Sequential Composition Research Articles

Analyzing Program Analyses

We want to prove that a static analysis of a given program is complete, namely, no imprecision arises when asking some query on the program behavior in the concrete (ie, for its concrete semantics) or in the abstract (ie, for its abstract interpretation). Completeness proofs are therefore useful to assign confidence to alarms raised by static analyses. We introduce the completeness class of an abstraction as the set of all programs for which the abstraction is complete. Our first result shows that for any nontrivial abstraction, its completeness class is not recursively enumerable. We then introduce a stratified deductive system to prove the completeness of program analyses over an abstract domain A. We prove the soundness of the deductive system. We observe that the only sources of incompleteness are assignments and Boolean tests --- unlikely a common belief in static analysis, joins do not induce incompleteness. The first layer of this proof system is generic, abstraction-agnostic, and it deals with the standard constructs for program composition, that is, sequential composition, branching and guarded iteration. The second layer is instead abstraction-specific: the designer of an abstract domain A provides conditions for completeness in A of assignments and Boolean tests which have to be checked by a suitable static analysis or assumed in the completeness proof as hypotheses. We instantiate the second layer of this proof system first with a generic nonrelational abstraction in order to provide a sound rule for the completeness of assignments. Orthogonally, we instantiate it to the numerical abstract domains of Intervals and Octagons, providing necessary and sufficient conditions for the completeness of their Boolean tests and of assignments for Octagons.

Evolution Time of Composed Stochastic Systems with Final Sequence of States and Independent Transitions

Abstract In this paper the composed stochastic systems with final sequence states and independent transitions are studied. The ordered and unordered sequential compositions and excludable and nonexcludable parallel compositions are analyzed. For these stochastic systems the problem of determining the main probabilistic characteristics of evolution time is considered. The elaborated algorithms are based on the properties of the homogeneous linear recurrences.

Structural transformations for data-enriched real-time systems

Abstract We investigate design-level structural transformations that aim at easier subsequent verification of real-time systems with shared data variables, modelled as networks of extended timed automata (ETA). Our contributions to this end are the following: (1) we first equip ETA with an operator for layered composition , intermediate between parallel and sequential composition. Under certain non-interference and/or precedence conditions imposed on the structure of the ETA networks, the communication closed layer (CCL) laws and associated partial-order (po-) and (layered) reachability equivalences are shown to hold. (2) Next, we investigate (under certain cycle conditions on the ETA) the (reachability preserving) transformations of separation and flattening aimed at reducing the number of cycles of the ETA. (3) We then show that our separation and flattening in (2) may be applied together with the CCL laws in (1), in order to restructure ETA networks such that the verification of layered reachability properties is rendered easier. This interplay of the three structural transformations (separation, flattening, and layering) is demonstrated on an enhanced version of Fischer’s real-time mutual exclusion protocol for access to multiple critical sections.

Laws of concurrent programming

The talk extends the Laws of Programming [1] by four laws governing concurrent composition of programs. This operator is associative and commutative and distributive through union; and it has the same unit (do nothing) as sequential composition. Furthermore, sequential and concurrent composition distribute through each other, in accordance with an exchange law; this permits an implementation of concurrency by partial interleaving.

Relational properties of sequential composition of coalgebras

In this paper we define a sequential composition for arbitrary coalgebras in a Dedekind category. We show some basic algebraic properties of this operation up to bisimulation. Furthermore, we consider certain recursive equations and provide an explicit solution, i.e., a solution not based on an iterative process.

NetKAT

Recent years have seen growing interest in high-level languages for programming networks. But the design of these languages has been largely ad hoc, driven more by the needs of applications and the capabilities of network hardware than by foundational principles. The lack of a semantic foundation has left language designers with little guidance in determining how to incorporate new features, and programmers without a means to reason precisely about their code. This paper presents NetKAT, a new network programming language that is based on a solid mathematical foundation and comes equipped with a sound and complete equational theory. We describe the design of NetKAT, including primitives for filtering, modifying, and transmitting packets; union and sequential composition operators; and a Kleene star operator that iterates programs. We show that NetKAT is an instance of a canonical and well-studied mathematical structure called a Kleene algebra with tests (KAT) and prove that its equational theory is sound and complete with respect to its denotational semantics. Finally, we present practical applications of the equational theory including syntactic techniques for checking reachability, proving non-interference properties that ensure isolation between programs, and establishing the correctness of compilation algorithms.

