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https://hdl.handle.net/10784/27391

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Declarative debugging of functional logic programs
(Elsevier BV, 2001-01-01) Alpuente, M.; Correa, F.J.; Falaschi, M.; Alpuente, M.; Correa, F.J.; Falaschi, M.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
We present a general framework for the declarative debugging of functional logic programs, which is valid both for eager as well as lazy programs. We associate to our programs a semantics based on a (continuous) immediate consequence operator which models computed answers. Then we show that, given the intended specification of a program P, it is possible to check the correctness of P by a single step of the immediate consequence operator. We also present a more effective methodology which is based on abstract interpretation. By approximating the intended specification of the success set we derive a finitely terminating debugging method, which can be used statically. Our framework is parametric w.r.t. to the chosen approximation of the success set. We present one specific example of approximation. We provide an implementation of our debugging system which shows experimentally on a wide set of benchmarks that we are able to find some common errors in the user programs. © 2001 Published by Elsevier Science B. V.
A debugging scheme for functional logic programs
(Elsevier BV, 2002-01-01) Alpuente, M.; Correa, F.; Falaschi, M.; Alpuente, M.; Correa, F.; Falaschi, M.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
We present a generic scheme for the declarative debugging of functional logic programs which is valid for eager as well as lazy programs. In particular we show that the framework extends naturally some previous work and applies to the most modern lazy strategies, such as needed narrowing. First we associate to our programs a semantics based on a (continuous) immediate consequence operator, TR, which models computed answers. We show that, given the intended specification of a program R, it is possible to check the correctness of R by a single step of TR. We consider then a more effective methodology which is based on abstract interpretation: by approximating the intended specification of the success set we derive a finitely terminating diagnosis method, which can be used statically and is parametric w.r.t. to the chosen approximation. In order to correct the bugs, we sketch a preliminary deductive approach which uses example-guided unfolding. We specialize the incorrect rules w.r.t. sets of positive and negative examples which are gathered (bottom-up) during the diagnosis process, so that all refutations of negative examples and no refutation of positive examples are excluded. Our debugging framework does not require the user to either provide error symptoms in advance or answer difficult questions concerning program correctness. We extend an implementation of our system to the case of needed narrowing and illustrate it through some examples which demonstrate the practicality of our approach. © 2002 Published by Elsevier Science B.V.
Numerical simulations of a possible hypercomputational quantum algorithm
(SPRINGER-VERLAG WIEN, 2005-01-01) Sicard, A; Ospina, J; Velez, M; Sicard, A; Ospina, J; Velez, M; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
The hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. Several numerical simulations of a possible hypercomputational algorithm based on quantum computations previously constructed by the authors are presented. The hypercomputability of our algorithm is based on the fact that this algorithm could solve a classically non-computable decision problem, the Hilbert's tenth problem. The numerical simulations were realized for three types of Diophantine equations: with and without solutions in non-negative integers, and without solutions by way of various traditional mathematical packages.
A possible hypercomputational quantum algorithm
(SPIE-INT SOC OPTICAL ENGINEERING, 2005-01-01) Sicard, A; Velez, M; Ospina, J; Sicard, A; Velez, M; Ospina, J; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
The term `hypermachine' denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically non-computable decision problem, Hilbert's tenth problem; more specifically, we present a possible hypercomputation model based on quantum computation. Our algorithm is inspired by the one proposed by Tien D. Kieu, but we have selected the infinite square well instead of the (one-dimensional) simple harmonic oscillator as the underlying physical system. Our model exploits the quantum adiabatic process and the characteristics of the representation of the dynamical Lie algebra su(1, 1) associated to the infinite square well.
