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114
Quantum Weakest Preconditions
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2005
"... We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stonetype duality between the usual statetransformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum comput ..."
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Cited by 27 (2 self)
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We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stonetype duality between the usual statetransformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example we give the semantics of Selinger’s language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilizers.
A probabilistic language based upon sampling functions
 In Conference Record of the 32nd Annual ACM Symposium on Principles of Programming Languages
, 2005
"... As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages which treat probability distributions as primitive datatypes. Most probabilistic languages, however, focus only on discrete distributions and have limited expressive p ..."
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Cited by 26 (1 self)
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As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages which treat probability distributions as primitive datatypes. Most probabilistic languages, however, focus only on discrete distributions and have limited expressive power. This paper presents a probabilistic language, called λ○, whose expressive power is beyond discrete distributions. Rich expressiveness of λ ○ is due to its use of sampling functions, i.e., mappings from the unit interval (0.0, 1.0] to probability domains, in specifying probability distributions. As such, λ ○ enables programmers to formally express and reason about sampling methods developed in simulation theory. The use of λ ○ is demonstrated with three applications in robotics: robot localization, people tracking, and robotic mapping. All experiments have been carried out with real robots.
Algebraic Approaches to Nondeterminism  an Overview
 ACM Computing Surveys
, 1997
"... this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University ..."
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Cited by 23 (3 self)
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this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University
Derivation of Randomized Sorting and Selection Algorithms, in Parallel Algorithm Derivation And Program Transformation, edited by
, 1993
"... In this paper we systematically derive randomized algorithms (both sequential and parallel) for sorting and selection from basic principles and fundamental techniques like random sampling. We prove several sampling lemmas which will find independent applications. The new algorithms derived here are ..."
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Cited by 22 (18 self)
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In this paper we systematically derive randomized algorithms (both sequential and parallel) for sorting and selection from basic principles and fundamental techniques like random sampling. We prove several sampling lemmas which will find independent applications. The new algorithms derived here are the most efficient known. From among other results, we have an efficient algorithm for sequential sorting. The problem of sorting has attracted so much attention because of its vital importance. Sorting with as few comparisons as possible while keeping the storage size minimum is a long standing open problem. This problem is referred to as ‘the minimum storage sorting ’ [10] in the literature. The previously best known minimum storage sorting algorithm is due to Frazer and McKellar [10]. The expected number of comparisons made by this algorithm is n log n + O(n log log n). The algorithm we derive in this paper makes only an expected n log n + O(n ω(n)) number of comparisons, for any function ω(n) that tends to infinity. A variant of this algorithm makes no more than n log n + O(n log log n) comparisons on any input of size n with overwhelming probability. We also prove high probability bounds for several randomized algorithms for which only expected bounds have been proven so far.
Quantitative Relations and Approximate Process Equivalences
, 2003
"... We introduce a characterisation of probabilistic transition systems (PTS) in terms of linear operators on some suitably defined vector space representing the set of states. Various notions of process equivalences can then be reformulated as abstract linear operators related to the concrete PTS sem ..."
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Cited by 22 (12 self)
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We introduce a characterisation of probabilistic transition systems (PTS) in terms of linear operators on some suitably defined vector space representing the set of states. Various notions of process equivalences can then be reformulated as abstract linear operators related to the concrete PTS semantics via a probabilistic abstract interpretation. These process equivalences can be turned into corresponding approximate notions by identifying processes whose abstract operators "differ" by a given quantity, which can be calculated as the norm of the difference operator. We argue that this number can be given a statistical interpretation in terms of the tests needed to distinguish two behaviours.
Concurrent Constraint Programming: Towards Probabilistic Abstract Interpretation
 Proc. of the 23rd International Symposium on Mathematical Foundations of Computer Science, MFCS'98, Lecture Notes in Computer Science
, 2000
"... We present a method for approximating the semantics of probabilistic programs to the purpose of constructing semanticsbased analyses of such programs. The method resembles the one based on Galois connection as developed in the Cousot framework for abstract interpretation. The main difference betwee ..."
