Calculi for interaction

Action structures have previously been proposed as an algebra for both the syntax and the semantics of interactive computation. Here, a class of concrete action structures called action calculi is identified, which can serve as a non-linear syntax for a wide variety of models of interactive behaviour. Each action in an action calculus is represented as an assembly of molecules; the syntactic binding of names is the means by which molecules are bound together. A graphical form, action graphs, is used to aid presentation. One action calculus differs from another only in its generators, called controls. Action calculi generalise a previously defined action structure for the -calculus. Several extensions to are given as action calculi, giving essentially the same power as the -calculus. An action calculus is also given for the typed -calculus, and for Petri nets parametrized on their places and transitions. An equational characterization of action calculi is given: each action calculus is the quotient of a term algebra by certain equations. The terms are generated by a set of operators, including those basic to all action structures as well as the controls specific to ; the equations are the basic axioms of action structures together with four additional axiom schemata. Received May 12, 1995 / August 7, 1995     

半边图模型之聚合归约演算

提出了实现自组织多层次归约的一个指导原则,即自组聚合与归约的协调准则,其核心思想是,基于多聚合准则的自组聚合演算中的各个聚合子集是可以相交的,相交的部分是下一步归约演算的基础。给出了符合上述协调准则的自组图聚合归约演算模型,聚合子图是聚合演算的结果,在归约演算中,聚合子图对应为归约顶点,子图的子边界对应为归约半边,而由子图相交部分抽象出的子图边界之间的关系则对应为归约边,从而构成了形式上完整统一的自组织多层次归约。     

A Rewriting Calculus for Cyclic Higher-order Term Graphs

Introduced at the end of the nineties, the Rewriting Calculus (ρ-calculus, for short) is a simple calculus that fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the obtained structured results are first class objects of the calculus. The evaluation mechanism, generalizing beta-reduction, strongly relies on term matching in various theories.In this paper we propose an extension of the ρ-calculus, handling graph like structures rather than simple terms. The transformations are performed by explicit application of rewrite rules as first class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.The calculus over terms is naturally generalized by using unification constraints in addition to the standard ρ-calculus matching constraints. This therefore provides us with the basics for a natural extension of an explicit substitution calculus to term graphs. Several examples illustrating the introduced concepts are given.     

Calculi for interaction

Action structures have previously been proposed as an algebra for both the syntax and the semantics of interactive computation. Here, a class of concrete action structures calledaction calculi is identified, which can serve as a non-linear syntax for a wide variety of models of interactive behaviour. Each action in an action calculus is represented as an assembly ofmolecules; the syntactic binding ofnames is the means by which molecules are bound together. A graphical form,action graphs, is used to aid presentation. One action calculus differs from another only in its generators, calledcontrols. Action calculi generalise a previously defined action structure PIC for the π- calculus. Several extensions to PIC are given as action calculi, giving essentially the same power as the π-calculus. An action calculus is also given for the typed λ-calculus, and for Petri nets parametrized on their places and transitions. An equational characterization of action calculi is given: each action calculusA is the quotient of a term algebra by certain equations. The terms are generated by a set of operators, including those basic to all action structures as well as the controls specific toA; the equations are the basic axioms of action structures together with four additional axiom schemata.     

Liberalized Variable Splitting

Variable splitting is a technique applicable to free variable tableaux, sequent calculi, and matrix characterizations that exploits a relationship between β- and γ-rules. Using contextual information to differentiate between occurrences of the same free variable in different branches, the technique admits conditions under which these occurrences may safely be assigned different values by substitutions. This article investigates a system of variable splitting and shows its consistency by a semantical argument. The splitting system is liberalized with respect to β-inferences analogously to a well-known liberalization of δ-rules, and this is used to show an exponential speedup compared to free variable systems without splitting.     

