An Algebra of Synchronous Scheduling Interfaces

In this paper we propose an algebra of synchronous scheduling interfaces which combines the expressiveness of Boolean algebra for logical and functional behaviour with the min-max-plus arithmetic for quantifying the non-functional aspects of synchronous interfaces. The interface theory arises from a realisability interpretation of intuitionistic modal logic (also known as Curry-Howard-Isomorphism or propositions-as-types principle). The resulting algebra of interface types aims to provide a general setting for specifying type-directed and compositional analyses of worst-case scheduling bounds. It covers synchronous control flow under concurrent, multi-processing or multi-threading execution and permits precise statements about exactness and coverage of the analyses supporting a variety of abstractions. The paper illustrates the expressiveness of the algebra by way of some examples taken from network flow problems, shortest-path, task scheduling and worst-case reaction times in synchronous programming.Comment: In Proceedings FIT 2010, arXiv:1101.426

WCRT Algebra and Scheduling Interfaces for Esterel-Style Synchronous Multithreading

The abstractions used in system design typically limit themselves to encapsulate and guarantee functionality, not timing. Hence, it is very difficult to transfer results on timing behavior across layers, e.g., from the application level through the operating system level to the hardware level. The choice of the model of computation plays a big role in facilitating this transfer. In the realm of reactive systems, the synchronous model of computation has some appeal here, as it inherently limits the number of operations per reaction, and addresses concurrency and preemptive behavior at the language level. Recently, reactive processing architectures have been proposed as execution platform for synchronous languages, notably Esterel. Initially, these architectures were driven by the desire for high performance with low resource usage, including low power consumption. However, by now they have also demonstrated their benefits in terms of predictability. Preliminary work on worst case reaction time (WCRT) analysis has been promising---fairly simple heuristics already achieve an accuracy typically in the 30--40% range. However, these methods so far lack formal grounding, and do not exploit knowledge about signal consistency etc. To provide a formal basis for WCRT analysis, we here propose a type-theoretic, algebraic approach. This approach not only allows to verify the correctness of WCRT analyses methods, but also opens the door for more exact analyses, as it allows to capture functionality and timing precisely and to trade off precision against analysis effort. This approach is still under development; this report presents first results on suitable interface types and the proper characterization of instantaneous nodes, delay nodes and concurrency. As a concrete application, it builds on a multi-threaded Esterel processor, the Kiel Esterel Processor (KEP)

Denotational Fixed-Point Semantics for Constructive Scheduling of Synchronous Concurrency

The synchronous model of concurrent computation (SMoCC) is well established for programming languages in the domain of safety-critical reactive and embedded systems. Translated into mainstream C/Java programming, the SMoCC corresponds to a cyclic execution model in which concurrent threads are synchronised on a logical clock that cuts system computation into a sequence of macro-steps. A causality analysis verifies the existence of a schedule on memory accesses to ensure each macro-step is deadlock-free and determinate. We introduce an abstract semantic domain I(D, P) and an associated denotational fixed point semantics for reasoning about concurrent and sequential variable accesses within a synchronous cycle-based model of computation. We use this domain for a new and extended behavioural definition of Berry’s causality analysis in terms of approximation intervals. The domain I(D, P) extends the domain I(D) from our previous work and fixes a mistake in the treatment of initialisations. Based on this fixed point semantics the notion of Input Berry-constructiveness (IBC) for synchronous programs is proposed. This new IBC class lies properly between strong (SBC) and normal Berry-constructiveness (BC) defined in previous work. SBC and BC are two ways to interpret the standard constructive semantics of synchronous programming, as exemplified by imperative SMoCC languages such as Esterel or Quartz. SBC is often too restrictive as it requires all variables to be initialised by the program. BC can be too permissive because it initialises all variables to a fixed value, by default. Where the initialisation happens through the memory, e.g., when carrying values from one synchronous tick to the next, then IBC is more appropriate. IBC links two levels of execution, the macro-step level and the micro-step level. We prove that the denotational fixed point analysis for IBC, and hence Berry’s causality analysis, is sound with respect to operational micro-level scheduling. The denotational model can thus be viewed as a compositional presentation of a synchronous scheduling strategy that ensures reactiveness and determinacy for imperative concurrent programming

A sequentially constructive circuit semantics for Esterel

Static Single Assignment (SSA) is an established concept that facilitates various program optimizations. However, it is typically restricted to sequential programming. We present an approach that extends SSA for concurrent, reactive programming, specifically for the synchronous language Esterel. This extended SSA transformation expands the class of programs that can be compiled by existing Esterel compilers without causality problems. It also offers a new, efficient solution for the well-studied signal reincarnation problem. Finally, our approach rules out speculation/backtracking, unlike the recently proposed sequentially constructive model of computation

Towards Constructive Description Logics for Abstraction and Refinement

This work explores some aspects of a new and natural semantical dimension that can be accommodated within the syntax of description logics which opens up when passing from the classical truth-value interpretation to a constructive interpretation. We argue that such a strengthened interpretation is essential to represent applications with partial information adequately and to achieve consistency under abstraction as well as robustness under refinement. We introduce a constructive version of ALC, called cALC, for which we give a sound and complete Hilbert axiomatisation and a Gentzen tableau calculus showing finite model property and decidability

On the Computational Interpretation of CKn for Contextual Information Processing – Ancillary Material

In the journal article we introduce a modal λ-calculus λCKn whose type system corresponds to the constructive multi-modal logic CKn. This logic is a constructive refinement of the classical multi-modal logic CKn which forms the heart of the class of description logics used in semantic information processing. λCKn constitutes the core of a functional language for information processing in this application domain. Being strongly typed, it offers static type checking to support safe contextual reasoning in relational structures like those treated by description logics. Here we provide detailed mathematical proofs for the results presented in [8]. Accordingly, this report is not meant as a self-contained technical exposition of the whys and hows of λCKn. It is merely a collection of ancillary material to complement the main article [8] to which the reader is referred for more information. An early version of this work appeared at the 3rd International Workshop on Logics, Agents and Mobility (LAM 2010). We are grateful for the support by the German Research Council (DFG) who funded this research as part of the project SPACMODL under grant No. ME 1427/4-1