We investigate the power of non-determinism in purely functional programming languages with higher-order types. Specifically, we consider cons-free programs of varying data orders, equipped with explicit non-deterministic choice. Cons-freeness roughly means that data constructors cannot occur in function bodies and all manipulation of storage space thus has to happen indirectly using the call stack. While cons-free programs have previously been used by several au thors to characterise complexity classes, the work on non-deterministic programs has almost exclusively considered programs of data order 0. Previous work has shown that adding explicit non-determinism to cons-free programs taking data of order 0 does not increase expressivity; we prove that this - dramatically - is not the case for higher data orders: adding non-determinism to programs with data order at least 1 allows for a characterisation of the entire class of elementary-time decidable sets.
Characterisation of non-deterministic complexity classes via programming languages remains a largely unexplored area. Bellantoni obtained a characterisation of @math in his dissertation @cite_4 using similar approaches as @cite_13 , but at the cost of having a minimisation operator (as in recursion theory), a restriction later removed by Oitavem @cite_12 . A general framework for implicitly characterising a larger hierarchy of non-deterministic classes remains an open problem.
@cite_13 @cite_4 @cite_12
2069792094 2016014073
Weighted automata are a generalisation of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterise weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterise weighted language equivalence. Relying on the second characterisation, we show three different procedures for computing weighted language equivalence. The first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages. We study the effect of polynomial interpretation termination proofs of deterministic (resp. non-deterministic) algorithms defined by con uent (resp. non-con uent) rewrite systems over data structures which include strings, lists and trees, and we classify them according to the interpretations of the constructors. This leads to the definition of six function classes which turn out to be exactly the deterministic (resp. non-deterministic) polynomial time, linear exponential time and linear doubly exponential time computable functions when the class is based on con uent (resp. non-con uent) rewrite systems. We also obtain a characterisation of the linear space computable functions. Finally, we demonstrate that functions with exponential interpretation termination proofs are super-elementary. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . In a document network such as a citation network of scientific documents, web-logs, etc., the content produced by authors exhibits their interest in certain topics. In addition some authors influence other authors' interests. In this work, we propose to model the influence of cited authors along with the interests of citing authors. Moreover, we hypothesize that apart from the citations present in documents, the context surrounding the citation mention provides extra topical information about the cited authors. However, associating terms in the context to the cited authors remains an open problem. We propose novel document generation schemes that incorporate the context while simultaneously modeling the interests of citing authors and influence of the cited authors. Our experiments show significant improvements over baseline models for various evaluation criteria such as link prediction between document and cited author, and quantitatively explaining unseen text. Our focus is on the semantics of programming and specification languages. Over the years, different approaches to give semantics to these languages have been put forward. We restrict ourselves to the operational and the denotational approach, two main streams in the field of semantics. Two notions which play an important role in this paper are (non)determinism and (non)termination. Nondeterminism arises naturally in concurrent languages and it is a key concept in specification languages. Nontermination is usually caused by recursive constructs which are crucial in programming. The operational models are based on labelled transition systems. The definition of these systems is guided by the structure of the language. Metric spaces are an essential ingredient of our denotational models. We exploit the metric structure to model recursive constructs and to define operators on infinite entities. Furthermore, we also employ the metric structure to relate operational and denotational models for a given language. On the basis of four toy languages, we develop some general theory for defining operational and denotational semantic models and for relating them. This theory is applicable to a wide variety of languages. We start with a very simple deterministic and terminating imperative programming language. By adding the recursive while statement, we obtain a deterministic and nonterminating language. Next, we augment the language with the parallel composition resulting in a bounded nondeterministic and nonterminating language. Finally, we add some timed constructs. We obtain an unbounded nondeterministic and nonterminating specification language. Abstract Laws in the Miranda programming language provide a means of implementing non-free algebraic types, by means of term rewriting. In this paper we investigate program verification in such a context. Specifically, we look at how to deduce properties of functions over these “lawful” types. After examining the general problem, we look at a particular class of functions, the faithful functions. For such functions we are able, in a direct manner, to transfer properties of functions from free types to non-free types. We introduce sufficient model theory to explain these transfer results, and then find characterisations of various classes of faithful functions. Then we investigate an application of this technique to general, unfaithful, situations. In conclusion we survey Wadler's work on views and assess the utility of laws and views. In 1992 Wang & Larsen extended the may- and must preorders of De Nicola and Hennessy to processes featuring probabilistic as well as nondeterministic choice. They concluded with two problems that have remained open throughout the years, namely to find complete axiomatisations and alternative characterisations for these preorders. This paper solves both problems for finite processes with silent moves. It characterises the may preorder in terms of simulation, and the must preorder in terms of failure simulation. It also gives a characterisation of both preorders using a modal logic. Finally it axiomatises both preorders over a probabilistic version of CSP. We introduce the notion of regular cost functions: a quantitative extension to the standard theory of regular languages. We provide equivalent characterisations of this notion by means of automata (extending the nested distance desert automata of Kirsten), of history-deterministic automata (history-determinism is a weakening of the standard notion of determinism, that replaces it in this context), and a suitable notion of recognisability by stabilisation monoids. We also provide closure and decidability results. Monads for global state and local state have been used to provide semantics of programming languages for many years. There is a computation- ally natural presentation of an ordinary Lawvere theory that corresponds to the monad on Set for global state, inevitably called