Action graphs and user performance analysis

A user operating an interactive system performs actions such as “pressing a button” and these actions cause state transitions in the system. However to perform an action, a user has to do what amounts to a state transition themselves, from the state of having completed the previous action to the state of starting to perform the next action; this user transition is out of step with the system's transition. This paper introduces action graphs, an elegant way of making user transitions explicit in the arcs of a graph derived from the system specification. Essentially, a conventional transition system has arcs labeled in the form “user performs action A” whereas an action graph has arcs labelled in the form “having performed action P, the user performs Q.” Action graphs support many modelling techniques (such as GOMS, KLM or shortest paths) that could have been applied to the user's actions or to the system graph, but because it combines both, the modelling techniques can be used more powerfully.Action graphs can be used to directly apply user performance metrics and hence perform formal evaluations of interactive systems. The Fitts Law is one of the simplest and most robust of such user modelling techniques, and is used as an illustration of the value of action graphs in this paper. Action graphs can help analyze particular tasks, any sample of tasks, or all possible tasks a device supports—which would be impractical for empirical evaluations. This is an important result for analyzing safety critical interactive systems, where it is important to cover all possible tasks in testing even when doing so is not feasible using human participants because of the complexity of the system.An algorithm is presented for the construction of action graphs. Action graphs are then used to study devices (a consumer device, a digital multimeter, an infusion pump) and results suggest that: optimal time is correlated with keystroke count, and that keyboard layout has little impact on optimal times. Many other applications of action graphs are suggested.     

Accelerating tableaux proofs using compact representations

In this article a modified form of tableau calculus, calledTableau Graph Calculus, is presented for overcoming the well-known inefficiencies of the traditional tableau calculus to a large extent. This calculus is based on a compact representation of analytic tableaux by using graph structures calledtableau graphs. These graphs are obtained from the input formula in linear time and incorporate most of the rule applications of normal tableau calculus during the conversion process. The size of this representation is linear with respect to the length of the input formula and the graph closely resembles the proof tree of tableau calculi thus retaining the naturalness of the proof procedure. As a result of this preprocessing step, tableau graph calculus has only a single rule which is repeatedly applied to obtain a proof. Many optimizations for the applications of this rule, to effectively prune the search space, are presented as well. Brief details of an implemented prover called FAUST, embedded within the higher-order theorem prover called HOL, are also given. Applications of FAUST within a hardware verification context are also sketched.This work has been partly financed by german national grant under the project Automated System Design, SFB No. 358.     

Fuzzy logic and the calculi of fuzzy rules, fuzzy graphs, and fuzzy probabilities

The past few years have witnessed a rapid growth in the number and variety of applications of fuzzy logic, ranging from consumer products and industrial process control to medical instrumentation, information systems, and decision analysis. The foundations of fuzzy logic have become firmer and its impact within the basic sciences—and especially in mathematical and physical sciences—has become more visible and more substantive. And yet, there are still many misconceptions about the aims of fuzzy logic and misjudgments of its strengths and limitations.One of the common misconceptions is rooted in semantics: as a label, fuzzy logic, FL, has two different meanings. More specifically, in a narrow sense, fuzzy logic, FLn, is a logical system which aims at a formalization of approximate reasoning. In this sense, fuzzy logic is an extension of multivalued logic but its agenda is quite different from that of conventional multivalued systems.In a wide sense, fuzzy logic, FLw, is coextensive with fuzzy set theory, FST. Flw is far broader than FLn and contains Fln as one of its branches. Today, the term fuzzy logic is used predominantly in its wide sense. Thus, effectively, FL = FLw = FST.Another important point is that any field X can be fuzzified by replacing crisp sets in X by fuzzy sets. For example, through fuzzification, arithmetic can be generalized to fuzzy arithmetic, topology to fuzzy topology, control theory to fuzzy control theory, etc.In this perspective, the calculi of fuzzy rules (CFR), fuzzy graphs (CFG), and fuzzy probabilities (CFP) may be viewed as generalizations of the calculi of rules, graphs, and probabilities. The importance of CFR, CFG, and CFP derives from the fact that they play a central role in most of the applications of FL. In particular, the calculus of fuzzy graphs, which is a subset of the calculus of fuzzy rules, accounts for most of the applications of fuzzy logic in control, systems analysis, and related fields.Central to the calculus of fuzzy rules is a language referred to as the Fuzzy Dependency and Command Language (FDCL). The syntax of FDCL is concerned with the form of rules, while the semantics of FDCL is concerned with their meaning. An important issue in CFR is that of the induction of rules from observations.In CFG, a fuzzy graph is defined as the disjunction of Cartesian products of fuzzy sets. In effect, a fuzzy graph may be viewed as a compressed representation of a functional or relational dependence. Operations on fuzzy graphs play an important role in CFG.In the calculus of fuzzy probabilities, probabilities are assumed to be represented as fuzzy rather than crisp numbers. In a related way, probability distributions are represented as fuzzy graphs. A major aim of CFP is to provide a framework for linguistic decision analysis—a type of qualitative analysis in which fuzzy numbers and fuzzy graphs are employed to represent both probabilities and utilities.In an essential way, the methodologies of fuzzy rules, fuzzy graphs, and fuzzy probabilities reflect the fact that

1. (a) imprecision and uncertainty are pervasive; and

2. (b) precision and certainty carry a cost.

