Action Calculi Research Articles

Live functional programming with typed holes

Live programming environments aim to provide programmers (and sometimes audiences) with continuous feedback about a program's dynamic behavior as it is being edited. The problem is that programming languages typically assign dynamic meaning only to programs that are complete, i.e. syntactically well-formed and free of type errors. Consequently, live feedback presented to the programmer exhibits temporal or perceptive gaps. This paper confronts this "gap problem" from type-theoretic first principles by developing a dynamic semantics for incomplete functional programs, starting from the static semantics for incomplete functional programs developed in recent work on Hazelnut. We model incomplete functional programs as expressions with holes, with empty holes standing for missing expressions or types, and non-empty holes operating as membranes around static and dynamic type inconsistencies. Rather than aborting when evaluation encounters any of these holes as in some existing systems, evaluation proceeds around holes, tracking the closure around each hole instance as it flows through the remainder of the program. Editor services can use the information in these hole closures to help the programmer develop and confirm their mental model of the behavior of the complete portions of the program as they decide how to fill the remaining holes. Hole closures also enable a fill-and-resume operation that avoids the need to restart evaluation after edits that amount to hole filling. Formally, the semantics borrows machinery from both gradual type theory (which supplies the basis for handling unfilled type holes) and contextual modal type theory (which supplies a logical basis for hole closures), combining these and developing additional machinery necessary to continue evaluation past holes while maintaining type safety. We have mechanized the metatheory of the core calculus, called Hazelnut Live, using the Agda proof assistant. We have also implemented these ideas into the Hazel programming environment. The implementation inserts holes automatically, following the Hazelnut edit action calculus, to guarantee that every editor state has some (possibly incomplete) type. Taken together with this paper's type safety property, the result is a proof-of-concept live programming environment where rich dynamic feedback is truly available without gaps, i.e. for every reachable editor state.

First-order μ-calculus over generic transition systems and applications to the situation calculus

We consider μL, μLa, and μLp, three variants of the first-order μ-calculus studied in verification of data-aware processes, that differ in the form of quantification on objects across states. Each of these three logics has a distinct notion of bisimulation. We show that the three notions collapse for generic dynamic systems, which include all state-based systems specified using a logical formalism, e.g., the situation calculus. Hence, for such systems, μL, μLa, and μLp have the same expressive power. We also show that, when the dynamic system stores only a bounded number of objects in each state (e.g., for bounded situation calculus action theories), a finite abstraction can be constructed that is faithful for μL (the most general variant), yielding decidability of verification. This contrasts with the undecidability for first-order ltl, and notably implies that first-order ltl cannot be captured by μL.

Bounded situation calculus action theories

In this paper,1 we investigate bounded action theories in the situation calculus. A bounded action theory is one which entails that, in every situation, the number of object tuples in the extension of fluents is bounded by a given constant, although such extensions are in general different across the infinitely many situations. We argue that such theories are common in applications, either because facts do not persist indefinitely or because the agent eventually forgets some facts, as new ones are learned. We discuss various classes of bounded action theories. Then we show that verification of a powerful first-order variant of the μ-calculus is decidable for such theories. Notably, this variant supports a controlled form of quantification across situations. We also show that through verification, we can actually check whether an arbitrary action theory maintains boundedness.

Verifying ConGolog Programs on Bounded Situation Calculus Theories

We address verification of high-level programs over situation calculus action theories that have an infinite object domain, but bounded fluent extensions in each situation. We show that verification of mu-calculus temporal properties against ConGolog programs over such bounded theories is decidable in general. To do this, we reformulate the transition semantics of ConGolog to keep the bindings of “pick variables” into a separate variable environment whose size is naturally bounded by the number of variables. We also show that for situation-determined ConGolog programs, we can compile away the program into the action theory itself without loss of generality. This can also be done for arbitrary programs, but only to check certain properties, such as if a situation is the result of a program execution, not for mu-calculus verification.

Affine connection form of Regge calculus

Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein–Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the three-simplices which play the role of a general discrete connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4, R) of the connection matrices. As a result, we have some action invariant w.r.t. arbitrary change of coordinates of the vertices (and related GL(4, R) transformations in the four-simplices). Excluding GL(4, R) connection from this action via the equations of motion we have exactly the Regge action for the considered spacetime.

