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FORWARD ANALYSIS FOR WSTS, PART I: COMPLETIONS
, 2009
"... Wellstructured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretic ..."
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Cited by 17 (9 self)
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Wellstructured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretical framework for manipulating downwardclosed sets was missing. We answer this problem, using insights from domain theory (dcpos and ideal completions), from topology (sobrifications), and shed new light on the notion of adequate domains of limits.
EXHAUSTIBLE SETS IN HIGHERTYPE COMPUTATION
, 2008
"... We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The C ..."
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Cited by 15 (12 self)
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We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela–Ascoli type characterization of compact subsets of function spaces. We also show that, in the nonempty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications.
Infinite sets that admit fast exhaustive search
 In Proceedings of the 22nd Annual IEEE Symposium on Logic In Computer Science
, 2007
"... Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive sea ..."
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Cited by 15 (8 self)
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Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive search over infinite sets be performed? Keywords. Highertype computability and complexity, Kleene–Kreisel functionals, PCF, Haskell, topology. 1.
Forward analysis for WSTS, part II: Complete WSTS
 In ICALP’09, volume 5556 of LNCS
, 2009
"... Abstract. We describe a simple, conceptual forward analysis procedure for ∞complete WSTS S. This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When S is the completion of a WSTS X, the clover in S is a finite description of the cover in X. We show ..."
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Cited by 12 (5 self)
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Abstract. We describe a simple, conceptual forward analysis procedure for ∞complete WSTS S. This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When S is the completion of a WSTS X, the clover in S is a finite description of the cover in X. We show that this applies exactly when X is an ω 2WSTS, a new robust class of WSTS. We show that our procedure terminates in more cases than the generalized KarpMiller procedure on extensions of Petri nets. We characterize the WSTS where our procedure terminates as those that are cloverflattable. Finally, we apply this to wellstructured counter systems. 1
A convenient category of locally preordered spaces
 Applied Categorical Structures
, 2008
"... Abstract. As a practical foundation for a homotopy theory of abstract spacetime, ..."
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Cited by 11 (0 self)
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Abstract. As a practical foundation for a homotopy theory of abstract spacetime,
Operational domain theory and topology of a sequential language
 In Proceedings of the 20th Annual IEEE Symposium on Logic In Computer Science
, 2005
"... A number of authors have exported domaintheoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact ..."
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Cited by 10 (6 self)
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A number of authors have exported domaintheoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact set and show that total programs with values on certain types are uniformly continuous on compact sets of total elements. We apply this and other conclusions to prove the correctness of nontrivial programs that manipulate infinite data. What is interesting is that the development applies to sequential programming languages, in addition to languages with parallel features. 1
A computational model for multivariable differential calculus
 Proc. FoSSaCS 2005, LNCS
, 2005
"... Abstract. We introduce a domaintheoretic computational model for multivariable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a subdomain of the pro ..."
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Abstract. We introduce a domaintheoretic computational model for multivariable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a subdomain of the product of n + 1 copies of the function space on the domain of intervals by tupling together consistent information about locally Lipschitz (piecewise differentiable) functions and their differential properties (partial derivatives). The main result of the paper is to show, in two stages, that consistency is decidable on basis elements, which implies that the domain can be given an effective structure. First, a domaintheoretic notion of line integral is used to extend Green’s theorem to intervalvalued vector fields and show that integrability of the derivative information is decidable. Then, we use techniques from the theory of minimal surfaces to construct the least and the greatest piecewise linear functions that can be obtained from a tuple of n + 1 rational step functions, assuming the integrability of the ntuple of the derivative part. This provides an algorithm to check consistency on the rational basis elements of the domain, giving an effective framework for multivariable differential calculus. 1
A Logic for Probabilities in Semantics
, 2003
"... Probabilistic computation has proven to be a challenging and interesting area of research, both from the theoretical perspective of denotational semantics and the practical perspective of reasoning about probabilistic algorithms. On the theoretical side, the probabilistic powerdomain of Jones and Pl ..."
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Cited by 9 (1 self)
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Probabilistic computation has proven to be a challenging and interesting area of research, both from the theoretical perspective of denotational semantics and the practical perspective of reasoning about probabilistic algorithms. On the theoretical side, the probabilistic powerdomain of Jones and Plotkin represents a significant advance. Further work, especially by AlvarezManilla, has greatly improved our understanding of the probabilistic powerdomain, and has helped clarify its relation to classical measure and integration theory. On the practical side, many researchers such as Kozen, Segala, Desharnais, and Kwiatkowska, among others, study problems of verification for probabilistic computation by defining various suitable logics for the classes of processes under study. The work reported here begins to bridge the gap between the domain theoretic and verification (model checking) perspectives on probabilistic computation by exhibiting sound and complete logics for probabilistic powerdomains that arise directly from given logics for the underlying domains. The category in which the construction is carried out generalizes Scott’s Information Systems by taking account of full classical sequents. Via Stone duality, following Abramsky’s Domain Theory in Logical Form, all known interesting categories of domains are embedded as subcategories. So the results reported here properly generalize similar constructions on specific categories of domains. The category offers a promising universe of semantic domains characterized by a very rich structure and good preservation properties of standard constructions. Furthermore, because the logical constructions make use of full classical sequents, the morphisms have a natural nondeterministic interpretation. Thus the category is a natural one in which to investigate the relationship between probabilistic and nondeterministic computation. We discuss the problem of integrating probabilistic and nondeterministic computation after presenting the construction of logics for probabilistic powerdomains.
Continuous Previsions
"... We define strong monads of continuous (lower, upper) previsions, and of forks, modeling both probabilistic and nondeterministic choice. This is an elegant alternative to recent proposals by Mislove, Tix, Keimel, and Plotkin. We show that our monads are sound and complete, in the sense that they m ..."
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Cited by 9 (6 self)
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We define strong monads of continuous (lower, upper) previsions, and of forks, modeling both probabilistic and nondeterministic choice. This is an elegant alternative to recent proposals by Mislove, Tix, Keimel, and Plotkin. We show that our monads are sound and complete, in the sense that they model exactly the interaction between probabilistic and (demonic, angelic, chaotic) choice.