1,085 research outputs found

    On the locality of indistinguishable quantum systems

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    This thesis investigates local realism in quantum indistinguishable particle systems, focusing on bosonic, fermionic, and 2D non-abelian anyonic systems. The local realism of quantum indistinguishable particle systems is asserted. It proves annihilation operators represent the local ontic states in these systems. It closes the literature gap on obtaining Deutsch-Hayden descriptors in indistinguishable particle systems. The prima facie paradox of action at a distance using fermionic annihilation operators as descriptors is resolved. The work provides examples of using and interpreting the annihilation operators as local ontic states. It contains the novel construction and characterisation of the annihilation operators for 2 D non-abelian anyonic systems. The explicit form of Fibonacci anyon annihilation operators is provided, and their usefulness is shown in expressing the anyonic Hubbard model Hamiltonian algebraically. By studying the indistinguishable particle systems’ local realistic structure, the thesis showcases the relevance of the choice of subsystem lattice and exotic possible compositions of subsystems

    Certificates for decision problems in temporal logic using context-based tableaux and sequent calculi.

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    115 p.Esta tesis trata de resolver problemas de Satisfactibilidad y Model Checking, aportando certificados del resultado. En ella, se trabaja con tres lógicas temporales: Propositional Linear Temporal Logic (PLTL), Computation Tree Logic (CTL) y Extended Computation Tree Logic (ECTL). Primero se presenta el trabajo realizado sobre Certified Satisfiability. Ahí se muestra una adaptación del ya existente método dual de tableaux y secuentes basados en contexto para satisfactibilidad de fórmulas PLTL en Negation Normal Form. Se ha trabajado la generación de certificados en el caso en el que las fórmulas son insactisfactibles. Por último, se aporta una prueba de soundness del método. Segundo, se ha optimizado con Sat Solvers el método de Certified Satisfiability para el contexto de Certified Model Checking. Se aportan varios ejemplos de sistemas y propiedades. Tercero, se ha creado un nuevo método dual de tableaux y secuentes basados en contexto para realizar Certified Satisfiability para fórmulas CTL yECTL. Se presenta el método y un algoritmo que genera tanto el modelo en el caso de que las fórmulas son satisfactibles como la prueba en el caso en que no lo sean. Por último, se presenta una implementación del método para CTL y una experimentación comparando el método propuesto con otro método de similares características

    Tradition and Innovation in Construction Project Management

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    This book is a reprint of the Special Issue 'Tradition and Innovation in Construction Project Management' that was published in the journal Buildings

    Canonical Algebraic Generators in Automata Learning

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    Many methods for the verification of complex computer systems require the existence of a tractable mathematical abstraction of the system, often in the form of an automaton. In reality, however, such a model is hard to come up with, in particular manually. Automata learning is a technique that can automatically infer an automaton model from a system -- by observing its behaviour. The majority of automata learning algorithms is based on the so-called L* algorithm. The acceptor learned by L* has an important property: it is canonical, in the sense that, it is, up to isomorphism, the unique deterministic finite automaton of minimal size accepting a given regular language. Establishing a similar result for other classes of acceptors, often with side-effects, is of great practical importance. Non-deterministic finite automata, for instance, can be exponentially more succinct than deterministic ones, allowing verification to scale. Unfortunately, identifying a canonical size-minimal non-deterministic acceptor of a given regular language is in general not possible: it can happen that a regular language is accepted by two non-isomorphic non-deterministic finite automata of minimal size. In particular, it thus is unclear which one of the automata should be targeted by a learning algorithm. In this thesis, we further explore the issue and identify (sub-)classes of acceptors that admit canonical size-minimal representatives. In more detail, the contributions of this thesis are three-fold. First, we expand the automata (learning) theory of Guarded Kleene Algebra with Tests (GKAT), an efficiently decidable logic expressive enough to model simple imperative programs. In particular, we present GL*, an algorithm that learns the unique size-minimal GKAT automaton for a given deterministic language, and prove that GL* is more efficient than an existing variation of L*. We implement both algorithms in OCaml, and compare them on example programs. Second, we present a category-theoretical framework based on generators, bialgebras, and distributive laws, which identifies, for a wide class of automata with side-effects in a monad, canonical target models for automata learning. Apart from recovering examples from the literature, we discover a new canonical acceptor of regular languages, and present a unifying minimality result. Finally, we show that the construction underlying our framework is an instance of a more general theory. First, we see that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on a category of subobjects with respect to an epi-mono factorisation system. Second, we explore the abstract theory of generators and bases for algebras over a monad: we discuss bases for bialgebras, the product of bases, generalise the representation theory of linear maps, and compare our ideas to a coalgebra-based approach