A unifying action calculus

McCarthy's Situation Calculus is arguably the oldest special-purpose knowledge representation formalism, designed to axiomatize knowledge of actions and their effects. Four decades of research in this area have led to a variety of alternative formalisms: While some approaches can be considered instances or extensions of the classical Situation Calculus, like Reiter's successor state axioms or the Fluent Calculus, there are also special planning languages like ADL and approaches based on a linear (rather than branching) time structure like the Event Calculus. The co-existence of many different calculi has two main disadvantages: The formal relations among them is a largely open issue, and a lot of today's research concerns the transfer of specific results from one approach to another. In this paper, we present a unifying action calculus, which encompasses (well-defined classes of) all of the aforementioned formalisms. Our calculus not only facilitates comparisons and translations between specific approaches, it also allows to solve interesting problems for various calculi at once. We exemplify this by providing a general, calculus-independent solution to a problem of practical relevance, which is intimately related to McCarthy's quest for elaboration tolerant formalisms: the modularity of domain axiomatizations.

An implementation of real-time 3D interactive drama

Interactive fiction and adventure video games are narrative genres which provide the player with the option of acting as the main character of the story. However these genres do not fully match the expectations of their authors and readers because the player cannot deeply affect the storyline. This article describes a system integrating highly interactive narrative structures in a real-time 3D environment. Based on a theoretical foundation of narrative and drama, an interactive drama engine (IDE) has been implemented. It comprises an action calculus system, a text-generation system, a behavior engine, an animation engine as well as an innovative adaptive user interface. The IDE is demonstrated with two scenarios.

Symmetric Monoidal Sketches and Categories of Wirings

We introduce a potential application of two-dimensional linear algebra to concurrency. Motivated by the structure of categories of wirings, in particular in action calculi but also in other models of concurrency, we investigate the notion of symmetric monoidal sketch for providing an abstract notion of category of wirings. Every symmetric monoidal sketch generates a generic model. If the sketch is single-sorted, the generic model can be characterised as a free structure on 1, with structure defined coalgebraically. We investigate how these results generalise results about categories of wirings given by Milner and others, and we outline how the constructs may be extended to model controls and dynamics.

Shallow Linear Action Graphs and their Embeddings

Abstract.Action calculi, which generalise process calculi such as Petri nets, π-calculusand ambient calculus, have been presented in terms ofaction graphs. We here offerlinearaction graphs as a primitive basis for action calculi. This paper presents the category of embeddings of undirected linear action graphs without nesting, using a novel form of graphical reasoning which simplifies some otherwise complex manipulations in regular algebra. The results are adapted in a few lines to directed graphs. This work is part of a long-term search for a uniform behavioural theory for process calculi.

Comparing logics for rewriting: rewriting logic, action calculi and tile logic

The large diffusion of concurrent and distributed systems has spawned in recent years a variety of new formalisms, equipped with features for supporting an easy specification of such systems. The aim of our paper is to analyze three proposals, namely rewriting logic, action calculi and tile logic, chosen among those formalisms designed for the description of rule-based systems. For each of these logics we first try to understand their foundations, then we briefly sketch some applications. The overall goal of our work is to find out a common layout where these logics can be recast, thus allowing for a comparison and an evaluation of their specific features.

Closed action calculi

Action calculi provide a framework for capturing many kinds of interactive behaviour by focussing on the primitive notion of names. We introduce a name-free account of action calculi, called the closed action calculi, and show that there is a strong correspondence between the original presentation and the name-free presentation. We also add free names plus natural axioms to the closed world, and show that the abstraction operator can be constructed as a derived operator. Our results show that in some sense names are inessential. However, the purpose of action calculi is to understand formalisms which mimic the behaviour of interactive systems. Perhaps more significantly therefore, these results highlight the important presentational role that names play.

