A semantics for a class of non-deterministic and causal production system programs

We define a class of function-free rule-based production system (PS) programs that exhibit non-deterministic and/or causal behavior. We develop a fixpoint semantics and an equivalent declarative semantics for these programs. The criterion to recognize the class of non-deterministic causal (NDC) PS programs is based upon extending and relaxing the concept of stratification, to partition the rules of the program. Unlike strict stratification, this relaxed stratification criterion allows a more flexible partitioning of the rules and admits programs whose execution is non-deterministic or causal or both. The fixpoint semantics is based upon a monotonic fixpoint operator which guarantees that the execution of the program will terminate. Each fixpoint corresponds to a minimal database of answers for the NDC PS program. Since the execution of the program is non-deterministic, several fixpoints may be obtained. To obtain a declarative meaning for the PS program, we associate a normal logic program with each NDC PS program. We use the generalized disjunctive well-founded semantics to provide a meaning to the normal logic program Through these semantics, a well-founded state is associated with and a set of possible extensions, each of which are minimal models for the well-founded state, are obtained. We show that the fixpoint semantics for the NDC PS programs is sound and complete with respect to the declarative semantics for the corresponding normal logic program .This research is partially sponsored by the National Science Foundation under grant IRI-9008208 and by the Institute for Advanced Computer Studies.     

A rewriting logic framework for operational semantics of membrane systems

Existing results in membrane computing refer mainly to P systems’ characterization of Turing computability, also to some polynomial solutions to NP-complete problems by using an exponential workspace created in a “biological way”. In this paper we define an operational semantics of a basic class of P systems, and give two implementations of the operational semantics using rewriting logic. We present some results regarding these implementations, including two operational correspondence results, and discuss why these implementations are relevant in order to take advantage of good features of both structural operational semantics and rewriting logic.     

A Rewriting Logic Approach to Operational Semantics (Extended Abstract)

This paper shows how rewriting logic semantics (RLS) can be used as a computational logic framework for operational semantic definitions of programming languages. Several operational semantics styles are addressed: big-step and small-step structural operational semantics (SOS), modular SOS, reduction semantics with evaluation contexts, and continuation-based semantics. Each of these language definitional styles can be faithfully captured as an RLS theory, in the sense that there is a one-to-one correspondence between computational steps in the original language definition and computational steps in the corresponding RLS theory. A major goal of this paper is to show that RLS does not force or pre-impose any given language definitional style, and that its flexibility and ease of use makes RLS an appealing framework for exploring new definitional styles.     

OCL 1.4/5 vs. 2.0 Expressions Formal semantics and expressiveness

A type inference system and a big-step operational semantics for expressions of the Object Constraint Language (OCL), the declarative and navigational constraint language for the Unified Modeling Language (UML), are provided; the account is mainly based on OCL 1.4/5, but also includes the main features of OCL 2.0. The formal systems are parameterised in terms of UML static structures and UML object models, which are treated abstractly. It is proved that the operational semantics satisfies a subject reduction property with respect to the type inference system. Proceeding from the operational semantics and providing a denotational semantics, pure OCL 2.0 expressions are shown to exactly represent the primitive recursive functions, whereas pure OCL 1.4/5 expressions are Turing complete.     

Reconciling statechart semantics

Statecharts are a visual technique for modelling reactive behaviour. Over the years, a plethora of statechart semantics have been proposed. The three most widely used are the fixpoint, Statemate, and UML semantics. These three semantics differ considerably from each other. In general, they interpret the same statechart differently, which impedes the communication of statechart designs among both designers and tools. In this paper, we identify a set of constraints on statecharts that ensure that the fixpoint, Statemate and UML semantics coincide, if observations are restricted to linear, stuttering-closed, separable properties. Moreover, we show that for a subset of these constraints, a slight variation of the Statemate semantics coincides for linear stuttering-closed properties with the UML semantics.     