the Lawvere theory for global state. Here, we introduce a notion of indexed Lawvere theory and use it to give a Lawvere-style account of local state, extending the theorem for global state to local state. En route, we develop the notion of comodel of a Lawvere theory and exploit a universal characterisation of the category of worlds for local state. Ultimately, we give both syntactic and semantic characterisations of the operation block that allows one to move between worlds and use them to characterise the monad for local state. We study termination proofs in order to (i) determine computational complexity of programs and (ii) generate efficient programs from the complexity analysis. For this, we construct a termination ordering, called light multiset path ordering (LMPO), which is a restriction of the multiset path ordering. We establish that the class of first order functional programs on lists which is terminating by LMPO characterises exactly the functions computable in polynomial time. We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBF — previously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energy-minimisation characterisation of polyharmonic splines result in a “smoothest” interpolant. This scale-independent characterisation is well-suited to reconstructing surfaces from non-uniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a non-interpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for real-world rangefinder data. We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schroder and Simpson. We show the two constructions coincide when restricted to ω-continuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domain-theoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countably-based spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countablybased spaces. We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid over the set of positive @math th powers, provided the equation has at least @math variables. We thus completely describe which diagonal forms are partition regular and which are not, given sufficiently many variables. In addition, we prove a supersaturated version of Rado's theorem for a linear equation restricted either to squares minus one or to logarithmically-smooth numbers. The permutations that can be sorted by two stacks in series are considered, subject to the condition that each stack remains ordered. A forbidden characterisation of such permutations is obtained and the number of permutations of each length is determined by a generating function. We give an algorithmic characterisation of finite cop-win digraphs. The case of k>1 cops and k>=l>=1 robbers is then reduced to the one cop case. Similar characterisations are also possible in many situations where the movements of the cops and or the robbers are somehow restricted. Normative systems (norms) have been widely proposed as a technique for coordinating multi-agent systems. The automated synthesis of norms for coordination remains an open and complex problem, which we tackle in this paper. We propose a novel mechanism called IRON (Intelligent Robust On-line Norm synthesis mechanism), for the on-line synthesis of norms. IRON aims to synthesise conflict-free norms without lapsing into over-regulation. Thus, IRON produces norms that characterise necessary conditions for coordination, without over-regulation. In addition to defining the norm synthesis problem formally, we empirically show that IRON is capable of synthesising norms that are effective even in the presence of non-compliance behaviours in a system. A central theme in distributed network algorithms concerns understanding and coping with the issue of locality . Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for . In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard @math model of computation and define @math (for local decision ) as the class of decision problems that can be solved in @math communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class @math , containing all languages for which there exists a randomized algorithm that runs in @math rounds, accepts correct instances with probability at least @math and rejects incorrect ones with probability at least @math . We show that @math is a threshold for the containment of @math in @math . More precisely, we show that there exists a language that does not belong to @math for any @math but does belong to @math for any @math such that @math . On the other hand, we show that, restricted to hereditary languages, @math , for any function @math and any @math such that @math . In addition, we investigate the impact of non-determinism on local decision, and establish some structural results inspired by classical computational complexity theory. Specifically, we show that non-determinism does help, but that this help is limited, as there exist languages that cannot be decided non-deterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with non-determinism that enables to decide languages . Finally, we introduce the notion of local reduction, and establish some completeness results. We study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function. We characterise when margin losses can be proper composite losses, explicitly show how to determine a symmetric loss in full from half of one of its partial losses, introduce an intrinsic parametrisation of composite binary losses and give a complete characterisation of the relationship between proper losses and "classification calibrated" losses. We also consider the question of the "best" surrogate binary loss. We introduce a precise notion of "best" and show there exist situations where two convex surrogate losses are incommensurable. We provide a complete explicit characterisation of the convexity of composite binary losses in terms of the link function and the weight function associated with the proper loss which make up the composite loss. This characterisation suggests new ways of "surrogate tuning" as well as providing an explicit characterisation of when Bregman divergences on the unit interval are convex in their second argument. Finally, in an appendix we present some new algorithm-independent results on the relationship between properness, convexity and robustness to misclassification noise for binary losses and show that all convex proper losses are non-robust to misclassification noise. We introduce a representation system for relational data based on algebraic factorisation using distributivity of product over union and commutativity of product and union. We give two characterisations of conjunctive queries based on factorisations of their results whose nesting structure is defined by so-called factorisation trees. The first characterisation concerns sizes of factorised representations. For any query, we derive a size bound that is asymptotically tight within our class of factorisations. We also characterise the queries by tight bounds on the readability of the provenance of result tuples and define syntactically the class of queries with bounded readability. The broadcast phase (downlink transmission) of the two-way relay network is studied in the source coding and joint source-channel coding settings. The rates needed for reliable communication are characterised for a number of special cases including: small distortions, deterministic distortion measures, and jointly Gaussian sources with quadratic distortion measures. The broadcast problem is also studied with common-reconstruction decoding constraints, and the rates needed for reliable communication are characterised for all discrete memoryless sources and per-letter distortion measures.