for one-letter regular expressions with multiplicities in the tropical semiring =-=[6]-=-. However, to the best of our knowledge, there has not been a systematic investigation of the equational theory of the di#erent exotic semirings studied in the literature. This is the aim of this pape... |

10 |
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(Show Context)
Citation Context ...ower property problem, we refer the reader to [31].) The tropical semiring also plays a key role in Simon's study of the nondeterministic complexity of a standard finite automaton [36]. In his thesis =-=[26]-=-, Leung introduced topological ideas into the study of the limitedness problem for distance automata (see also [27]). For an improved treatment of his solutions, and further references, we refer the r... |

3 |
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(Show Context)
Citation Context ...on is standard multiplication (Sect. 6.2), the semirings studied by Mascle and Leung in [30] and [26, 27], respectively, (Sect. 6.3), and a minplus algebra based on the ordinals proposed by Mascle in =-=[29]-=- (Sect. 6.4). For all of these structures, we o#er results to the e#ect that their equational theory is not finitely based, and has no axiomatization in a bounded number of variables. 4 Throughout the... |

2 |
olfsd ottir, The max-plus algebra of the natural numbers has no finite equational basis
- Aceto, Esik, et al.
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(Show Context)
Citation Context ... been a systematic investigation of the equational theory of the di#erent exotic semirings studied in the literature. This is the aim of this paper. Our starting points are the results we obtained in =-=[1, 2]-=-. In [1] we studied the equational theory of the max-plus algebra of the natural numbers N# = (N, #, +, 0), and proved that not only its equational theory is not finitely based, but, for every n, the ... |

2 |
languages of star height one
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(Show Context)
Citation Context ...nite order [7].) The tropical semiring was also used by Hashiguchi in his independent solution to the aforementioned problem [16], and in his study of the star height of regular languages (see, e.g., =-=[17, 18, 19]-=-). (For a tutorial introduction on how the tropical semiring is used to solve the finite power property problem, we refer the reader to [31].) The tropical semiring also plays a key role in Simon's st... |

2 |
Recent BRICS Report Series Publications RS-01-21 Luca Aceto, Zolt an Esik, and Anna Ing olfsd ottir. Equational Theories of Tropical Semirings. June 2001. 52 pp. Extended abstracts of parts of this paper have appeared in Honsell and Miculan
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(Show Context)
Citation Context ...mpotent semirings arose in the 1950s through the observation that some problems in discrete optimization could be linearized over such structures (see, e.g., [10] for some of the early references and =-=[38]-=- for a survey). Since then, the study of idempotent semirings has forged productive connections with such diverse fields as, e.g., performance evaluation of manufacturing systems, discrete event syste... |