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

Polynomial Invariants for Affine Programs

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate

Modular Descriptions of Regular Functions

We discuss various formalisms to describe string-to-string transformations. Many are based on automata and can be seen as operational descriptions, allowing direct implementations when the input scanner is deterministic. Alternatively, one may use more human friendly descriptions based on some simple basic transformations (e.g., copy, duplicate, erase, reverse) and various combinators such as function composition or extensions of regular operations.Comment: preliminary version appeared in CAI 2019, LNCS 1154

Sequentiality of String-to-Context Transducers

Transducers extend finite state automata with outputs, and describe transformations from strings to strings. Sequential transducers, which have a deterministic behaviour regarding their input, are of particular interest. However, unlike finite-state automata, not every transducer can be made sequential. The seminal work of Choffrut allows to characterise, amongst the functional one-way transducers, the ones that admit an equivalent sequential transducer. In this work, we extend the results of Choffrut to the class of transducers that produce their output string by adding simultaneously, at each transition, a string on the left and a string on the right of the string produced so far. We call them the string-to-context transducers. We obtain a multiple characterisation of the functional string-to-context transducers admitting an equivalent sequential one, based on a Lipschitz property of the function realised by the transducer, and on a pattern (a new twinning property). Last, we prove that given a string-to-context transducer, determining whether there exists an equivalent sequential one is in coNP

Algebraic Recognition of Regular Functions

We consider regular string-to-string functions, i.e. functions that are recognized by copyless streaming string transducers, or any of their equivalent models, such as deterministic two-way automata. We give yet another characterization, which is very succinct: finiteness-preserving functors from the category of semigroups to itself, together with a certain output function that is a natural transformation

On Two-Pass Streaming Algorithms for Maximum Bipartite Matching

We study two-pass streaming algorithms for \textsf{Maximum Bipartite Matching} (\textsf{MBM}). All known two-pass streaming algorithms for \textsf{MBM} operate in a similar fashion: They compute a maximal matching in the first pass and find 3-augmenting paths in the second in order to augment the matching found in the first pass. Our aim is to explore the limitations of this approach and to determine whether current techniques can be used to further improve the state-of-the-art algorithms. We give the following results: We show that every two-pass streaming algorithm that solely computes a maximal matching in the first pass and outputs a $(2/3+\epsilon)$-approximation requires $n^{1+\Omega(\frac{1}{\log \log n})}$ space, for every $\epsilon > 0$, where $n$ is the number of vertices of the input graph. This result is obtained by extending the Ruzsa-Szemer\'{e}di graph construction of [GKK, SODA'12] so as to ensure that the resulting graph has a close to perfect matching, the key property needed in our construction. This result may be of independent interest. Furthermore, we combine the two main techniques, i.e., subsampling followed by the \textsc{Greedy} matching algorithm [Konrad, MFCS'18] which gives a $2-\sqrt{2} \approx 0.5857$-approximation, and the computation of \emph{degree-bounded semi-matchings} [EHM, ICDMW'16][KT, APPROX'17] which gives a $\frac{1}{2} + \frac{1}{12} \approx 0.5833$-approximation, and obtain a meta-algorithm that yields Konrad's and Esfandiari et al.'s algorithms as special cases. This unifies two strands of research. By optimizing parameters, we discover that Konrad's algorithm is optimal for the implied class of algorithms and, perhaps surprisingly, that there is a second optimal algorithm. We show that the analysis of our meta-algorithm is best possible. Our results imply that further improvements, if possible, require new techniques