A Generalised Twinning Property for Minimisation of Cost Register Automata

Weighted automata (WA) extend finite-state automata by associating with transitions weights from a semiring S, defining functions from words to S. Recently, cost register automata (CRA) have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine. Unambiguous WA over a monoid (M, ⊗) can equivalently be described by cost register automata whose registers take their values in M, and are updated by operations of the form x: = y ⊗ c, with c ∈ M. This class is denoted by CRA⊗c(M). We introduce a twinning property and a bounded variation property parametrised by an integer k, such that the corresponding notions introduced originally by Choffrut for finite-state transducers are obtained for k = 1. Given an unambiguous weighted automaton W over an infinitary group (G, ⊗) realizing some function f, we prove that the three following properties are equivalent: i) W satisfies the twinning property of order k, ii) f satisfies the k-bounded variation property, and iii) f can be described by a CRA⊗c(G) with at most k registers. In the spirit of tranducers, we actually prove this result in a more general setting by considering machines over the semiring of finite sets of elements from (G, ⊗): the three properties are still equivalent for such finite-valued weighted automata, that is the ones associating with words subsets of G of cardinality at most ℓ, for some natural ℓ. Moreover, we show that if the operation ⊗ of G is commutative and computable, then one can decide whether a WA satisfies the twinning property of order k. As a corollary, this allows to decide the register minimisation problem for the class CRA⊗c(G). Last, we prove that a similar result holds for finite-valued finite-state transducers, and that the register minimisation problem for the class CRA.c (B*) is Pspace-complete

LIPIcs

Repeated idempotent elements are commonly used to characterise iterable behaviours in abstract models of computation. Therefore, given a monoid M, it is natural to ask how long a sequence of elements of M needs to be to ensure the presence of consecutive idempotent factors. This question is formalised through the notion of the Ramsey function R_M associated to M, obtained by mapping every k ∈ ℕ to the minimal integer R_M(k) such that every word u ∈ M^* of length R_M(k) contains k consecutive non-empty factors that correspond to the same idempotent element of M. In this work, we study the behaviour of the Ramsey function R_M by investigating the regular -length of M, defined as the largest size L(M) of a submonoid of M isomorphic to the set of natural numbers {1,2, …, L(M)} equipped with the max operation. We show that the regular -length of M determines the degree of R_M, by proving that k^L(M) ≤ R_M(k) ≤ (k|M|⁴)^L(M). To allow applications of this result, we provide the value of the regular -length of diverse monoids. In particular, we prove that the full monoid of n × n Boolean matrices, which is used to express transition monoids of non-deterministic automata, has a regular -length of (n²+n+2)/2

Bidding Graph Games with Partially-Observable Budgets

Two-player zero-sum "graph games" are a central model, which proceeds as follows. A token is placed on a vertex of a graph, and the two players move it to produce an infinite "play", which determines the winner or payoff of the game. Traditionally, the players alternate turns in moving the token. In "bidding games", however, the players have budgets and in each turn, an auction (bidding) determines which player moves the token. So far, bidding games have only been studied as full-information games. In this work we initiate the study of partial-information bidding games: we study bidding games in which a player's initial budget is drawn from a known probability distribution. We show that while for some bidding mechanisms and objectives, it is straightforward to adapt the results from the full-information setting to the partial-information setting, for others, the analysis is significantly more challenging, requires new techniques, and gives rise to interesting results. Specifically, we study games with "mean-payoff" objectives in combination with "poorman" bidding. We construct optimal strategies for a partially-informed player who plays against a fully-informed adversary. We show that, somewhat surprisingly, the "value" under pure strategies does not necessarily exist in such games.Comment: The full version of a paper published at AAAI 2