Bottom-up Evaluation of Datalog with Negation

Declarative semantics gives the meaning of a logic program in terms of properties,while the procedural semantics gives the meaning in terms of the execution or evaluation of the program.From the database point of view,the procedural semantics of the program is equally important.This paper focuses on the study of the bottom-up evaluation of the WFM semantics of datalog‘ programs.To compute the WFM,first,the stability transformation is revisited,and a new operator Op and its fixpoint are defined. Based on this,a fixpoint semantics,called oscillating fixpoint model semantics,is defined.Then,it is shown that for any datalog‘ program the oscillating fixpoint model is identical to its WFM.So,the oscillating fixpoint model can be viewed as an alternative (constructive) definition of WFM.The underlying operation (or transformation) for reaching the oscillating fixpoint provides a potential of bottom-up evaluation.For the sake of computational feasibility,the strongly range-restricted program is considered,and an algorithm used to compute the oscillating fixpoint is described.     

A Constructor-Based EI-Model Semantics of EI-CTRS

This paper investigates the semantics of conditional term rewriting systems with negation (denoted by EI-CTRS),called constructor-based EI-model semantics.The introduction of “≠” in EI-CTRS make EI-CTRS more difficult to study.This is in part because of a failure of EI-CTRS to guarantee that there exist least Herbrand models in classical logical point of views.The key idea of EI-model is to explain that t≠s” means that the two concepts represented by t and s respectively actually belong to distinguished basic concepts represented by two constructor-ground terms.We define the concept of EI-model,and show that there exist least Herbrand EI-models for EI-satisfiable EI-CTRS.From algebraic and logic point of view,we show that there are very strong reasons for regarding the least Herbrand EI-models as the intended semantics of EI-CTRS.According to fixpoint theory,we develop a method to construct least Herbrand EI-models in a bottom-up manner.Moreover,we discuss soundness and completeness of EI-rewrite for EI-model semantics.     

基于函数式语义的循环和递归程序结构通用证明技术

各类安全攸关系统的可靠运行离不开软件程序的正确执行.程序的演绎验证技术为程序执行的正确性提供高度保障.程序语言种类繁多,且用途覆盖高可靠性场景的新式语言不断涌现,难以为每种语言设计支撑其程序验证任务的整套逻辑规则,并证明其相对于形式语义的可靠性和完备性.语言无关的程序验证技术提供以程序语言的语义为参数的验证过程及其可靠性结果.对每种程序语言,提供其形式语义后可直接获得面向该语言的程序验证过程.提出一种面向大步操作语义的语言无关演绎验证技术,其核心是对不同语言中循环、递归等可导致无界行为的语法结构进行可靠推理的通用方法.特别地,借助大步操作语义的一种函数式形式化提供表达程序中子结构所执行计算的能力,从而允许借助辅助信息对子结构进行推理.证明所提出验证技术的可靠性和相对完备性,通过命令式、函数式语言中的程序验证实例初步评估了该技术的有效性,并在Coq辅助证明工具中形式化了所有理论结果和验证实例,为基于辅助证明工具实现面向大步语义的语言无关程序验证工具提供了基础.     

A formal library of set relations and its application to synchronous languages

Set relations are particularly suitable for specifying the small-step operational semantics of synchronous languages. In this paper, a formal library of set relations for the definition, verification of properties, and execution of binary set relations is presented. The formal library consists of a set of theories written in the Prototype Verification System (PVS) that contains definitions and proofs of properties, such as determinism and compositionality, for synchronous relations. The paper also proposes a serialization procedure that enables the simulation of synchronous set relations via set rewriting systems. The library and the serialization procedure are illustrated with the rewriting logic semantics of the Plan Execution Interchange Language (PLEXIL), a rich synchronous plan execution language developed by NASA to support autonomous spacecraft operations.     

Trace-Based Abstract Interpretation of Operational Semantics

We present trace-based abstract interpretation, a unification of severallines of research on applying Cousot-Cousot-style abstract interpretation a.i. tooperational semantics definitions (such as flowchart, big-step, and small-step semantics)that express a programs semantics as a concrete computation tree of trace paths. Aprograms trace-based a.i. is also a computation tree whose nodes contain abstractions ofstate and whose paths simulate the paths in the programs concrete computation tree.Using such computation trees, we provide a simple explanation of the central concept of collecting semantics, and we distinguish concrete from abstract collectingsemantics and state-based from path-based collecting semantics. We also expose therelationship between collecting semantics extraction and results garnered from flow-analytic and model-checking-based analysis techniques. We adapt concepts fromconcurrency theory to formalize safe and live a.i.s for computation trees; in particular, coinduction techniques help extend fundamental results to infinite computation trees.Problems specific to the various operational semantics methodologies are discussed: Big-step semantics cannot express divergence, so we employ a mixture of induction andcoinduction in response; small-step semantics generate sequences of programconfigurations unbounded in size, so we abstractly interpret source language syntax.Applications of trace-based a.i. to data-flow analysis, model checking, closure analysis,and concurrency theory are demonstrated.     