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Cited by 20 (8 self)
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We present a method for approximating the semantics of probabilistic programs to the purpose of constructing semanticsbased analyses of such programs. The method resembles the one based on Galois connection as developed in the Cousot framework for abstract interpretation. The main difference between our approach and the standard theory of abstract interpretation is the choice of linear space structures instead of ordertheoretic ones as semantical (concrete and abstract) domains. We show that our method generates "best approximations" according to an appropriate notion of precision defined in terms of a norm. Moreover, if recasted in a ordertheoretic setting these approximations are correct in the sense of classical abstract interpretation theory. We use Concurrent ...
Probabilistically accurate program transformations
 In SAS
, 2011
"... Abstract. The standard approach to program transformation involves the use of discrete logical reasoning to prove that the transformation does not change the observable semantics of the program. We propose a new approach that, in contrast, uses probabilistic reasoning to justify the application of t ..."
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Cited by 18 (12 self)
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Abstract. The standard approach to program transformation involves the use of discrete logical reasoning to prove that the transformation does not change the observable semantics of the program. We propose a new approach that, in contrast, uses probabilistic reasoning to justify the application of transformations that may change, within probabilistic accuracy bounds, the result that the program produces. Our new approach produces probabilistic guarantees of the form P(D  ≥ B) ≤ ɛ, ɛ ∈ (0, 1), where D is the difference between the results that the transformed and original programs produce, B is an acceptability bound on the absolute value of D, and ɛ is the maximum acceptable probability of observing large D. We show how to use our approach to justify the application of loop perforation (which transforms loops to execute fewer iterations) to a set of computational patterns. 1
Proofs of randomized algorithms in Coq
 Sci. Comput. Program
"... Abstract. Randomized algorithms are widely used either for finding efficiently approximated solutions to complex problems, for instance primality testing, or for obtaining good average behavior, for instance in distributed computing. Proving properties of such algorithms requires subtle reasoning bo ..."
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Cited by 15 (1 self)
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Abstract. Randomized algorithms are widely used either for finding efficiently approximated solutions to complex problems, for instance primality testing, or for obtaining good average behavior, for instance in distributed computing. Proving properties of such algorithms requires subtle reasoning both on algorithmic and probabilistic aspects of the programs. Providing tools for the mechanization of reasoning is consequently an important issue. Our paper presents a new method for proving properties of randomized algorithms in a proof assistant based on higherorder logic. It is based on the monadic interpretation of randomized programs as probabilistic distribution [18]. It does not require the definition of an operational semantics for the language nor the development of a complex formalization of measure theory, but only use functionals and algebraic properties of the unit interval. Using this model, we show the validity of general rules for estimating the probability for a randomized algorithm to satisfy certain properties, in particular in the case of general recursive functions. We apply this theory for formally proving a program implementing a Bernoulli distribution from a coin flip and the termination of a random walk. All the theories and results presented in this paper have been fully formalized and proved in the Coq proof assistant [19]. 1
LambdaUpsilonOmega  The 1989 Cookbook
, 1989
"... LambdaUpsilonOmega ( \Upsilon\Omega ) is a research tool designed to assist the average case analysis of some well defined classes of algorithms and data structures. This cookbook consists of an informal introduction to the system together with eighteen examples of programmes that are automatica ..."
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Cited by 15 (6 self)
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LambdaUpsilonOmega ( \Upsilon\Omega ) is a research tool designed to assist the average case analysis of some well defined classes of algorithms and data structures. This cookbook consists of an informal introduction to the system together with eighteen examples of programmes that are automatically analyzed. Amongst the applications treated here, we find: addition chains, quantitative concurrency analysis of simple systems, symbolic manipulation algorithms such as formal differentiation, simplification and rewriting systems, as well as combinatorial models including various tree and permutation statistics and functional graphs with applications to integer factorisation.