LIPIcs

A deterministic finite automaton (DFA) is composite if its language L() can be decomposed into an intersection ⋂_{i = 1}^k L(_i) of languages of smaller DFAs. Otherwise, is prime. This notion of primality was introduced by Kupferman and Mosheiff in 2013, and while they proved that we can decide whether a DFA is composite, the precise complexity of this problem is still open, with a doubly-exponential gap between the upper and lower bounds. In this work, we focus on permutation DFAs, i.e., those for which the transition monoid is a group. We provide an NP algorithm to decide whether a permutation DFA is composite, and show that the difficulty of this problem comes from the number of non-accepting states of the instance: we give a fixed-parameter tractable algorithm with the number of rejecting states as the parameter. Moreover, we investigate the class of commutative permutation DFAs. Their structural properties allow us to decide compositionality in NL, and even in LOGSPACE if the alphabet size is fixed. Despite this low complexity, we show that complex behaviors still arise in this class: we provide a family of composite DFAs each requiring polynomially many factors with respect to its size. We also consider the variant of the problem that asks whether a DFA is k-factor composite, that is, decomposable into k smaller DFAs, for some given integer k ∈ ℕ. We show that, for commutative permutation DFAs, restricting the number of factors makes the decision computationally harder, and yields a problem with tight bounds: it is NP-complete. Finally, we show that in general, this problem is in PSPACE, and it is in LOGSPACE for DFAs with a singleton alphabet

LIPIcs

A regular language L of finite words is composite if there are regular languages L₁,L₂,…,L_t such that L = ⋂_{i = 1}^t L_i and the index (number of states in a minimal DFA) of every language L_i is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [O. Kupferman and J. Mosheiff, 2015], where the complexity of deciding the primality of the language of a given DFA was left open, with a doubly-exponential gap between the upper and lower bounds. We study primality for unary regular languages, namely regular languages with a singleton alphabet. A unary language corresponds to a subset of ℕ, making the study of unary prime languages closer to that of primality in number theory. We show that the setting of languages is richer. In particular, while every composite number is the product of two smaller numbers, the number t of languages necessary to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness for a unary language L, namely a word that is not in L but is in all products of languages that contain L and have an index smaller than L’s, may be of exponential length. Still, we are able to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace algorithm for primality checking of unary DFAs

Improving Indigenous access to healthcare services through interprofessional collaboration

In northwestern Ontario, interprofessional collaboration is needed to improve access to healthcare services for Indigenous people. The Truth and Reconciliation Commission of Canada identified 94 Calls to Action, four of which guided the design of this community action research project (No.18: acknowledge previous health policy is responsible for Indigenous health, No.19: identify appropriate health services, No.22: recognize the value of traditional health practices; No.23: provide competency training for health professionals). The purpose of this study was to examine whether the teaching of six interprofessional competencies to healthcare teams servicing northern First Nation communities enhances: 1) interprofessional collaboration and 2) Indigenous healthcare access. A two-eyed seeing approach supported an interprofessional collaboration (IPC) training intervention involving 30 participants. A convergent parallel mixed methods design, including a post-post test design survey and second-order narratives, supported the generation of community action-oriented goals. A statistically significant difference in each of the six interprofessional competency domains was found following the training. Qualitative analysis demonstrated that access to healthcare services does improve following collaboration training. The mixed analysis demonstrated that sustainable community resources focused on healthcare access were developed as a result of the action-oriented goals. Northern First Nation communities can benefit directly and indirectly from interprofessional competency training for the purpose of improving access to healthcare services. By incorporating Indigenous ways of knowing within a community action research framework, Calls to Action can be enacted

Une traduction castillane anonyme de l’Éthique à Nicomaque dans la bibliothèque du Marquis de Santillane (ms BNE 10268)

Cet article étudie deux types d’inscriptions marginales dans le manuscrit 10268 de la Biblioteca Nacional de España, manuscrit du XVe siècle ayant appartenu à la bibliothèque du Marquis de Santillane et contenant une traduction castillane de l’Éthique à Nicomaque d’Aristote : j’analyse d’une part le rapport entre le texte traduit et les annotations marginales qui l’accompagnent, d’autre part les citations éparses copiées sur les premiers et derniers feuillets du manuscrit, afin de mettre en évidence l’importance qu’y revêt le thème de la prudence, comme vertu morale mais aussi et surtout comme vertu politique.Este artículo estudia dos tipos de inscripciones marginales en el manuscrito 10268 de la Biblioteca Nacional de España, manuscrito del siglo XV que perteneció a la biblioteca del Marqués de Santillana y que contiene una traducción castellana de la Ética a Nicómaco de Aristóteles : analizo por una parte la relación entre el texto traducido y las notas marginales que lo acompañan, por otra parte las citas dispersas copiadas en los primeros y últimos folios del manuscrito, para poner de manifiesto la importancia atribuida al tema de la prudencia, como virtud moral pero también y sobre todo como virtud política