Reduction Semantics and Formal Analysis of Orc Programs

Orc is a language for orchestration of web services developed by J. Misra that offers simple, yet powerful and elegant, constructs to program sophisticated web orchestration applications. The formal semantics of Orc poses interesting challenges, because of its real-time nature and the different priorities of external and internal actions. In this paper, building upon our previous SOS semantics of Orc in rewriting logic, we present a much more efficient reduction semantics of Orc, which is provably equivalent to the SOS semantics thanks to a strong bisimulation. We view this reduction semantics as a key intermediate stage towards a future, provably correct distributed implementation of Orc, and show how it can naturally be extended to a distributed actor-like semantics. We show experiments demonstrating the much better performance of the reduction semantics when compared to the SOS semantics. Using the Maude rewriting logic language, we also illustrate how the reduction semantics can be used to endow Orc with useful formal analysis capabilities, including an LTL model checker. We illustrate these formal analysis features by means of an online auction system, which is modeled as a distributed system of actors that perform Orc computations.     

There is no fully abstract fixpoint semantics for non-deterministic languages with infinite computations

It is well known that for many non-deterministic programming languages there is no continuous fully abstract fixpoint semantics. This is usually attributed to “problems with continuity”, that is, the assumption that the semantic functions should be continuous supposedly plays a role in the difficulties of giving a fully abstract fixpoint semantics. We show that for a large class of non-deterministic programming languages there is no fully abstract least fixpoint semantics even if one considers arbitrary functions (not necessarily continuous) over arbitrary partial orders (not necessarily complete).     

An initial algebra approach to term rewriting systems with variable binders

We present an extension of first-order term rewriting systems. It involves variable binding in the term language. We develop systems called binding term rewriting systems (BTRSs) in a stepwise manner. First we present the term language, then formulate equational logic. Finally, we define rewriting systems. This development is novel because we follow the initial algebra approach in an extended notion of Σ-algebras in various functor categories. These are based on Fiore-Plotkin-Turi’s presheaf semantics of variable binding and Lüth-Ghani’s monadic semantics of term rewriting systems. We characterise the terms, equational logic and rewrite systems for BTRSs as initial algebras in suitable categories. Then, we show an important rewriting property of BTRSs: orthogonal BTRSs are confluent. Moreover, by using the initial algebra semantics, we give a complete characterisation of termination of BTRSs. Finally, we discuss our design choice of BTRSs from a semantic perspective. An erlier version appeared in Proc. Fifth ACM-SIGPLAN International Conference on Principles and Practice of Declarative Programming (PPDP2003).     

A logic programming system for nonmonotonic reasoning

The evolution of logic programming semantics has included the introduction of a new explicit form of negation, beside the older implicit (or default) negation typical of logic programming. The richer language has been shown adequate for a spate of knowledge representation and reasoning forms.The widespread use of such extended programs requires the definition of a correct top-down querying mechanism, much as for Prolog wrt. normal programs. One purpose of this paper is to present and exploit a SLDNF-like derivation procedure, SLX, for programs with explicit negation under well-founded semantics (WFSX) and prove its soundness and completeness. (Its soundness wrt. the answer-sets semantics is also shown.) Our choice ofWFSX as the base semantics is justi-fied by the structural properties it enjoys, which are paramount for top-down query evaluation.Of course, introducing explicit negation requires dealing with contradiction. Consequently, we allow for contradiction to appear, and show moreover how it can be removed by freely changing the truth-values of some subset of a set of predefined revisable literals. To achieve this, we introduce a paraconsistent version ofWFSX, WFSX _p , that allows contradictions and for which our SLX top-down procedure is proven correct as well.This procedure can be used to detect the existence of pairs of complementary literals inWESX _p simply by detecting the violation of integrity rulesf L, -L introduced for eachL in the language of the program. Furthermore, integrity constraints of a more general form are allowed, whose violation can likewise be detected by SLX.Removal of contradiction or integrity violation is accomplished by a variant of the SLX procedure that collects, in a formula, the alternative combinations of revisable literals' truth-values that ensure the said removal. The formulas, after simplification, can then be satisfied by a number of truth-values changes in the revisable, among true, false, and undefined. A notion of minimal change is defined as well that establishes a closeness relation between a program and its revisions. Forthwith, the changes can be enforced by introducing or deleting program rules for the revisable literals.To illustrate the usefulness and originality of our framework, we applied it to obtain a novel logic programming approach, and results, in declarative debugging and model-based diagnosis problems.     