Parametric effect monads and semantics of effect systems

We study fundamental properties of a generalisation of monad called parametric effect monad , and apply it to the interpretation of general effect systems whose effects have sequential composition operators. We show that parametric effect monads admit analogues of the structures and concepts that exist for monads, such as Kleisli triples, the state monad and the continuation monad, Plotkin and Power's algebraic operations, and the categorical ┬┬-lifting. We also show a systematic method to generate both effects and a parametric effect monad from a monad morphism. Finally, we introduce two effect systems with explicit and implicit subeffecting, and discuss their denotational semantics and the soundness of effect systems.

Полиномиальный алгоритм проверки эквивалентности в модели программ с перестановочными и подавляемыми операторами

In this paper we study the equivalence problem in the model of sequential programs which assumes that some instructions are commutative and absorbing. Two instructions are commutative if the result of their executions does not depend on an order of their execution. An instruction b absorbs an instruction a if the sequential composition a;b yields the same result as the single instruction b. A.A. Letichevskij in 1971 proved the decidability of equivalence checking problem in this model of programs. Nevertheless a possibility of building polynomial time equivalence checking procedures remains an open problem till nowadays. The main result of this paper is the description of a polynomial time algorithm for checking the equivalence of sequential programs with commutative and absorbing instructions. The paper includes 9 sections. In Section 1 we introduce informally the model of programs under consideration, the equivalence checking problem, and give a brief overview of the preceding results in the study of equivalence checking problem for this model. In Sections 2 and 3 the syntax and a semigroup-based semantics of propositional model of sequential programs are formally defined. The algebraic properties of semigroups of commutative and absorbing program instructions are studied in Section 4. In Section 5 we introduce a graph of joined computations which is the key structure in designing our equivalence checking techniques. In Section 6 and 7 we show that the cumulative absorbing effect of finite sequences of commutative instructions can be specified by means of finite automata; this is another key step in building the decision procedure. In Section 8 we show that equivalence checking of two programs is reducible to a traversal of a bounded fragment of the graph of joint computations of these programs; the latter can be performed in time polynomial of the size of programs to be checked.

Functional architectures and structured flows on manifolds: a dynamical framework for motor behavior.

We outline a dynamical framework for sequential sensorimotor behavior based on the sequential composition of basic behavioral units. Basic units are conceptualized as temporarily existing low-dimensional dynamical objects, or structured flows, emerging from a high-dimensional system, referred to as structured flows on manifolds. Theorems from dynamical system theory allow for the unambiguous classification of behaviors as represented by structured flows, and thus provide a means to define and identify basic units. The ensemble of structured flows available to an individual defines his or her dynamical repertoire. We briefly review experimental evidence that has identified a few basic elements likely to contribute to each individual's repertoire. Complex behavior requires the involvement of a (typically high-dimensional) dynamics operating at a time scale slower than that of the elements in the dynamical repertoire. At any given time, in the competition between units of the repertoire, the slow dynamics temporarily favor the dominance of one element over others in a sequential fashion, binding together the units and generating complex behavior. The time scale separation between the elements of the repertoire and the slow dynamics define a time scale hierarchy, and their ensemble defines a functional architecture. We illustrate the approach with a functional architecture for handwriting as proof of concept and discuss the implications of the framework for motor control.

Commutativity of Quantum Weakest Liberal Precondition and Its Properties

首先给出了量子最弱自由前置条件(weakest liberal precondition,简称wlp)wlp (A,B,C)-可交换的定义,研究了wlp (A,B,C)-可交换的充分必要条件;其次,得到了wlp 不是良好的谓词转换,验证了wlp 是比量子最弱前置条件(weakest precondition,简称wp)更弱的谓词转换,揭示了wlp 和wp 的本质区别;最后证明了wlp 的序列合成、并行合成和块结构等性质.;First, the definition of quantum weakest liberal precondition (termed wlp) wlp (A,B,C)-commutativity is proposed, some necessary, and sufficient conditions of wlp (A,B,C)-commutativity are presented. Secondly, it has been shown that wlp is not a healthy predicate transformer: it is verified that wlp is a weaker predicate transformer than the quantum weakest precondition (termed wp). The essential differences of wlp and wp are disclosed. Finally, the properties for sequential composition, parallel composition and block structure of wlp are investigated.

