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A hierarchical completeness proof for propositional temporal logic
 Verification: Theory and Practice: Essays Dedicated to Zohar Manna on the Occasion of His 64th Birthday, volume 2772 of LNCS
, 2004
"... Abstract. We present a new proof of axiomatic completeness for Proposition Temporal Logic (PTL) for discrete, linear time for both finite and infinite time (without pasttime). This makes use of a natural hierarchy of logics and notions and is an interesting alternative to the proofs in the literatu ..."
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Abstract. We present a new proof of axiomatic completeness for Proposition Temporal Logic (PTL) for discrete, linear time for both finite and infinite time (without pasttime). This makes use of a natural hierarchy of logics and notions and is an interesting alternative to the proofs in the literature based on tableaux, filtration, game theory and other methods. In particular we exploit the deductive completeness of a sublogic in which the only temporal operator is ○ (“next”). This yields a proof which is in certain respects more direct and higherlevel than previous ones. The presentation also reveals unexpected fundamental links to a natural and preexisting framework for intervalbased reasoning and fixpoints of temporal operators. 1
On the Equational Definition of the Least Prefixed Point
, 2003
"... We propose a method to axiomatize by equations the least prefixed point of an order preserving function. We discuss its domain of application and show that the Boolean Modal µCalculus has a complete equational axiomatization. The method relies on the existence of a "closed structure" ..."
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We propose a method to axiomatize by equations the least prefixed point of an order preserving function. We discuss its domain of application and show that the Boolean Modal µCalculus has a complete equational axiomatization. The method relies on the existence of a "closed structure" and its relationship to the equational axiomatization of Action Logic is made explicit. The implication operation of a closed strucure is not monotonic in one of its variables; we show that the existence of such a term that does not preserve the order is an essential condition for defining by equations the least prefixed point. We stress the interplay between closed structures and fixed point operators by showing that the theory of Boolean modal µalgebras is not a conservative extension of the theory of modal µalgebras. The latter is shown to lack the finite model property.
Flat Coalgebraic Fixed Point Logics
"... Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µcalculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact ..."
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Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µcalculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the µcalculus. The family of such flat fixed point logics includes, e.g., CTL, the ∗nestingfree fragment of PDL, and the logic of common knowledge. Here, we extend this notion to the generic semantic framework of coalgebraic logic, thus covering a wide range of logics beyond the standard µcalculus including, e.g., flat fragments of the graded µcalculus and the alternatingtime µcalculus (such as ATL), as well as probabilistic and monotone fixed point logics. Our main results are completeness of the KozenPark axiomatization and a timedout tableaux method that matches EXPTIME upper bounds inherited from the coalgebraic µcalculus but avoids using automata.
A Dynamic 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 for modeling and understanding many applications, including embedded ..."
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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 for 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 dynamics with hybrid systems. We survey dynamic logics for specifying and verifying properties for each of those classes of dynamical systems. A dynamic logic is a firstorder modal logic with a pair of parametrized modal operators for each dynamical system to express necessary or possible properties of their transition behavior. Due to their full basis of firstorder modal logic operators, dynamic logics can express a rich variety of system properties, including safety, controllability, reactivity, liveness, and quantified parametrized properties, even about
Abstract ANNALS OF PURE AND APPLIED LOGIC
"... A new process logic is defined, called computation paths logic (CPL). which treats lbnn~~la~ and programs essentially alike. CPL is a pathwlse extension of PDL. following the basic ptocess logic of Harel. Kozen and Parikh. and is close in spirit to the logic R of Hare1 and Peleg. It enjoys most of ..."
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A new process logic is defined, called computation paths logic (CPL). which treats lbnn~~la~ and programs essentially alike. CPL is a pathwlse extension of PDL. following the basic ptocess logic of Harel. Kozen and Parikh. and is close in spirit to the logic R of Hare1 and Peleg. It enjoys most of the advantages of previous process logics. yet is decidable in elementary tlmc. We also ofrcr extensions for modeling asynchronouaisynchronous concurrency and infinite computanons. All extensions are also shown to be decidable in elementary time.:g 1999 Elsevicr Scicncr
A Revised Version: Belief Revision and Epistemic Acts MSc Thesis (Afstudeerscriptie)
"... It is customary to open this sort of dissertation with some words of thanks to those who have helped one along the path to achieving it. Even if it were not, I like to think that I would find the space in these mostly rather dry pages to do so. For in this case, things would not have been possible w ..."
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It is customary to open this sort of dissertation with some words of thanks to those who have helped one along the path to achieving it. Even if it were not, I like to think that I would find the space in these mostly rather dry pages to do so. For in this case, things would not have been possible without the help of a few, and would not have been so enjoyable without the presence of many. More specifically, to both of my supervisors, Krister and Eric, I owe thanks. Krister has revealed a great amount of knowledge, explaining (often patiently) many of the subtleties involved in the topics addressed in this thesis. Beyond that much more: He and his wife Anita have proved themselves very successful hosts and good company at dinners and other events. I hope to become as sharp, insightful and openminded as Krister is by the time I am seventy. Sometimes he even laughed at my jokes. Eric has suggested a number of directions for the research contained (or not contained) in this thesis, and has provided a great deal of enthusiasm. Even better, he always laughed at my jokes. A number of fruitful discussions with Johan, in the period before I started working with Krister and Eric, should be acknowledged for having been particularly inspiring.
A languagetheoretic view of verification
"... 1.1 Setting the stage The use of automata in verification goes back a long way, to Büchi [Bü60], Elgot [Elg61] and Trakhtenbrot [Tra61] who, in the early 1960s, used the theory of automata on finite words to give an algorithm to check whether a sentence of monadic second order logic on such structu ..."
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1.1 Setting the stage The use of automata in verification goes back a long way, to Büchi [Bü60], Elgot [Elg61] and Trakhtenbrot [Tra61] who, in the early 1960s, used the theory of automata on finite words to give an algorithm to check whether a sentence of monadic second order logic on such structures is valid, true in all models. They showed that this validity problem can be reduced to checking whether the language accepted by such a finite automaton is empty. Büchi went on to develop [Bü62] a theory of finite automata on infinite words and proved the same result (which was considerably harder). This would come into use in verification nearly 25 years later! These days, the verification question is posed as follows. We are given a model of a system (which can be hardware, software, or even mixed), typically as some kind of transition system. More precisely, we might be provided with a run of the system, modelled as a word over a suitable alphabet. We are given a system property, specified by a logical formula, typically in
Edinburgh Research Explorer
"... Modal mucalculi Citation for published version: Bradfield, J & Stirling, C 2007, 'Modal mucalculi'. in Handbook of Modal Logic. Elsevier, pp. 721756. ..."
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Modal mucalculi Citation for published version: Bradfield, J & Stirling, C 2007, 'Modal mucalculi'. in Handbook of Modal Logic. Elsevier, pp. 721756.