Basic reproductive rate of a spatial epidemic model using computer algebra software
(2005-01-01) Doracelly Hincapié, P.; Juan Ospina, G.; Doracelly Hincapié, P.; Juan Ospina, G.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Using computer algebra software we obtain the basic reproductive rate corresponding to the propagation of a directly transmitted disease in a circular habitat when the disease is endemic at the boundary. The method used is the Laplace Transform Technique and calculus of residues. The results that were obtained include both the explicit form of the R0 for the boundary condition that was considered, as the explicit symbolic solution of the model equation. The method that was used can be extended to other more complex problems such as indirectly transmitted diseases with one or more intermediary hosts or effects of genetic, immunological, geographical or social heterogeneity in the human population. This application indicates that the computer algebra software for symbolic computation has a very promissory future in mathematical epidemiology.
Mackendrick: A Maple Package oriented to symbolic computational epidemiology
(SPRINGER, 2006-01-01) Ospina, Juan; Hincapie, Doracelly; Ospina, Juan; Hincapie, Doracelly; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
A Maple Package named Mackendrick is presented. Such package is oriented to symbolic computational epidemiology. © Springer-Verlag Berlin Heidelberg 2006.
Using computer algebra for Yang-Baxterization applied to quantum computing
(SPIE-INT SOC OPTICAL ENGINEERING, 2006-05-12) Velez, Mario; Ospina, Juan; Velez, Mario; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Using Computer Algebra Software (Mathematica and Maple), the recently introduced topic of Yang-Baxterization applied to quantum computing, is explored from the mathematical and computational views. Some algorithms of computer algebra were elaborated with the aim to make the calculations to obtain some of results that were originally presented in the paper by Shang-Kauffman-Ge. Also certain new results about computational Yang-baxterization are presented. We obtain some Hamiltonians for hypothetical physical systems which can be realized within the domain of spin chains and certain diffusion process. We conclude that it is possible to have real physical systems on which implement, via Yang-baxterization, the standard quantum gates with topological protection. Finally some lines for future research are deligned.
Spatial epidemic patterns recognition using computer algebra
(SPRINGER, 2007-01-01) Hincapié, D.; Ospina, J.; Hincapié, D.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
An exploration in Symbolic Computational bio-surveillance is showed. The main obtained results are that the geometry of the habitat determines the critical parameters via the zeroes of the Bessel functions and the explicit forms of the static and non-static spatial epidemic patterns. © Springer-Verlag Berlin Heidelberg 2007.
Universal quantum gates Via Yang-baxterization of dihedral quantum double
(SPRINGER, 2007-01-01) Velez, Mario; Ospina, Juan; Velez, Mario; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
The recently discovered Yang-Baxterization process for the quantum double of the dihedral group algebra, is presented keeping on mind the quantum computation. The products resultant from Yang-Baxterization process are interpreted as universal quantum gates using the Bryslinski's theorem. Results are obtained for two-qubits and two-qutrits gates. Using the Zhang-Kauffman-Ge method (ZKGM), certain Hamiltonians responsible for the quantum evolution of the quantum gates are obtained. Possible physical systems such as anyons systems are mentioned as referents for practical implementation. © Springer-Verlag Berlin Heidelberg 2007.
Analytical solution for transient flow of a generalized bingham fluid with memory in a movable tube using computer algebra
(SPRINGER, 2007-01-01) Ospina, Juan; Velez, Mario; Ospina, Juan; Velez, Mario; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
A rheological linear model for a certain generalized Bingham fluid with rheological memory, which flows in a movable tube is proposed and analytically solved. The model is a system of two linear and coupled partial differential equations with integral memory. We apply the Laplace transform method making the inverse transform by means of the Bromwich integral and the theorem of residues and the analytical solution are obtained using computer algebra. We deduce the explicit forms of the velocity and stress profiles for the generalized Bingham fluid in terms of Bessel and Struve functions. Various limit cases are obtained and the standard Hagen-Poiseuille and Buckingham-Reiner equations are recovered from more general equations. This works shows the powerful of Maple to solve complex rheological problems in an analytical form as it is presented here by the first time. © Springer-Verlag Berlin Heidelberg 2007.
