A HOL Basis for Reasoning about Functional Programs

Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system. Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided. The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily

Quantum entanglement of identical particles by standard information-theoretic notions

Quantum entanglement of identical particles is essential in quantum information theory. Yet, its correct determination remains an open issue hindering the general understanding and exploitation of many-particle systems. Operator-based methods have been developed that attempt to overcome the issue. We introduce a state-based method which, as second quantization, does not label identical particles and presents conceptual and technical advances compared to the previous ones. It establishes the quantitative role played by arbitrary wave function overlaps, local measurements and particle nature (bosons or fermions) in assessing entanglement by notions commonly used in quantum information theory for distinguishable particles, like partial trace. Our approach furthermore shows that bringing identical particles into the same spatial location functions as an entangling gate, providing fundamental theoretical support to recent experimental observations with ultracold atoms. These results pave the way to set and interpret experiments for utilizing quantum correlations in realistic scenarios where overlap of particles can count, as in Bose-Einstein condensates, quantum dots and biological molecular aggregates.Comment: 6+3 pages, 3 Figures. Stories on: Physics World (http://physicsworld.com/cws/article/news/2016/feb/12/theorists-disentangle-particle-identity); Phys.org (http://phys.org/news/2016-02-entanglement-identical-particles-doesnt-textbook.html). Invited article on 2Physics.com, presenting key developments in physics (http://www.2physics.com/2016/03/a-new-approach-to-quantum-entanglement.html

A HOL basis for reasoning about functional programs

Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system. Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided. The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily

Photoabsorption in formaldehyde: Intensities and assignments in the discrete and continuous spectral intervals

Theoretical investigations of total and partial‐channel photoabsorption cross sections in molecular formaldehyde are reported employing the Stieltjes–Tchebycheff (S–T) technique and separated‐channel static‐exchange (IVO) calculations. Vertical one‐electron dipole spectra for the 2b_2(n), 1b_1(π), 5a_1(σ), 1b_2, and 4a_1 canonical molecular orbitals are obtained using Hartree–Fock frozen‐core functions and large basis sets of compact and diffuse normalizable Gaussians to describe the photoexcited and ejected electrons. The calculated discrete excitation spectra provide reliable zeroth‐order approximations to both valence and Rydberg transitions, and, in particular, the 2b_2(n) →nsa_1, npa_1, npb_2, and nda_2 IVO spectra are in excellent accord with recent experimental assignments and available intensity measurements. Convergent (S–T) photoionization cross sections in the static‐exchange (IVO) approximation are obtained for the 15 individual partial channels associated with ionization of the five occupied molecular orbitals considered. Resonance features in many of the individual‐channel photoionization cross sections are attributed to contributions from valencelike a_1σ^∗ (CO), a_1σ^∗ (CH), and b_2σ^∗ (CH)/π_y^∗ (CO) molecular orbitals that appear in the photoionization continua, rather than in the corresponding one‐electron discrete spectral intervals. The vertical electronic cross sections for ^1A_1→^1B_1, ^1B_2, and ^1A_1 excitations are in generally good accord with previously reported CI (S–T) predictions of continuum orbital assignments and intensities, although some discrepancies due to basis‐set differences are present in the ^1B_1 and ^1B_2 components, and larger discrepancies apparently due to channel coupling are present in the ^1A_1→^1A_1 cross section. Partial‐channel vertical electronic cross sections for the production of the five lowest parent‐ion electronic states are found to be in general agreement with the results of very recent synchrotron‐radiation photoelectron branching‐ratio measurements in the 20 to 30 eV excitation energy interval. Most important in this connection is the tentative verification of the predicted orderings in intensities of the partial‐ channel cross sections, providing support for the presence of a strong ka_1σ^∗ (CO) resonance in the (5a_1^(−1))^2A_1 channel. Finally, the total vertical electronic cross sections for absorption and ionization are in general accord with photoabsorption measurements, photoionization–mass–spectrometric studies, and the previously reported CI (S–T) calculations. Although further refined calculations including vibrational degrees of freedom and autoionization line shapes are required for a more precise quantitative comparison between theory and experiment, the present study should provide a reliable zeroth‐order account of discrete and continuum electronic dipole excitations in molecular formaldehyde

