Boris A. Trakhtenbrot

Introduction to the Theory of Finite Automata

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From Denotational to Operational and Axiomatic Semantics for ALGOL-like Languages: an Overview

The advantages of denotational over operational semantics are argued. A denotational semantics is provided for an Algol-like language with finite-mode procedures, blocks with local storage, and sharing (aliasing). Procedure declarations are completely explained in the usual framework of complete partial orders, but cpos are inadequate for the semantics of blocks, and a new class of store models is developed. Partial correctness theory over store models is developed for commands which may contain calls to global procedures, but do not contain function procedures returning storable values.

Communication among relations

We investigate the relationship between three kinds of interpreted nets: nets of processes, nets of relations and nets of functions (or of sets of functions). The respective semantic definitions are based on three fundamental constructs: synchronization, strong conjunction and least fixed point operator. In this way we relate notions from concurrency, logic and domain theory and explain phenomena like Kahns Principle and the Brock-Ackerman anomaly for data flow networks. We advocate the analysis of these phenomena on the abstract level of processes, relations and functions over arbitrary F-domains instead of dealing only with concrete models of stream processing. In this way we get results which subsume and improve previous results on data flow networks and may be applied also to other formalisms describing distributed systems.

Nets and data flow interpreters

The authors investigate and compare two ways of specifying stream relations (in particular, stream functions). The first uses relational programs, i.e., netlike program schemes in which the signature primitives are interpreted as relations over a given CPO. No stream domains are assumed; semantics is in fixed-point style. The second is through data flow nets, i.e., nets whose nodes are interpreted as processes (computational stations). The authors prove the existence of an adequate data flow interpreter for relational programs over all relations and its uniqueness. When dealing with functions the interpreter is modular and obeys the Kahn principle, the authors identify two kinds of anomalies. The first (meagerness anomaly) is caused by the defect of the used processes (computational stations) and holds in fact for arbitrary input-output behaviors. The second (ambiguity anomaly) is rooted in the semantics of relational nets over arbitrary CPO. It is unavoidable in any extension beyond functional behaviors.<<ETX>>

Nets of processes and data flow

Nets of Processes provide a unifying approach to semantics of concurrently executing agents. In this framework modularity and Kahns principle are investigated for Data Flow Networks.

The semantics of local storage, or what makes the free-list free?(Preliminary Report)

Denotational semantics for an ALGOL-like language with finite-mode procedures, blocks with local storage, and sharing (aliasing) is given by translating programs into an appropriately typed λ-calculus. Procedures are entirely explained at a purely functional level - independent of the interpretation of program constructs - by continuous models for λ-calculus. However, the usual (cpo) models are not adequate to model local storage allocation for blocks because storage overflow presents an apparent discontinuity. New domains of store models are offered to solve this problem.

Origins and metamorphoses of the Trinity: logic, nets, automata

Synthesis and verification of systems with finite state space are well established problems in logic and computer science and have a long history. Precise formulations are based on preliminary formalization of two languages: SPEC (for specifications), IMP (for implementations) and a (satisfaction) relation, sat, included in IMP x SPEC. Clearly, one can consider different languages which may reflect a variety of abstraction levels. It may well happen that an object at a given level may serve as implementation for a higher level and also as specification for a lower level. From this perspective the three level paradigm is instructive and there is a proliferation of its versions. Though many of them are relevant to the subject, in this lecture the author singles out only one to which he refers to as The Trinity, namely: at the highest level-specifications expressed as formulas based on second order monadic logic (SOML); at the intermediate level-formalization of transducers (i.e. transformers of input signals into output signals) via finite sequential automata; at the lower level-formalization of discrete synchronous hardware via logical nets.

From Finite Automata toward Hybrid Systems (Extended Abstract)

We consider two orthogonal extensions of the basic finite automaton paradigm and clarify to what degree and in what form do they preserve fundamental facts from the theory of finite automata. Hopefully, this approach facilitates a lucid adaptation of Automata Theory to Hybrid Systems.

Automata and Their Interaction: Definitional Suggestions

There is a growing feeling in the community that the current literature on reactive and hybrid systems is plagued by a Babel of models, constructs and formalisms, and by an amazing discord of terminology and notation. Further models and formalisms are engendered, and it is not clear where to stop. Hence, the urge toward a pithy conceptual/notational setting, supported by a consistent and comprehensive taxonomy for a wide range of formalisms and models. The paper outlines an automata-based approach to this challenge, which emerged in previous research [PRT, RT] and in teaching experience [T1, T2]. We compare our definitional suggestions with similar background in the current literature, where the subject is sometimes complicated by a premature mixture of semantics, syntax and pragmatics.

Discerning causality in interleaving behavior

We examine situations where for a given system N there is a strong intuition and a general consensus about its interleaving behavior but the inherently causal aspects of the behavior are still to be discerned. As an application to the theory we aim at a better understanding of the semantics of Place-Transition systems.