In the final analysis, the principal aim of these methodologies is to exploit the tolerance for imprecision and uncertainty to achieve tractability, robustness, and low solution cost.     

Beta-binders for Biological Quantitative Experiments

The similarities between biological systems and distributed and mobile systems suggest that the theory of process calculi could be a useful starting point for understanding, if not predicting, the behaviour of complex biological systems.To formally model in vitro or in vivo experiments, appropriate quantitative extensions of process calculi have to be investigated. This paper focuses on Beta-binders, a language of processes with typed interaction sites which has been recently introduced to accurately represent biological entities.Here the syntax and semantics of Beta-binders are enriched to achieve a stochastic version of it, in order to obtain quantitative measures on biological phenomena. The quantitative parameters are derived from typed interaction sites introducing the concept of affinity. The relevance of quantitative reasoning is outlined running a biological example.     

A Calculus of Lambda Calculus Contexts

The calculus c serves as a general framework for representing contexts. Essential features are control over variable capturing and the freedom to manipulate contexts before or after hole filling, by a mechanism of delayed substitution. The context calculus c is given in the form of an extension of the lambda calculus. Many notions of context can be represented within the framework; a particular variation can be obtained by the choice of a pretyping, which we illustrate by three examples.     

Comparing Instance Generation Methods for Automated Reasoning

The clause-linking technique of Lee and Plaisted proves the unsatisfiability of a set of first-order clauses by generating a sufficiently large set of instances of these clauses that can be shown to be propositionally unsatisfiable. In recent years, this approach has been refined in several directions, leading to both tableau-based methods, such as the disconnection tableau calculus, and saturation-based methods, such as primal partial instantiation and resolution-based instance generation. We investigate the relationship between these calculi and answer the question to what extent refutation or consistency proofs in one calculus can be simulated in another one. This work was partly supported by the German Research Council (DFG) as part of the Transregional Collaborative Research Center “Automatic Verification and Analysis of Complex Systems” (SFB/TR 14 AVACS). See for more information.     

A sound and complete theory of graph transformations for service programming with sessions and pipelines

Graph transformation techniques, the Double-Pushout (DPO) approach in particular, have been successfully applied in the modeling of concurrent systems. In this area, a research thread has addressed the definition of concurrent semantics for process calculi. In this paper, we propose a theory of graph transformations for service programming with sophisticated features such as sessions and pipelines. Through graph representation of CaSPiS, a recently proposed process calculus, we show how graph transformations can cope with advanced features of service-oriented computing, such as several logical notions of scoping together with the interplay between linking and containment. We first exploit a graph algebra and set up a graph model that supports graph transformations in the DPO approach. Then, we show how to represent CaSPiS processes as hierarchical graphs in the graph model and their behaviors as graph transformation rules. Finally, we provide the soundness and completeness results of these rules with respect to the reduction semantics of CaSPiS.     

An enhanced bitstring encoding for exact maximum clique search in sparse graphs

This paper describes BBMCW, a new efficient exact maximum clique algorithm tailored for large sparse graphs which can be bit-encoded directly into memory without a heavy performance penalty. These graphs occur in real-life problems when some form of locality may be exploited to reduce their scale. One such example is correspondence graphs derived from data association problems. The new algorithm is based on the bit-parallel kernel used by the BBMC family of published exact algorithms. BBMCW employs a new bitstring encoding that we denote ‘watched’, because it is reminiscent of the ‘watched literal’ technique used in satisfiability and other constraint problems. The new encoding reduces the number of spurious operations computed by the BBMC bit-parallel kernel in large sparse graphs. Moreover, BBMCW also improves on bound computation proposed in the literature for bit-parallel solvers. Experimental results show that the new algorithm performs better than prior algorithms over data sets of both real and synthetic sparse graphs. In the real data sets, the improvement in performance averages more than two orders of magnitude with respect to the state-of-the-art exact solver IncMaxCLQ.     