Modified Green-Hyperbolic Operators

Green-hyperbolic operators - partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators - play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which $K$-nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The $K$-nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described

Marital Status and Burdensomeness as Risk Factors of Suicide Ideation in Poststroke Patients

Suicide ideation, suicide attempts, and suicide (SISAS) are increased in poststroke patients, yet not everyone who has suffered a stroke is at risk for SISAS. Two risk factors for SISAS, marital status and burdensomeness, may be of particular relevance to poststroke patients. The majority of poststroke patients have a disability that may require help from a family member with basic functions such as dressing and bathing. It was not known if being married decreases risk of SISAS for stoke victims as shown in studies with nonpoststroke subjects or increases risk for SISAS due to its influence on feelings of burdensomeness. Guided by the interpersonal psychological theory of suicidal behavior, the purpose of this study was to examine if marital status moderates the association between burdensomeness (measured by disability level) and suicide ideation. A secondary analysis was performed of the Outcome and Assessment Information Set data, which was collected by the National Centers for Medicare and Medicaid Services. A data sample of 1,596,962 records was obtained. This data sample included 5% of the Home Health Outcome Information and Assessment Set for the year 2008. Of those, 8,6381 (5.4%) individuals had suffered a stroke. The results suggested partial support for the hypotheses presented in this study. However, a significant moderation was found. As burdensomeness increased, suicide ideation increased in patients who were married. High levels of burdensomeness increase suicide risk to those who are married. Identifying a vulnerable population can provide potential positive social change by serving as basis for future research regarding program implementation in reducing suicide rates

