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A HigherOrder Duration Calculus
 Millenial Perspectives in Computer Science. Proceedings of the 1999 OxfordMicrosoft Symposium in Honour of Professor Sir Anthony Hoare, Palgrave
, 1999
"... Calculus (DC) which can specify realtime requirements of computing system. This paper investigates how realtime behaviour of programs can be described within this logical framework. In order to describe local variable declaration, quantifications over program variables are inevitable, and therefor ..."
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Cited by 28 (7 self)
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Calculus (DC) which can specify realtime requirements of computing system. This paper investigates how realtime behaviour of programs can be described within this logical framework. In order to describe local variable declaration, quantifications over program variables are inevitable, and therefore a higherorder DC is established in the paper. This higherorder DC has a complete proof system, if we assume finite variability of program variables. Zhou Chaochen is the Director of UNU/IIST, on leave of absence from the Software Institute, the Chinese Academy of Sciences, where he is a Professor. Address: UNU/IIST, P.O. Box 3158, Macau. Email: zcc@iist.unu.edu Dimitar P. Guelev is a PhD student of logic at the Department of Mathematical Logic and its Applications, Faculty of Mathematics and Informatics, Sofia University. He was a fellow of UNU/IIST from March until August 1998. His scientific interests include modal logic, temporal logic and probabilistic logic. Email: gelevdp@fmi.unisofia.bg Zhan Naijun is a Fellow of UNU/IIST (July 1998 to August 1999), on leave from Institute of Software, Chinese Academy of Sciences, where he is a PhD student. Address: Institute of Software, P.O. Box 8718, Beijing, 100080, China. Email: znj@ox.ios.ac.cn Copyright c fl 1999 by UNU/IIST, Zhou Chaochen, Dimitar P. Guelev Contents i Contents 1
An Adequate First Order Interval Logic
 In COMPOS'97, volume 1536 of LNCS
, 1996
"... The paper uses left and right neighbourhoods as primitive interval modalities to define other unary and binary modalities of intervals in a first order logic with interval length. A complete first order logic for the neighbourhood modalities is presented. The paper demonstrates how the logic can sup ..."
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Cited by 18 (2 self)
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The paper uses left and right neighbourhoods as primitive interval modalities to define other unary and binary modalities of intervals in a first order logic with interval length. A complete first order logic for the neighbourhood modalities is presented. The paper demonstrates how the logic can support formal specification and verification of liveness and fairness, and also of various notions of real analysis. 1 Introduction Interval temporal logics, based on ITL [11], have shown to be useful for the specification and verification of safety properties of realtime systems. In these logics one can succinctly express properties like: "for all intervals of a given size, OE must hold", and "if OE holds for an interval, then there is a subinterval where / holds", and so on. However, these logics cannot express more abstract liveness properties like "eventually there is an interval where OE holds" and "OE will hold infinitely often in the future". The reason for this limitation is that the...
A Road Map on Interval Temporal Logics and Duration Calculi
 Journal of Applied NonClassical Logics
, 2003
"... We survey main developments, results, and open problems on interval temporal logics and duration calculi. We present various formal systems studied in the literature and discuss their distinctive features, emphasizing on expressiveness, axiomatic systems, and (un)decidability results. ..."
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Cited by 12 (5 self)
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We survey main developments, results, and open problems on interval temporal logics and duration calculi. We present various formal systems studied in the literature and discuss their distinctive features, emphasizing on expressiveness, axiomatic systems, and (un)decidability results.
Stochastic Differential Dynamic Logic for Stochastic Hybrid Programs
, 2011
"... should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. A conference version of this report has appeared at CADE [Pla11].Keywords: Dynamic logic, proof calculus, stochastic differential equations, stochastic hybrid Lo ..."
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Cited by 11 (10 self)
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should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. A conference version of this report has appeared at CADE [Pla11].Keywords: Dynamic logic, proof calculus, stochastic differential equations, stochastic hybrid Logic is a powerful tool for analyzing and verifying systems, including programs, discrete systems, realtime systems, hybrid systems, and distributed systems. Some applications also have a stochastic behavior, however, either because of fundamental properties of nature, uncertain environments, or simplifications to overcome complexity. Discrete probabilistic systems have been studied using logic. But logic has been chronically underdeveloped in the context of stochastic hybrid systems, i.e., systems with interacting discrete, continuous, and stochastic dynamics. We aim at overcoming this deficiency and introduce a dynamic logic for stochastic hybrid systems. Our results indicate that logic is a promising tool for understanding stochastic hybrid systems and can help taming some of their complexity. We introduce a compositional model for stochastic hybrid systems. We prove adaptivity, càdlàg, and Markov time properties, and prove that the semantics
Neighbourhood Logics
 Journal of Logic and Computation
, 1997
"... Abstract. This paper presents a completeness result for a firstorder interval temporal logic, called Neighbourhood Logic (NL) which has two neighbourhood modalities. NL can support the specification of liveness and fairness properties of computing systems as well as formalisation of many concepts o ..."