A Type-theoretic Description of Action Calculi

Action calculi, introduced by Milner, provide a framework for investigating models of interaction. This talk will focus on the connection between action calculi and known concepts arising from type theory. The aim of this work is to isolate what is distinctive about action calculi, and to investigate the potential of action calculi as an underlying framework for many kinds of computational behaviour.The first part of the talk will introduce action calculi. In the second part, I'll give a type-theoretic account of action calculi, using the general binding operators of Aczel. I will discuss two extensions: higher-order action calculi which correspond to Moggi's commutative computational lambda-calculus, and linear action calculi which correspond to the linear type theories of Barber and Benton.This talk is based on joint work with Andrew Barber, Masahito Hasegawa and Gordon Plotkin. If time, I will also describe current work arising from the connections described above.

Premonoidal categories and flow graphs

We give two presentations of the semantics of programs: a categorical semantics based on Power and Robinson's symmetric premonoidal categories and Joyal, Street and Verity's traced monoidal categories, and a graphical semantics based on mixed control flow and data flow graphs. We show how these semantics are related, and sketch how the 2-categorical versions could be used to give an operational semantics for programs. The semantics is similar to Hasegawa's presentation of Milner and Gardner's name-free action calculi.

Categorical logic of names and abstraction in action calculi

Milner's action calculus implements abstraction in monoidal categories, so that familiar λ-calculi can be subsumed together with the π-calculus and the Petri nets. Variables are generalised to names, which allow only a restricted form of substitution.In the present paper, the well-known categorical semantics of the λ-calculus is generalised to the action calculus. A suitable functional completeness theorem for symmetric monoidal categories is proved: we determine the conditions under which the abstraction is definable. Algebraically, the distinction between the variables and the names boils down to the distinction between the transcendental and the algebraic elements. The former lead to polynomial extensions, like, for example, the ring ℤ[x]; the latter lead to algebraic extensions like ℤ[√2] or ℤ[i].Building upon the work of P. Gardner, we introduce action categories, and show that they are related to the static action calculus in exactly the same way as cartesian closed categories are related to the λ-calculus. Natural examples of this structure arise from allegories and cartesian bicategories. On the other hand, the free algebras for any commutative Moggi monad form an action category. The general correspondence of action calculi and Moggi monads will be worked out in a sequel to this work.

Categorical logic of names and abstraction in action calculi

Milner's action calculus implements abstraction in monoidal categories, so that familiar -calculi can be subsumed together with the -calculus and the Petri nets. Variables are generalised to names,...

Ramification and causality

In formal systems for reasoning about actions, the ramification problem denotes the problem of handling indirect effects. These effects are not explicitly represented in action specifications but follow from general laws describing dependencies among components of the world state. An adequate treatment of indirect effects requires a suitably weakened version of the general law of persistence. It also requires a method to avoid unintuitive changes suggested by the aforementioned dependency laws. We propose a solution to the ramification problem that uses directed relations between two single effects, stating the circumstances under which the occurrence of the first causes the second. We argue for the necessity of an approach based on causality by elaborating the limitations of common paradigms employed to handle ramifications—the principle of categorization and the policy of minimal change. Our abstract solution is realized on the basis of a particular action calculus, namely, the fluent calculus.

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 ?- calculus. Several extensions to PIC 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 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.

A Name-free Account of Action Calculi

Action calculi provide a unifying framework for representing a variety of models of communication, such as CCS, Petri nets and the π-calculus, within a unified setting. A central idea is to model the interaction between actions using names. We introduce a name-free account of action calculi, called the closed action calculi, and show that there is a strong correspondence between the original presentation and the name-free presentation. These results show that, although names play an important presentational role, they are in some sense inessential.

On the continuum limit of curvature squared actions in the Regge calculus

We evaluate the continuum limit of a family of curvature squared actions for the Regge calculus proposed by Hamber and Williams. The answers depend on how the continuum limit is defined. When the link lengths are defined as the distance in an embedding space between the endpoints of the link, we find that no member of this family approaches the continuum limit correctly. Defining the link lenghts as the length of a geodesic between the endpoints of the link, we find that a unique member is selected, and we prove for the general two dimensional compact manifold that this Regge calculus action converges to ∫ R 2√ gd 2 x.