Bisimulation for Higher-Order Process Calculi

Ahigher-order process calculusis a calculus for communicating systems which contains higher-order constructs like communication of terms. We analyse the notion ofbisimulationin these calculi. We argue that both the standard definition of bisimulation (i.e., the one for CCS and related calculi), as well ashigher-order bisimulation[E. Astesiano, A. Giovini, and G. Reggio,in“STACS '88,” Lecture Notes in Computer Science, Vol. 294, pp. 207–226, Springer-Verlag, Berlin/New York, 1988; G. Boudol,in“TAPSOFT '89,” Lecture Notes in Computer Science, Vol. 351, pp. 149–161, Springer-Verlag, Berlin/New York, 1989; B. Thomsen, Ph.D. thesis, Dept. of Computing, Imperial College, 1990] are in general unsatisfactory, because of their over-discrimination. We propose and study a new form of bisimulation for such calculi, calledcontext bisimulation, which yields a more satisfactory discriminanting power. A drawback of context bisimulation is the heavy use of universal quantification in its definition, which is hard to handle in practice. To resolve this difficulty we introducetriggered bisimulationandnormal bisimulation, and we prove that they both coincide with context bisimulation. In the proof, we exploit thefactorisation theorem: When comparing the behaviour of two processes, it allows us to “isolate” subcomponents which might give differences, so that the analysis can be concentrated on them     

Finding Maximum Induced Matchings in Subclasses of Claw-Free and <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs,and in Graphs with Matching and Induced Matching of Equal Maximum Size

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P ₅,D _m )-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.     

A Simple Calculus for Proteins and Cells

The use of process calculi to represent biological systems has led to the design of different calculi such as brane calculi [Luca Cardelli. Brane calculi. In CMSB, pages 257–278, 2004] and κ-calculus [Vincent Danos and Cosimo Laneve. Formal molecular biology. Theoritical Computer Science, 325(1):69–110, 2004]. Both have proved to be useful to model different types of biological systems.As an attempt to unify the two directions, we introduce the bioκ-calculus, a simple calculus for describing proteins and cells, in which bonds are represented by means of shared names and interactions are modelled at the domain level. Protein-protein interactions have to be at most binary and cell interactions have to fit with sort constraints.We define the semantics of bioκ-calculus, analyse its properties, and discuss its expressiveness by modelling two significant examples: a signalling pathway and a virus infection.     

Absolute versus Relative Time in Process Algebras

Timed process algebras are useful tools for the specification and verification of real-time systems. We study the relationships between two of these algebras, I (closed interval process Algebra) and TCCS (temporal CCS), which deal with temporal aspects of concurrent systems by following very different interpretations: durational actions versus durationless actions, absolute time versus relative time, timed functional behavior versus time and functional behavior, local clocks versus global clocks. We show that these different choices are not irreconcilable by presenting simple mappings from I to TCCS which preserve the behavioral equivalences over the two timed calculi. These results hold whenever basic actions are interpreted as either eager or lazy, whenever the starting time of action execution is observed rather than their completion time. A study on the size of the labelled transition systems describing the transitional semantics of cIpa processes and those describing the transitional semantics of their translated versions is also presented.     

Process Calculi à la Bird-Meertens

This paper is an attempt to apply the reasoning principles and calculational style underlying the so-called Bird-Meertens formalism to the design of process calculi, parametrized by a behaviour model. In particular, basically equational and pointfree proofs of process properties are given, relying on the universal characterisation of anamorphisms and therefore avoiding the explicit construction of bisimulations. The developed calculi can be directly implemented on a functional language supporting coinductive types, which provides a convenient way to prototype processes and assess alternative design decisions.     

About permutation algebras, (pre)sheaves and named sets

In this paper we survey some well-known approaches proposed as general models for calculi dealing with names (like for example process calculi with name-passing). We focus on (pre)sheaf categories, nominal sets, permutation algebras and named sets, studying the relationships among these models, thus allowing techniques and constructions to be transferred from one model to the other. Research partially supported by the EU IST-2004-16004 SENSORIA.