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Cited by 11 (1 self)
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Abstract. This paper presents a completeness result for a firstorder interval temporal logic, called Neighbourhood Logic (NL) which has two neighbourhood modalities. NL can support the specification of liveness and fairness properties of computing systems as well as formalisation of many concepts of real analysis. These two modalities are also adequate in the sense that they can derive other important unary and binary modalities of interval temporal logic. We prove the completeness result for NL by giving a Kripke model semantics and then mapping the Kripke models to the interval models for NL. 1
Propositional interval neighborhood temporal logics
 Journal of Universal Computer Science
, 2003
"... Abstract: Logics for time intervals provide a natural framework for dealing with time in various areas of computer science and artificial intelligence, such as planning, natural language processing, temporal databases, and formal specification. In this paper we focus our attention on propositional i ..."
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Cited by 7 (4 self)
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Abstract: Logics for time intervals provide a natural framework for dealing with time in various areas of computer science and artificial intelligence, such as planning, natural language processing, temporal databases, and formal specification. In this paper we focus our attention on propositional interval temporal logics with temporal modalities for neighboring intervals over linear orders. We study the class of propositional neighborhood logics (PNL) over two natural semantics, respectively admitting and excluding pointintervals. First, we introduce interval neighborhood frames and we provide representation theorems for them; then, we develop complete axiomatic systems and semantic tableaux for logics in PNL.
Signed Interval Logic
 In CSL'99, volume 1683 of LNCS
, 1999
"... Signed Interval Logic (SIL) is an extension of Interval Temporal Logic (ITL) with the introduction of the notion of a direction of an interval. We develop syntax, semantics, and proof system of SIL, and show that this proof system is sound and complete. The proof system of SIL is not more complicate ..."
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Cited by 5 (4 self)
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Signed Interval Logic (SIL) is an extension of Interval Temporal Logic (ITL) with the introduction of the notion of a direction of an interval. We develop syntax, semantics, and proof system of SIL, and show that this proof system is sound and complete. The proof system of SIL is not more complicated than that of ITL but SIL is (contrary to ITL) capable of specifying liveness properties. Other interval logics capable of this (such as Neighbourhood Logic) have more complicated proof systems. We discuss how to de ne future intervals in SIL for the specification of liveness properties. To characterize the expressive power of SIL we relate SIL to arrow logic and relational algebra.
Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded ..."
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Cited by 4 (4 self)
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyberphysical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multiagent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic
Some Results on the Decidability of Duration Calculus under Synchronous Interpretation
 LNCS
, 1996
"... Duration Calculus (or DC in short) presents a formal notation to specify properties of realtime systems and a calculus to formally prove such properties. Decidability is the underlying foundation to automated reasoning. But, excepting some of its simple fragments, DC has been shown to be undecidable ..."
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Cited by 3 (2 self)
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Duration Calculus (or DC in short) presents a formal notation to specify properties of realtime systems and a calculus to formally prove such properties. Decidability is the underlying foundation to automated reasoning. But, excepting some of its simple fragments, DC has been shown to be undecidable. DC takes the set of real numbers to represent time. The main reason of undecidability comes from the assumption that, in a realtime system, state changes can occur at any time point. But an implementation of a specification is ultimately executed on a computer, and there states change according to a system clock. Under such an assumption, it has been shown that the decidability results can be extended to cover relatively richer subsets of DC. In this report, we extend such decidability results to still richer subsets of DC.
Completeness of HigherOrder Duration Calculus
 UNU/IIST Report No.175, UNU/IIST, International Institute for Software Technology, P.O. Box 3058
, 1999
"... In order to describe the realtime behaviours of programs in terms of Duration Calculus, proposed by Zhou Chaochen, C.A.R. Hoare and A.P. Ravn in [5], which can specify realtime requirements of computing systems, quantifications over program variables are inevitable, e.g. to describe local variable ..."
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Cited by 2 (1 self)
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In order to describe the realtime behaviours of programs in terms of Duration Calculus, proposed by Zhou Chaochen, C.A.R. Hoare and A.P. Ravn in [5], which can specify realtime requirements of computing systems, quantifications over program variables are inevitable, e.g. to describe local variable declaration, to declare local channel and so on. So a higherorder duration calculus (abbreviated HDC) is established in [2]. This paper proves the completeness of HDC on abstract domains by encoding HDC to first order twosorted interval temporal logic. This idea is hinted by [13]. All results shown in this paper are done under the assumption that every program variable has finite variability. Zhan Naijun is a Fellow of UNU/IIST (July 1998 to August 1999), on leave from Institute of Software, the Chinese Academy of Sciences, where he is a PhD student. Address: Institute of Software, P.O. Box 8718, Beijing, 100080, China. Email: znj@ox.ios.ac.cn Copyright c fl 1999 by UNU/IIST, Zhan Naiju...