William C. Rounds

Behavioural Equivalence Relations Induced by Programming Logics

In this paper we compare the descriptive power of three programming logics by studying the elementary equivalence relations which the logics induce on nondeterministic state-transition systems. In addition, we compare these relations with other natural state-equivalence relations for nondeterministic systems. We find that the notions of bisimilarity (Park [P], Ogden [O]) and observation equivalence (Milner [M]) are very strong equivalences compared with those induced by the logics. These three comprise regular trace logic (RTL), propositional dynamic logic (PDL), and Hennessy-Milner logic (HML). Regular trace logic is a new logic which can be used to give behavioural specifications for concurrent systems (e.g. Wolper [W], but with significant differences). It is a way of formalising those properties of programs which have been given informally in terms of path expressions [CH]. The model theory and axiomatics of this logic are interesting in their own right. Propositional dynamic logic is well-known; our treatment differs from the standard one only in that we regard the modalities as specifying intended behaviour instead of being programs. Hennessy-Milner logic is a simplified modal logic which those authors used as a characterisation of their notion of observation equivalence, which we call weak observation equivalence in this paper. We also include a brief treatment in this context of two other natural equivalences for nondeterministic systems: failure equivalence [HBR] and trace equivalence [H], both of which are weaker than the relations induced by the logics but can be characterised using appropriate logical subsets.

Possible futures, acceptances, refusals, and communicating processes

Two distinct models for the notion of communicating processes are introduced, developed and related. The first, called the possible-futures model, is a generalization to nondeterministic systems of the familiar derivative (Nerode equivalence class) construction. The second, called the acceptance-refusals model, is a slight strengthening of a model introduced by Hoare, Brookes, and Roscoe. The PF model can be mapped onto the AR model homomorphically, and the equivalence classes of this map can be characterized by imposing a very natural equivalence relation on the PF model. The resulting quotient algebra admits a complete partial order structure in which the algebraic operations are continuous.

The φ-calculus: a language for distributed control of reconfigurable embedded systems

The Φ-calculus extends Milners π-calculus by adding active environments which flow continuously over time. This allows us to extend hybrid automata to specify systems of physical agents which can reconfigure themselves. We prove a theorem stating that processes (weakly) bisimilar in the process-algebraic sense, when placed in the same active environment, control it in the same way.

The logic of unification in grammar

By unification, we understand a family of algorithms employed by compu tational versions of certain grammar formalisms to combine information in feature structures. The use of these formalisms has become widespread, and several extensions to the basic notion of feature structure have been proposed. Although algorithms for unification of these extended feature structures have been written, they are complicated, and a precise model of feature structures is desirable to give an adequate specification of what the algorithms do. We have developed a model in which descriptions of feature structures can be regarded as logical formulas, and interpreted by sets of directed graphs which satisfy them. We identify a feature structure with such a directed graph, but our mathematical work is facilitated by considering the graphs to be transition graphs for a special type of deter ministic finite automaton. This semantics for feature structures extends the ideas of Pereira and Shieber [11], by providing a way to model feature values which are speci fied by disjunctions and non-local path values embedded within disjunc tions. Our interpretation differs from that of Pereira and Shieber by using a logical model in place of a denotational semantics. The model yields a calculus of equivalences between formulas, which can be used to simplify them. A similar use of logic to describe feature structures was first pro posed in Generalized Phrase Structure Grammar (GPSG)[3], in order to describe feature co-occurrence restrictions. Our formulation, which was developed independently, is for the purpose of understanding the proper ties of unification. Recently Gazdar et al. [2] have extended their logic in order to give uniform descriptions of linguistic categories. The two logics have much in common when presented formally, and it should be possible to combine them in a uniform way. This, however, will be left for future work.

Complexity of recognition in intermediate Level languages

Complexity of sentence recognition is studied for one-way stack languages, indexed languages, and tree transducer languages. The problem is shown to be polynomial-complete in each case. A class of naturallanguage grammars is formalized and the sentence-recognition problem is shown to be polynomial-hard although the languages are context-sensitive. The proofs give new language-theoretic characterizations of the set of satisfiable propositional formulas and the set of prepositional tautologies.

A logic for partially specified data structures

This paper investigates the use of logic to reason about partially specified data structures. Our original motivation is in the use in computational linguistics of so-called jealure structures. These structures, however, have been used, especially by Ait-Kaci and Nasr [AIT86], to characterize partially defined concrete data types in programming languages. Also, feature structures can be used to impose purely syntactic constraints on programs to ensure welltypedness as in Milner [MIL78]. In fact, we can characterize Milner’s decision procedure for syntactic well-typedness as an instance of the satisfiability problem in our logic. Feature structures have been used in computational linguistics to enforce global dependencies between constituents, such as number agreement between subject and predicate. Similar phenomena, such as type agreement and data initialisation, occur in programming languages. The main difference between previous systems, such as Milner’s, and ours is that by being able to express the notion of type checking in the linguistic formalism, we can obtain eoundness results of the form: If a string is a syntactically correct program, then it is semantically well-typed. We believe that expressing so-called static semantics directly in a syntactical formalism is a superior way of viewing compile-time constraints on programs. (The need for static semantics may have been a result of too great a dependence on the formalism of context-free grammars.) Kasper and Rounds [KAS86a] develop a logic to describe records and variant records. (The problem of type consistency is the same as the aatisfiability problem in thier logic.) The primary contribution of this paper is an extension of the logic to express implication and negation. This extension makes a novel use of intuitionistic techniques. We present a sound and complete proof system for our logic, and show the provability and satisliability problems to be PSPACEcomplete. 2 Feature Structures in Programming and Natural Languages

Clausal logic and logic programming in algebraic domains

We introduce a domain-theoretic foundation for disjunctive logic programming. This foundation is built on clausal logic, a representation of the Smyth powerdomain of any coherent algebraic dcpo. We establish the completeness of a resolution rule for inference in such a clausal logic; we introduce a natural declarative semantics and a fixed-point semantics for disjunctive logic programs, and prove their equivalence; finally, we apply our results to give both a syntax and semantics for default logic in any coherent algebraic dcpo. 2001 Elsevier Science

Connections between two theories of concurrency: metric spaces and synchronization trees

A connection is established between the semantic theories of concurrency and communication in the works of de Bakker and Zucker, who develop a denotational semantics of concurrency using metric spaces instead of complete partial orders, and Milner, who develops an algebraic semantics of communication based upon observational equivalence between processes. His rigid synchronization trees (RSTs) are endowed with a simple pseudometric distance induced by Milners weak equivalence relation and the quotient space is shown to be complete. An isometry between this space and the solution to a domain equation of de Bakker and Zucker is established, presenting their solution in a conceptually simpler framework. Under an additional assumption, the equivalence between the weak equivalence relation over RSTs and the elementary equivalence relation induced by the sentences of a modal logic due to Hennessy and Milner is established.

A LOGICAL VERSION OF FUNCTIONAL GRAMMAR

Kays functional-unification grammar notation [5] is a way of expressing grammars which relies on very few primitive notions. The primary syntactic structure is the feature structure, which can be visualised as a directed graph with arcs labeled by attributes of a constituent, and the primary structure-building operation is unification. In this paper we propose a mathematical formulation of FUG, using logic to give a precise account of the strings and the structures defined by any grammar written in this notation.

On the relationships between Scott domains, synchronization trees, and metric spaces

We use Scotts idea of information systems to provide a complete partial order semantics for concurrency involving Milners synchronization tree model. Several connections are investigated between different models; our principal technique in establishing these connections is the use of compact metric space methods.