    Internal Yoneda Ext Groups, Central H-spaces, and Banded Types

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    We develop topics in synthetic homotopy theory using the language of homotopy type theory, and study their semantic counterparts in an ∞-topos. Specifically, we study Grothendieck categories and Yoneda Ext groups in this setting, as well as a novel class of central H-spaces along with their associated bands. The former are fundamental notions from homological algebra that support important computations in traditional homotopy theory. We develop these tools with the goal of supporting similar computations in our setting. In contrast, our results about central H-spaces and bands are new, even when interpreted into the ∞-topos of spaces. In Chapter 2 we define and study Grothendieck categories in HoTT, which are abelian categories that satisfy additional axioms. The main subtlety in this development is the construction of left-exact coproducts (the AB4 axiom) from exactness of filtered colimits (AB5) in a cocomplete abelian category. In particular, it follows that coproducts of modules over a ring are left-exact, and this is one of our main results. Traditionally, it is easy to deduce AB4 from AB5 using that the finite subsets of a set X form a filtered category. However, the concept of finite bifurcates into multiple nonequivalent concepts in our constructive setting. Instead we consider lists of elements of X, inspired by Roswitha Harting\u27s construction of internal coproducts of abelian groups in an elementary topos [Har82]. These results lay the foundation for our study of Yoneda Ext groups [Yon60, Mac63]. Chapter 3 describes our formalization of higher Ext groups of abelian groups, and their expected (contravariant) long exact sequence. We give a novel proof of the usual six-term exact sequence from a fibre sequence of spaces of short exact sequences (Theorem 3.4.1). We also emphasize that our formalization can be adapted to modules over a ring, and that these higher Ext groups are interesting even for abelian groups. In Chapter 4 we further develop the theory of our Ext groups and relate their semantic counterparts to sheaf Ext [Gro57] in certain ∞-topos models. We also carry out a detailed study of internal injectivity and projectivity of modules an ∞-topos, and show that our Ext groups can be computed using resolutions of such in certain cases. The final Chapter 5 is mostly independent. In it, we study generalizations of Eilenberg–Mac Lane spaces called central H-spaces. Such H-spaces admit an astonishingly simple notion of torsor (independently studied in [Wär23]), namely that of a banded type. The type of such torsors form a delooping of a central H-space, analogously to how the type of torsors of a group G form a K(G,1). Using centrality, we define a tensor product on banded types, producing an H-space structure which makes the type of torsors into a central H-space itself. Iterating this procedure, we obtain arbitrary deloopings of A (and also of pointed self-maps of A) which are in fact unique

    Views from a peak:Generalisations and descriptive set theory

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    This dissertation has two major threads, one is mathematical, namely descriptive set theory, the other is philosophical, namely generalisation in mathematics. Descriptive set theory is the study of the behaviour of definable subsets of a given structure such as the real numbers. In the core mathematical chapters, we provide mathematical results connecting descriptive set theory and generalised descriptive set theory. Using these, we give a philosophical account of the motivations for, and the nature of, generalisation in mathematics.In Chapter 3, we stratify set theories based on this descriptive complexity. The axiom of countable choice for reals is one of the most basic fragments of the axiom of choice needed in many parts of mathematics. Descriptive choice principles are a further stratification of this fragment by the descriptive complexity of the sets. We provide a separation technique for descriptive choice principles based on Jensen forcing. Our results generalise a theorem by Kanovei.Chapter 4 gives the essentials of a generalised real analysis, that is a real analysis on generalisations of the real numbers to higher infinities. This builds on work by Galeotti and his coauthors. We generalise classical theorems of real analysis to certain sets of functions, strengthening continuity, and disprove other classical theorems. We also show that a certain cardinal property, the tree property, is equivalent to the Extreme Value Theorem for a set of functions which generalize the continuous functions.The question of Chapter 5 is whether a robust notion of infinite sums can be developed on generalisations of the real numbers to higher infinities. We state some incompatibility results, which suggest not. We analyse several candidate notions of infinite sum, both from the literature and more novel, and show which of the expected properties of a notion of sum they fail.In Chapter 6, we study the descriptive set theory arising from a generalization of topology, κ-topology, which is used in the previous two chapters. We show that the theory is quite different from that of the standard (full) topology. Differences include a collapsing Borel hierarchy, a lack of universal or complete sets, Lebesgue’s ‘great mistake’ holds (projections do not increase complexity), a strict hierarchy of notions of analyticity, and a failure of Suslin’s theorem.Lastly, in Chapter 7, we give a philosophical account of the nature of generalisation in mathematics, and describe the methodological reasons that mathematicians generalise. In so doing, we distinguish generalisation from other processes of change in mathematics, such as abstraction and domain expansion. We suggest a semantic account of generalisation, where two pieces of mathematics constitute a generalisation if they have a certain relation of content, along with an increased level of generality

    The Subgraph Isomorphism Problem for Port Graphs and Quantum Circuits

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    We study a variant of the subgraph isomorphism problem that is of high interest to the quantum computing community. Our results give an algorithm to perform pattern matching in quantum circuits for many patterns simultaneously, independently of the number of patterns. After a pre-computation step in which the patterns are compiled into a decision tree, the running time is linear in the size of the input quantum circuit. More generally, we consider connected port graphs, in which every edge ee incident to vv has a label Lv(e)L_v(e) unique in vv. Jiang and Bunke showed that the subgraph isomorphism problem HGH \subseteq G for such graphs can be solved in time O(V(G)V(H))O(|V(G)| \cdot |V(H)|). We show that if in addition the graphs are directed acyclic, then the subgraph isomorphism problem can be solved for an unbounded number of patterns simultaneously. We enumerate all mm pattern matches in time O(P)P+3/2V(G)+O(m)O(P)^{P+3/2} \cdot |V(G)| + O(m), where PP is the number of vertices of the largest pattern. In the case of quantum circuits, we can express the bound obtained in terms of the maximum number of qubits NN and depth δ\delta of the patterns : O(N)N+1/2δlogδV(G)+O(m)O(N)^{N + 1/2} \cdot \delta \log \delta \cdot |V(G)| + O(m)
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