Synthesis and analysis of distributed ensemble control strategies for allocation to multiple tasks

SUMMARYWe present the synthesis and analysis of distributed ensemble control policies to enable a team of robots to control their distribution across a collection of tasks. We assume that individual robot controllers are modeled as a sequential composition of individual task controllers. A macroscopic description of the team dynamics is then used to synthesize ensemble feedback control strategies that maintain the desired distribution of robots across the tasks. We present a distributed implementation of the ensemble feedback strategy that can be implemented with minimal communication requirements. Different from existing strategies, the approach results in individual robot control policies that maintain the desired mean and the variance of the robot populations at each task. We present the stability properties of the ensemble feedback strategy, verify the feasibility of the distributed ensemble controller through high-fidelity simulations, and examine the robustness of the strategy to sensing and/or actuation failures. Specifically, we consider the case when robots are subject to estimation and navigation errors resulting from lossy inter-agent wireless communication links and localization errors.

Composing least-change lenses

Non-trivial bidirectional transformations (BXs) are inherently ambiguous, as there are in general many different ways to consistently translate an update from one side to the other. Existing BX languages and frameworks typically satisfy fundamental first principles which ensure acceptable and stable (well-behaved) translation. Unfortunately, these give little insight about how a particular update translation is chosen among the myriad possible. From the user perspective, such unpredictability may hinder the adoption of BX frameworks. The problem can be remedied by imposing a “principle of least change” which, in a state-based framework, amounts to translating each update in a way such that its result is as close as possible to the original state, according to some distance measure. Starting by formalizing such BXs focusing on the particular framework of lenses, this paper discusses whether such least-change lenses can be defined by composition, an essential construct of BX frameworks. For sequential composition, two (dual) update translation alternatives are presented: a classical deterministic one and a nondeterministic. A key ingredient of the approach is the elegant formalization of the main concepts in relation algebra, which exposes several similarities and dualities.

Refinement algebra with dual operator

Algebras of imperative programming languages have been successful in reasoning about programs. In general an algebra of programs is an algebraic structure with programs as elements and with program compositions (sequential composition, choice, skip) as algebra operations. Various versions of these algebras were introduced to model partial correctness, total correctness, refinement, demonic choice, and other aspects. We introduce here an algebra which can be used to model total correctness, refinement, demonic and angelic choice. The basic model of our algebra are monotonic Boolean transformers (monotonic functions from a Boolean algebra to itself).

Concurrency and local reasoning under reverse exchange

Quite a number of aspects of concurrency are reflected by the inequational exchange law (P⁎Q);(R⁎S)⩽(P;R)⁎(Q;S) between sequential composition ; and concurrent composition ⁎. In particular, recent research has shown that, under a certain semantic definition, validity of this law is equivalent to that of the familiar concurrency rule for Hoare triples. Unfortunately, while the law holds in the standard model of concurrent Kleene algebra, its is not true in the relationally based setting of algebraic separation logic. However, we show that under mild conditions the reverse inequation (P;R)⁎(Q;S)⩽(P⁎Q);(R⁎S) still holds there. From this reverse exchange law we derive slightly restricted but still reasonably useful variants of the concurrency rule. Moreover, using a corresponding definition of locality, we obtain also a variant of the frame rule, where ⁎ now is interpreted as separating conjunction. These results allow using the relational setting also for modular and concurrency reasoning. Finally, we interpret the results further by discussing several variations of the approach.

Compositional reasoning using intervals and time reversal

Interval Temporal Logic (ITL) is an established formalism for reasoning about time periods. We investigate some simple kinds of ITL formulas which have application to compositional reasoning and furthermore are closed under conjunction and the conventional temporal operator known both as "box" and "always". Such closures help us modularly construct formulas from simple building blocks in a way which preserves useful compositional properties. The most important class considered here is called the 2-to-1 formulas. They offer an attractive framework for analysing sequential composition in ITL and provide the formal basis for most of the subsequent presentation. A key contribution of this work concerns a useful and apparently new and quite elementary mathematical theorem that 2-to-1 formulas are closed under "box". We also use a natural form of time symmetry with 2-to-1 formulas. This extends known facts about such formulas by looking at them in reverse. An important example involves showing that 2-to-1 formulas are also closed under a variant of "box" for prefix subintervals rather than suffix ones. We then apply the compositional formulas obtained with time symmetry to analyse concurrent behaviour involving mutual exclusion in both Peterson's algorithm and a new and more abstract one. At present, our study of mutual exclusion mainly serves as a kind of experimental "proof of concept" and research tool to develop and illustrate some of the logical framework's promising features. We also discuss how time symmetry sometimes assists in reducing reasoning in ITL to conventional linear-time temporal logic.