Gravitational topological quantum computation
(SPRINGER, 2007-01-01) Velez, Mario; Ospina, Juan; Velez, Mario; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
A new model in topological quantum computing, named Gravitational Topological Quantum Computing (GTQC), is introduced as an alternative respect to the Anyonic Topological Quantum Computing and DNA Computing. In the new model the quantum computer is the quantum space-time itself and the corresponding quantum algorithms refer to the computation of topological invariants for knots, links and tangles. Some applications of GTQC in quantum complexity theory and computability theory are discussed, particularly it is conjectured that the Khovanov polynomial for knots and links is more hard than #P-hard; and that the homeomorphism problem, which is noncomputable, maybe can be computed after all via a hyper-computer based on GTQC. © Springer-Verlag Berlin Heidelberg 2007.
Two qutrits universal quantum gates from the nine-dimensional unitary solutions of the Yang-Baxter equation
(SPIE-INT SOC OPTICAL ENGINEERING, 2007-04-25) Vélez, M.; Ospina, J.; Vélez, M.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Using the Kauffman-Lomonaco method, some two-qutrits universal quantum gates are derived from the nine-dimensional unitary solutions of the Yang-Baxter equations associated with algebraic structures like the partial transpose operator and the dihedral group, which admit three dimensional representations. The Yang-Baxterization method given by Zhang-Kauffman-Ge is continuously used to obtain two-qutrits quantum gates and certain Hamiltonians for the evolution of the quantum gates are obtained, being such Hamiltonians interpreted as physical Hamiltonians of chain of particles of spin 1. Finally, the generalization for systems of two qudits is presented in the case of Yang-Baxterization of representations of braided monoidal algebra like the BH algebra and the bicolored Birman-Wenzl-Muraki algebra For these algebras the corresponding two-qudits quantum gates are constructed jointly with the associated Hamiltonians interpreted like physical chains of particles with spin d . It is conjectured that the derived two-qdits quantum gates and the Hamiltonians may be implemented over bi-dimensional lattice systems like anyons systems or more generally over any physical systems ruled by the Yang-Baxter equations.
Tutte polynomials and topological quantum algorithms in social network analysis for epidemiology, bio-surveillance and bio-security
(SPRINGER, 2008-01-01) Velez, Mario; Ospina, Juan; Hincapie, Doracelly; Velez, Mario; Ospina, Juan; Hincapie, Doracelly; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
The Tutte polynomial and the Aharonov-Arab-Ebal-Landau algorithm are applied to Social Network Analysis (SNA) for Epidemiology, Biosurveillance and Biosecurity. We use the methods of Algebraic Computational SNA and of Topological Quantum Computation. The Tutte polynomial is used to describe both the evolution of a social network as the reduced network when some nodes are deleted in an original network and the basic reproductive number for a spatial model with bi-networks, borders and memories. We obtain explicit equations that relate evaluations of the Tutte polynomial with epidemiological parameters such as infectiousness, diffusivity and percolation. We claim, finally, that future topological quantum computers will be very important tools in Epidemiology and that the representation of social networks as ribbon graphs will permit the full application of the Bollobás-Riordan-Tutte polynomial with all its combinatorial universality to be epidemiologically relevant. © 2008 Springer Berlin Heidelberg.
Epidemic thresholds in SIR and SIIR models applying an algorithmic method
(SPRINGER, 2008-01-01) Hincapié P., D.; Ospina G., J.; Afuwape, A.U.; Gómez A., R.D.; Hincapié P., D.; Ospina G., J.; Afuwape, A.U.; Gómez A., R.D.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Epidemic thresholds were deduced and simulated from SIR models of Susceptible - Infected - Recovered individuals, through local stability analysis of the disease free and endemic equilibrium, with an algorithmic method. One and two types of infected individuals were modeled, considering the influence of sub clinical, undiagnosed or unrecognized infected cases in disease transmission. © 2008 Springer Berlin Heidelberg.