fixed point

The study of the dual complexity space, introduced by S. Romaguera and M. P. Schellekens [Quasi-metric properties of complexity spaces, Topol. Appl. 98 (1999), pp. 311-322], constitutes a part of the interdisciplinary research on Computer Science and Topology. The relevance of this theory is given by the fact that it allows one to apply fixed point techniques of denotational semantics to complexity analysis. Motivated by this fact and with the intention of obtaining a mixed framework valid for both disciplines, a new complexity space formed by partial functions was recently introduced and studied by S. Romaguera and O. Valero [On the structure of the space of complexity partial functions, Int. J. Comput. Math. 85 (2008), pp. 631-640]. An application of the complexity space of partial functions to model certain processes that arise, in a natural way, in symbolic computation was given in the aforementioned reference. In this paper, we enter more deeply into the relationship between semantics and complexity analysis of programs. We construct an extension of the complexity space of partial functions and show that it is, at the same time, an appropriate mathematical tool for the complexity analysis of algorithms and for the validation of recursive definitions of programs. As applications of our complexity framework, we show the correctness of the denotational specification of the factorial function and give an alternative formal proof of the asymptotic upper bound for the average case analysis of Quicksort.The first and the third authors acknowledge the support of the Spanish Ministry of Science and Innovation, and FEDER, grant MTM2009-12872-C02-01 (subprogram MTM), and the support of Generalitat Valenciana, grant ACOMP2009/005. The second author acknowledges the support of the Science Foundation Ireland, SFI Principal Investigator Grant 07/IN.1/I977.Romaguera Bonilla, S.; Schellekens, M.; Valero Sierra, Ó. (2011). The complexity space of partial functions: A connection between Complexity Analysis and Denotational Semantics. International Journal of Computer Mathematics. 88(9):1819-1829. https://doi.org/10.1080/00207161003631885S18191829889De Bakker, J. W., & de Vink, E. P. (1998). Denotational models for programming languages: applications of Banach’s Fixed Point Theorem. Topology and its Applications, 85(1-3), 35-52. doi:10.1016/s0166-8641(97)00140-5Emerson, E. A., & Jutla, C. S. (1999). The Complexity of Tree Automata and Logics of Programs. SIAM Journal on Computing, 29(1), 132-158. doi:10.1137/s0097539793304741Flajolet, P., & Golin, M. (1994). Mellin transforms and asymptotics. Acta Informatica, 31(7), 673-696. doi:10.1007/bf01177551García-Raffi, L. M., Romaguera, S., & Sánchez-Pérez, E. A. (2002). Sequence spaces and asymmetric norms in the theory of computational complexity. Mathematical and Computer Modelling, 36(1-2), 1-11. doi:10.1016/s0895-7177(02)00100-0García-Raffi, L. M., Romaguera, S., & Sánchez-Pérez, E. A. (2003). The supremum asymmetric norm on sequence algebras. Electronic Notes in Theoretical Computer Science, 74, 39-50. doi:10.1016/s1571-0661(04)80764-3García-Raffi, L. M., Romaguera, S., Sánchez-Pérez, E. A. and Valero, O. Normed Semialgebras: A Mathematical Model for the Complexity Analysis of Programs and Algorithms. Proceedings of The 7th World Multiconference on Systemics, Cybernetics and Informatics (SCI 2003), Orlando, Florida, USA. Edited by: Callaos, N., Di Sciullo, A. M., Ohta, T. and Liu, T.K. Vol. II, pp.55–58. Orlando, FL: International Institute of Informatics and Systemics.Den Hartog, J. I., de Vink, E. P., & de Bakker, J. W. (2001). Metric Semantics and Full Abstractness for Action Refinement and Probabilistic Choice. Electronic Notes in Theoretical Computer Science, 40, 72-99. doi:10.1016/s1571-0661(05)80038-6Künzi, H.-P. A. (2001). Nonsymmetric Distances and Their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology. History of Topology, 853-968. doi:10.1007/978-94-017-0470-0_3Medina, J., Ojeda-Aciego, M. and Ruiz-Calviño, J. A fixed point theorem for multi-valued functions with an application to multilattice-based logic programming. Applications of Fuzzy Sets Theory: 7th International Workshop on Fuzzy Logic and Applications, WILF 2007, Camogli, Italy, July 7–10, 2007, Proceedings. Edited by: Masulli, F., Mitra, S. and Pasi, G. Vol. 4578, pp.37–44. Berlin: Springer-Verlag. Notes in Artificial IntelligenceO’Keeffe, M., Romaguera, S., & Schellekens, M. (2003). Norm-weightable Riesz Spaces and the Dual Complexity Space. Electronic Notes in Theoretical Computer Science, 74, 105-121. doi:10.1016/s1571-0661(04)80769-2Rodríguez-López, J., Romaguera, S., & Valero, O. (2004). Asymptotic Complexity of Algorithms via the Nonsymmetric Hausdorff Distance. Computing Letters, 2(3), 155-161. doi:10.1163/157404006778330816Rodríguez-López, J., Romaguera, S., & Valero, O. (2008). Denotational semantics for programming languages, balanced quasi-metrics and fixed points. International Journal of Computer Mathematics, 85(3-4), 623-630. doi:10.1080/00207160701210653Romaguera, S., & Schellekens, M. (1999). Quasi-metric properties of complexity spaces. Topology and its Applications, 98(1-3), 311-322. doi:10.1016/s0166-8641(98)00102-3Romaguera, S., & Schellekens, M. (2000). The quasi-metric of complexity convergence. Quaestiones Mathematicae, 23(3), 359-374. doi:10.2989/16073600009485983Romaguera, S., & Schellekens, M. P. (2002). Duality and quasi-normability for complexity spaces. Applied General Topology, 3(1), 91. doi:10.4995/agt.2002.2116Romaguera, S., & Valero, O. (2008). On the structure of the space of complexity partial functions. International Journal of Computer Mathematics, 85(3-4), 631-640. doi:10.1080/00207160701210117Romaguera, S., Sánchez-Pérez, E. A., & Valero, O. (2003). The complexity space of a valued linearly ordered set. Electronic Notes in Theoretical Computer Science, 74, 158-171. doi:10.1016/s1571-0661(04)80772-2Schellekens, M. (1995). The Smyth Completion. Electronic Notes in Theoretical Computer Science, 1, 535-556. doi:10.1016/s1571-0661(04)00029-5Schellekens, M. 1995. “The smyth completion: A common topological foundation for denotational semantics and complexity analysis”. Pittsburgh: Carnegie Mellon University. Ph.D. thesisSeda, A. K., & Hitzler, P. (2008). Generalized Distance Functions in the Theory of Computation. The Computer Journal, 53(4), 443-464. doi:10.1093/comjnl/bxm108Straccia, U., Ojeda-Aciego, M., & Damásio, C. V. (2009). On Fixed-Points of Multivalued Functions on Complete Lattices and Their Application to Generalized Logic Programs. SIAM Journal on Computing, 38(5), 1881-1911. doi:10.1137/070695976Tennent, R. D. (1976). The denotational semantics of programming languages. Communications of the ACM, 19(8), 437-453. doi:10.1145/360303.360308Tix, R., Keimel, K., & Plotkin, G. (2005). RETRACTED: Semantic Domains for Combining Probability and Non-Determinism. Electronic Notes in Theoretical Computer Science, 129, 1-104. doi:10.1016/j.entcs.2004.06.06