The sequential semantics of producer effect systems

Effects are fundamental to programming languages. Even the lambda calculus has effects, and consequently the two famous evaluation strategies produce different semantics. As such, much research has been done to improve our understanding of effects. Since Moggi introduced monads for his computational lambda calculus, further generalizations have been designed to formalize increasingly complex computational effects, such as indexed monads followed by layered monads followed by parameterized monads. This succession prompted us to determine the most general formalization possible. In searching for this formalization we came across many surprises, such as the insufficiencies of arrows, as well as many unexpected insights, such as the importance of considering an effect as a small component of a whole system rather than just an isolated feature. In this paper we present our semantic formalization for producer effect systems, which we call a productor, and prove its maximal generality by focusing on only sequential composition of effectful computations, consequently guaranteeing that the existing monadic techniques are specializations of productors.

Composition pattern oriented tag extraction from short documents using a structural learning method

With the rapid growth of web, automatic tagging that detects informative terms from a document becomes an important problem for information aggregation and sharing services. In particular, automatic tagging for short documents becomes more interesting as many users are increasingly publishing information through social media services which encourage users to create the documents of short length. In this paper, we propose a novel automatic tagging model for short text documents from social media services, following the framework of supervised learning. We redefine traditional frequency-based term features so that they can address the properties of the documents created under length limitation and consider sequential dependencies between successive terms in a document based on a structural support vector machine. In addition, our proposed approach incorporates composition patterns by which users put informative terms into their documents. Extensive experiments have been conducted to validate the presented approach, and it was found that the proposed term features were effective for extracting tags, and the tag extractor trained by considering the sequential dependencies and composition patterns achieved superior performance results over the existing alternative methods.

Decomposition of Relations and Concept Lattices

We introduce the decomposition of an arbitrary relation into a sequential composition of three relations, viz. of a mapping with a partial order and then the transpose of a mapping. After presenting some basic properties, we investigate the specific classes of junkfree, irreducible and minimal decompositions and show that for all relations a minimal decomposition exists. We also study decompositions with regard to DedekindMacNeille completions and concept lattices. These constructions are closely related to decompositions of relations. In our setting the fundamental theorem of concept lattices states that concept lattices are minimal-complete decompositions and all such decompositions are isomorphic. As a further main result we prove that the cutDedekindMacNeille completion of the order that belongs to the minimal decomposition of a relation is isomorphic to the concept lattice of that relation. Instead of considering binary relations on sets, we will work point-free within the general framework of allegories. This complement-free approach implies that the results of the paper can be applied to all models of these algebraic structures, including, for instance, lattice-valued fuzzy relations.

Control and Planning of 3-D Dynamic Walking With Asymptotically Stable Gait Primitives

In this paper, we present a hierarchical framework that enables motion planning for asymptotically stable 3-D bipedal walking in the same way that planning is already possible for zero moment point walking. This framework is based on the construction of asymptotically stable gait primitives for a class of hybrid dynamical systems with impacts. Each primitive corresponds to an asymptotically stable hybrid limit cycle that admits rules a priori for sequential composition with other primitives, reducing a high-dimensional feedback motion planning problem into a low-dimensional discrete tree search. As a constructive example, we develop this planning framework for the 3-D compass-gait biped, where each primitive corresponds to walking along an arc of constant curvature for a fixed number of steps. We apply a discrete search algorithm to plan a sequence of these primitives, taking the 3-D biped stably from start to goal in workspaces with obstacles. We finally show how this framework generalizes to more complex models by planning walking paths for an underactuated five-link biped.

ADVANCED SYNCHRONIZATION OF AUDIO OR SYMBOLIC MUSICAL PATTERNS: AN ALGEBRAIC APPROACH

We define a generic model for finite audio or symbolic musical patterns that structurally encodes a rich and abstract synchronization mechanism. This is achieved by distinguishing for each pattern a realization window, describing what the pattern is, from a synchronization window, describing how the pattern can be used. The sequential composition of patterns is defined and studied. An algebra of musical patterns is introduced in a mathematically well-founded approach. We propose several high level operators that can be used either in audio processing or in musical analysis and composition. Practical uses and experiments conducted in both fields are described.