92 research outputs found
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Containment and equivalence of weighted automata: Probabilistic and max-plus cases
This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain
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Register complexity and determinisation of max-plus automata
We survey some results about the sequentiality problem for max-plus automata and its generalisation, the register complexity problem for cost register automata. We compare classes of functions computed by maxplus automata and by cost register automata with respect to the notion of ambiguity. The two models are introduced gently, so the novice reader is welcome
Revisiting the growth of polyregular functions: output languages, weighted automata and unary inputs
Polyregular functions are the class of string-to-string functions definable
by pebble transducers (an extension of finite automata) or equivalently by MSO
interpretations (a logical formalism). Their output length is bounded by a
polynomial in the input length: a function computed by a -pebble transducer
or by a -dimensional MSO interpretation has growth rate .
Boja\'nczyk has recently shown that the converse holds for MSO
interpretations, but not for pebble transducers. We give significantly
simplified proofs of those two results, extending the former to first-order
interpretations by reduction to an elementary property of -weighted
automata. For any , we also prove the stronger statement that there is some
quadratic polyregular function whose output language differs from that of any
-fold composition of macro tree transducers (and which therefore cannot be
computed by any -pebble transducer).
In the special case of unary input alphabets, we show that pebbles
suffice to compute polyregular functions of growth . This is obtained
as a corollary of a basis of simple word sequences whose ultimately periodic
combinations generate all polyregular functions with unary input. Finally, we
study polyregular and polyblind functions between unary alphabets (i.e. integer
sequences), as well as their first-order subclasses.Comment: 27 pages, not submitted ye
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Comparison of max-plus automata and joint spectral radius of tropical matrices
Weighted automata over the max-plus semiring S are closely related to finitely generated semigroups of matrices over S. In this paper, we use results in automata theory to study two quantities associated with sets of matrices: the joint spectral radius and the ultimate rank. We prove that these two quantities are not computable over the tropical semiring, i.e. there is no algorithm that takes as input a finite set of matrices M and provides as output the joint spectral radius (resp. the ultimate rank) of M. On the other hand, we prove that the joint spectral radius is nevertheless approximable and we exhibit restricted cases in which the joint spectral radius and the ultimate rank are computable. To reach this aim, we study the problem of comparing functions computed by weighted automata over the tropical semiring. This problem is known to be undecidable and we prove that it remains undecidable in some specific subclasses of automata
An Upper Bound on the Complexity of Recognizable Tree Languages
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular
tree language of infinite trees is in a class
for some natural number , where is the game quantifier. We
first give a detailed exposition of this result. Next, using an embedding of
the Wadge hierarchy of non self-dual Borel subsets of the Cantor space
into the class , and the notions of Wadge degree
and Veblen function, we argue that this upper bound on the topological
complexity of regular tree languages is much better than the usual
Unambiguous Separators for Tropical Tree Automata
In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that f ? g, there exists effectively an unambiguous tropical automaton computing h such that f ? h ? g.
This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different
On the Sensitivity Conjecture for Read-k Formulas
Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and degree over R, are all polynomially related to one another. The sensitivity conjecture states that there is also a polynomial relationship between sensitivity and block sensitivity, thus supplying the "missing link".
Since its introduction in 1991, the sensitivity conjecture has remained a challenging open question in the study of Boolean functions. One natural approach is to prove it for special classes of functions. For instance, the conjecture is known to be true for monotone functions, symmetric functions, and
functions describing graph properties.
In this paper, we consider the conjecture for Boolean functions computable by read-k formulas. A read-k formula is a tree in which each variable appears at most k times among the leaves and has Boolean gates at its internal nodes. We show that the sensitivity conjecture holds for read-once formulas with gates computing symmetric functions. We next consider regular formulas with OR and AND gates. A formula is regular if it is a leveled tree with all gates at a given level having the same fan-in and computing the same function. We prove the sensitivity conjecture for constant depth regular read-k formulas for constant k
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