Exact and Approximate Determinization of Discounted-Sum Automata

A discounted-sum automaton (NDA) is a nondeterministic finite automaton with edge weights, valuing a run by the discounted sum of visited edge weights. More precisely, the weight in the i-th position of the run is divided by $\lambda^i$, where the discount factor $\lambda$ is a fixed rational number greater than 1. The value of a word is the minimal value of the automaton runs on it. Discounted summation is a common and useful measuring scheme, especially for infinite sequences, reflecting the assumption that earlier weights are more important than later weights. Unfortunately, determinization of NDAs, which is often essential in formal verification, is, in general, not possible. We provide positive news, showing that every NDA with an integral discount factor is determinizable. We complete the picture by proving that the integers characterize exactly the discount factors that guarantee determinizability: for every nonintegral rational discount factor $\lambda$, there is a nondeterminizable $\lambda$-NDA. We also prove that the class of NDAs with integral discount factors enjoys closure under the algebraic operations min, max, addition, and subtraction, which is not the case for general NDAs nor for deterministic NDAs. For general NDAs, we look into approximate determinization, which is always possible as the influence of a word's suffix decays. We show that the naive approach, of unfolding the automaton computations up to a sufficient level, is doubly exponential in the discount factor. We provide an alternative construction for approximate determinization, which is singly exponential in the discount factor, in the precision, and in the number of states. We also prove matching lower bounds, showing that the exponential dependency on each of these three parameters cannot be avoided. All our results hold equally for automata over finite words and for automata over infinite words

Rabin vs. Streett Automata

The Rabin and Streett acceptance conditions are dual. Accordingly, deterministic Rabin and Streett automata are dual. Yet, when adding nondeterminsim, the picture changes dramatically. In fact, the state blowup involved in translations between Rabin and Streett automata is a longstanding open problem, having an exponential gap between the known lower and upper bounds. We resolve the problem, showing that the translation of Streett to Rabin automata involves a state blowup in $Theta(n^2)$, whereas in the other direction, the translations of both deterministic and nondeterministic Rabin automata to nondeterministic Streett automata involve a state blowup in $2^{Theta(n)}$. Analyzing this substantial difference between the two directions, we get to the conclusion that when studying translations between automata, one should not only consider the state blowup, but also the emph{size} blowup, where the latter takes into account all of the automaton elements. More precisely, the size of an automaton is defined to be the maximum of the alphabet length, the number of states, the number of transitions, and the acceptance condition length (index). Indeed, size-wise, the results are opposite. That is, the translation of Rabin to Streett involves a size blowup in $Theta(n^2)$ and of Streett to Rabin in $2^{Theta(n)}$. The core difference between state blowup and size blowup stems from the tradeoff between the index and the number of states. (Recall that the index of Rabin and Streett automata might be exponential in the number of states.) We continue with resolving the open problem of translating deterministic Rabin and Streett automata to the weaker types of deterministic co-B"uchi and B"uchi automata, respectively. We show that the state blowup involved in these translations, when possible, is in $2^{Theta(n)}$, whereas the size blowup is in $Theta(n^2)$

On the (In)Succinctness of Muller Automata

There are several types of finite automata on infinite words, differing in their acceptance conditions. As each type has its own advantages, there is an extensive research on the size blowup involved in translating one automaton type to another. Of special interest is the Muller type, providing the most detailed acceptance condition. It turns out that there is inconsistency and incompleteness in the literature results regarding the translations to and from Muller automata. Considering the automaton size, some results take into account, in addition to the number of states, the alphabet length and the number of transitions while ignoring the length of the acceptance condition, whereas other results consider the length of the acceptance condition while ignoring the two other parameters. We establish a full picture of the translations to and from Muller automata, enhancing known results and adding new ones. Overall, Muller automata can be considered less succinct than parity, Rabin, and Streett automata: translating nondeterministic Muller automata to the other nondeterministic types involves a polynomial size blowup, while the other way round is exponential; translating between the deterministic versions is exponential in both directions; and translating nondeterministic automata of all types to deterministic Muller automata is doubly exponential, as opposed to a single exponent in the translations to the other deterministic types

Increasing Accountability for Rape in Liberia: The Need for a Forensic System to Increase the Success Rates of Prosecution

The need for a fully functioning forensic system has been identified by the Liberian government and international partners, but it has not been addressed. This Article argues that despite a robust framework put in place to create accountability for rape, Liberia needs a system of collecting and processing forensic evidence to increase the success rate of prosecutions that currently fail due to the inadequacy of non-forensic evidence

Development and validation of the CAM Health Belief Questionnaire (CHBQ) and CAM use and attitudes amongst medical students

BACKGROUND: The need for Complementary and Alternative Medicine (CAM) and holistic approaches in allopathic medical school curricula has been well articulated. Despite increased CAM instruction, feasible and validated instruments for measuring learner outcomes in this content area do not widely exist. In addition, baseline attitudes or beliefs of medical students towards CAM, and the factors that may have formed them, including use of CAM itself, remain unreported. METHODS: A 10-item measure (CHBQ – CAM Health Belief Questionnaire) was constructed and administered to three successive classes of medical students simultaneously with the previously validated 29-item Integrative Medicine Attitude Questionnaire (IMAQ). Both measures were imbedded in a baseline needs assessment questionnaire. Demographic and other data were collected on students' use of CAM modalities and their awareness and use of primary CAM information resources. Analysis of CHBQ items was performed and its reliability and criterion-related validity were established. RESULTS: Response rate was 96.5% (272 of 282 students studied). The shorter CHBQ compared favorably with the longer IMAQ in internal consistency reliability. Cronbach's coefficient alpha was 0.75 and 0.83 for the CHBQ and IMAQ respectively. Students showed positive attitudes/beliefs towards CAM and high levels of self-reported CAM use. The majority (73.5%) of students reported using at least one CAM modality, and 54% reported using at least two modalities. Eighty-one percent use the internet as a primary source of information for CAM. CONCLUSIONS: The CHBQ is a practical, valid and reliable instrument for measuring medical student attitudes/beliefs and has potential utility for measuring the impact of CAM instruction. Medical students showed a high self-reported rate of CAM use and positive attitudes towards CAM. Short, didactic exposure to CAM instruction in the first year of medical school did not additionally impact these already positive attitudes. Unlike the IMAQ, which was intended for use with physicians, the CHBQ is generic in design and content and applicable to a variety of learner types. Evaluation measures must be appropriate for specific CAM instructional outcomes