Previous results on proving confluence for Constraint Handling Rules are extended in two ways in order to allow a larger and more realistic class of CHR programs to be considered confluent. Firstly, we introduce the relaxed notion of confluence modulo equivalence into the context of CHR: while confluence for a terminating program means that all alternative derivations for a query lead to the exact same final state, confluence modulo equivalence only requires the final states to be equivalent with respect to an equivalence relation tailored for the given program. Secondly, we allow non-logical built-in predicates such as var/1 and incomplete ones such as is/2, that are ignored in previous work on confluence. To this end, a new operational semantics for CHR is developed which includes such predicates. In addition, this semantics differs from earlier approaches by its simplicity without loss of generality, and it may also be recommended for future studies of CHR. For the purely logical subset of CHR, proofs can be expressed in first-order logic, that we show is not sufficient in the present case. We have introduced a formal meta-language that allows reasoning about abstract states and derivations with meta-level restrictions that reflect the non-logical and incomplete predicates. This language represents subproofs as diagrams, which facilitates a systematic enumeration of proof cases, pointing forward to a mechanical support for such proofs. The Project is supported by The DanishCouncil for IndependentResearch, Natural Sciences, Grant No. DFF4181-00442. The second author’s contribution received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 318337, ENTRA—Whole-Systems Energy Transparency.

Previous results on confluence for Constraint Handling Rules, CHR, are generalized to take into account user-defined state equivalence relations. This allows a much larger class of programs to enjoy the advantages of confluence, which include various optimization techniques and simplified correctness proofs. A new operational semantics for CHR is introduced that significantly reduces notational overhead and allows to consider confluence for programs with extra-logical and incomplete builtin predicates. Proofs of confluence are demonstrated for programs with redundant data representation, e.g., sets-as-lists, for dynamic programming algorithms with pruning as well as a Union-Find program, which are not covered by previous confluence notions for CHR. M.H. Kirkeby-The second author's contribution has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no 318337, ENTRA -Whole-Systems Energy Transparency.

Promoting energy efficiency to a first class system design goal is an important research challenge. Although more energy-efficient hardware can be designed, it is software that controls the hardware; for a given system the potential for energy savings is likely to be much greater at the higher levels of abstraction in the system stack. Thus the greatest savings are expected from energy-aware software development, which is the vision of the EU ENTRA project. This article presents the concept of energy transparency as a foundation for energyaware software development. We show how energy modelling of hardware is combined with static analysis to allow the programmer to understand the energy consumption of a program without executing it, thus enabling exploration of the design space taking energy into consideration. The paper concludes by summarising the current and future challenges identified in the ENTRA project.

Convergence of an abstract reduction system (ARS) is the property that any derivation from an initial state will end in the same final state, a.k.a. normal form. We generalize this for probabilistic ARS as almost-sure convergence, meaning that the normal form is reached with probability one, even if diverging derivations may exist. We show and exemplify properties that can be used for proving almost-sure convergence of probabilistic ARS, generalizing known results from ARS. IntroductionProbabilistic abstract reduction systems, PARS, are general models of systems that develop over time in discrete steps [7]. In each non-final state, the choice of successor state is governed by a probability distribution, which in turn induces a global, probabilistic behaviour of the system. Probabilities make termination more than a simple yes-no question, and the following criteria have been proposed: probabilistic termination -a derivation terminates with some probability > 0 -and almost-sure termination -a derivation terminates with probability = 1, even if infinite derivations may exist (and whose total probability thus amounts to 0). When considering a PARS as a computational system, almostsure termination may be the most interesting, and there exist well-established methods for proving this property [6,10].PARS cover a variety of probabilistic algorithms and programs, scheduling strategies and protocols [5,7,23], and PARS is a well-suited abstraction level for better understanding their termination and correctness properties. Randomized or probabilistic algorithms (e.g., [4,20,21]) come in two groups: Monte Carlo Algorithms that allow a set of alternative outputs (typically only correct with a certain probability or within a certain accuracy), e.g., , Monte Carlo integration and Simulated Annealing [19]; and Las Vegas Algorithms, that provide one (correct) output and that may be simpler and on average more efficient than their deterministic counterparts, e.g., Randomized Quicksort [11], checking equivalence of circular lists [17], probabilistic modular GCD [30]. We focus on results that are relevant for the latter kind of systems, and here the property of convergence is interesting, as it may be a necessary condition for correctness: a system is convergent if it is guaranteed to terminate

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