Inferring Rankings Using Constrained Sensing

We consider the problem of recovering a function over the space of permutations (or, the symmetric group) over $n$ elements from given partial information; the partial information we consider is related to the group theoretic Fourier Transform of the function. This problem naturally arises in several settings such as ranked elections, multi-object tracking, ranking systems, and recommendation systems. Inspired by the work of Donoho and Stark in the context of discrete-time functions, we focus on non-negative functions with a sparse support (support size $\ll$ domain size). Our recovery method is based on finding the sparsest solution (through $\ell_0$ optimization) that is consistent with the available information. As the main result, we derive sufficient conditions for functions that can be recovered exactly from partial information through $\ell_0$ optimization. Under a natural random model for the generation of functions, we quantify the recoverability conditions by deriving bounds on the sparsity (support size) for which the function satisfies the sufficient conditions with a high probability as $n \to \infty$. $\ell_0$ optimization is computationally hard. Therefore, the popular compressive sensing literature considers solving the convex relaxation, $\ell_1$ optimization, to find the sparsest solution. However, we show that $\ell_1$ optimization fails to recover a function (even with constant sparsity) generated using the random model with a high probability as $n \to \infty$. In order to overcome this problem, we propose a novel iterative algorithm for the recovery of functions that satisfy the sufficient conditions. Finally, using an Information Theoretic framework, we study necessary conditions for exact recovery to be possible.Comment: 19 page

Theory Morphisms in Church's Type Theory with Quotation and Evaluation

${\rm CTT}_{\rm qe}$ is a version of Church's type theory with global quotation and evaluation operators that is engineered to reason about the interplay of syntax and semantics and to formalize syntax-based mathematical algorithms. ${\rm CTT}_{\rm uqe}$ is a variant of ${\rm CTT}_{\rm qe}$ that admits undefined expressions, partial functions, and multiple base types of individuals. It is better suited than ${\rm CTT}_{\rm qe}$ as a logic for building networks of theories connected by theory morphisms. This paper presents the syntax and semantics of ${\rm CTT}_{\rm uqe}$, defines a notion of a theory morphism from one ${\rm CTT}_{\rm uqe}$ theory to another, and gives two simple examples that illustrate the use of theory morphisms in ${\rm CTT}_{\rm uqe}$.Comment: 17 page

Automata theory in nominal sets

We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we generalize nominal sets due to Gabbay and Pitts