On stratified disjunctive programs

We address the problem of a consistent fixpoint semantics for general disjunctive programs restricted to stratifiable programs which do not recurse through negative literals. We apply the nonmonotonic fixpoint theory developed by Apt, Blair and Walker to a closure operatorT ^c and develop a fixpoint semantics for stratified disjunctive programs. We also provide an iterative definition for negation, called the Generalized Closed World Assumption for Stratified programs (GCWAS), and show that our semantics captures this definition. We develop a model-theoretic semantics for stratified disjunctive programs and show that the least state characterized by the fixpoint semantics corresponds to a stable-state defined in a manner similar to the stable-models of Gelfond and Lifschitz. We also discuss a weaker stratification semantics for general disjunctive programs based on the Weak Generalized Closed World Assumption.     

The rewriting logic semantics project

Rewriting logic is a flexible and expressive logical framework that unifies algebraic denotational semantics and structural operational semantics (SOS) in a novel way, avoiding their respective limitations and allowing succinct semantic definitions. The fact that a rewrite logic theory’s axioms include both equations and rewrite rules provides a useful “abstraction dial” to find the right balance between abstraction and computational observability in semantic definitions. Such semantic definitions are directly executable as interpreters in a rewriting logic language such as Maude, whose generic formal tools can be used to endow those interpreters with powerful program analysis capabilities.     

On the verification of finite failure

In Gori [An abstract interpretation framework to reason on finite failure and other properties of finite and infinite computations, Theoret. Comput. Sci. 290(1) (2003) 863-936] a new fixpoint semantics which correctly models finite failure has been defined. This semantics is And-compositional, compositional w.r.t. instantiation and is based on a co-continuous operator. Based on this fixpoint semantics a new inductive method able to verify a program w.r.t. the property of finite failure can be defined. In this paper we show how Ferrand's approach, using both a least fixpoint and greatest fixpoint semantics, can be adapted to finite failure. The verification method is not effective. Therefore, we consider an approximation from above and an approximation from below of our semantics, which give two different finite approximations. These approximations are used for effective program verification.     

Coherence approach to logic program revision

In this paper, we present a new approach to the problem of revising extended programs; we base this approach on the coherence theory initially advocated by Gardenfors for belief revision. Our approach resolves contradiction by removing only conflicting information, not the believed source of it, and therefore, keeps information loss minimal. Furthermore, since there is no need to search for problematic assumptions, as is done in the traditional assumption-removal approach, our approach provides a skeptical revision semantics that is tractable. We define the skeptical and credulous coherence semantics and show that both semantics can be characterized in terms of the fixpoint semantics of a revised program using a simple program-revision technique. These semantics provide a suitable framework for knowledge and belief revision in the context of logic programs. Semantical properties and advantages of the proposed revision semantics are also analyzed     

Bi-inductive Structural Semantics: (Extended Abstract)

We propose a simple order-theoretic generalization of set-theoretic inductive definitions. This generalization covers inductive, co-inductive and bi-inductive definitions and is preserved by abstraction. This allows the structural operational semantics to describe simultaneously the finite/terminating and infinite/diverging behaviors of programs. This is illustrated on the structural bifinitary small/big-step trace/relational/operational semantics of the call-by-value λ-calculus.