Possible quantum algorithms for the Bollobás-Riordan-Tutte polynomial of a ribbon graph
(SPIE-INT SOC OPTICAL ENGINEERING, 2008-01-01) Velez, M.; Ospina, J.; Velez, M.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Three possible quantum algorithms, for the computation of the Bollobás-Riordan-Tutte polynomial of a given ribbon graph, are presented and discussed. The first possible algorithm is based on the spanning quasi-trees expansion for generalized Tutte polynomials of generalized graphs and on a quantum version of the Binary Decision Diagram (BDD) for quasi-trees . The second possible algorithm is based on the relation between the Kauffman bracket and the Tutte polynomial; and with an application of the recently introduced Aharonov-Arad-Eban-Landau quantum algorithm. The third possible algorithm is based on the relation between the HOMFLY polynomial and the Tutte polynomial; and with an application of the Wocjan-Yard quantum algorithm. It is claimed that these possible algorithms may be more efficient that the best known classical algorithms. These three algorithms may have interesting applications in computer science at general or in computational biology and bio-informatics in particular. A line for future research based on the categorification project is mentioned.
Symbolic solution for generalized quantum cylindrical wells using computer algebra
(2008-01-01) Pulgarin, E.Y.V.; Pulgarin, E.Y.V.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
This paper present how to solve the problem of cylindrical quantum wells with potential energy different from zero and with singularity of the energy on the axis of the cylinder. The solution to the problem was obtained using methods of computer algebra. The results depend of Bessel and Kummer functions. This paper present energy levels and wave functions in some of the cases with an exactly form and in other cases with an approximated form, this form depended on the possibility of integrating the special functions and calculating the zeros of these functions. Here we can see the power of the method in the applications concerning complex problems of quantum mechanics, and the possibility of being able to apply this method in order to solve other problems in science and also in engineering.
The epidemic threshold theorem with social and contact heterogeneity
(SPIE-INT SOC OPTICAL ENGINEERING, 2008-03-17) Palacio, D.H.; Giraldo, J.O.; Arias, R.D.G.; Palacio, D.H.; Giraldo, J.O.; Arias, R.D.G.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
The threshold theorem of an epidemic SIR model was compared when infectious and susceptible individuals have homogeneous mixing and heterogeneous social status and when individuals of random networks have contact heterogeneity. Particularly the effect of vaccination in such models is considered when: individuals or nodes are exposed to impoverished, vaccination and loss of immunity. An equilibrium analysis and local stability of small perturbations about the equilibrium values were implemented using computer algebra. Numerical simulations were executed in order to describe the dynamic of transmission of diseases and changes of the basic reproductive rate. The implications of these results are examined around the threats to the global public health security.
Possible topological quantum computation via khovanov homology: D-brane topological quantum computer
(SPIE-INT SOC OPTICAL ENGINEERING, 2009-01-01) Vélez, M.; Ospina, J.; Vélez, M.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
A model of a D-Brane Topological Quantum Computer (DBTQC) is presented and sustained. The model isbased on four-dimensional TQFTs of the Donaldson-Witten and Seiber-Witten kinds. It is argued that the DBTQC is able to compute Khovanov homology for knots, links and graphs. The DBTQC physically incorporates the mathematical process of categorification according to which the invariant polynomials for knots, links and graphs such as Jones, HOMFLY, Tutte and Bollobás-Riordan polynomials can be computed as the Euler characteristics corresponding to special homology complexes associated with knots, links and graphs. The DBTQC is conjectured as a powerful universal quantum computer in the sense that the DBTQC computes Khovanov homology which is considered like powerful that the Jones polynomial. © 2009 SPIE.
Dynamic analysis of heat transfer through a fin of constant cross-sectional area: Specified fin tip temperature
(2009-01-01) Carranza, R.G.; Ospina Giraldo, J.F.; Carranza, R.G.; Ospina Giraldo, J.F.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
A dynamic analysis is performed for heat transfer through a fin with constant cross-sectional area and with a specified fin tip temperature. The process starts with a dynamic energy balance around the fin. Laplace transforms and the Bromwich Integral are used to solve analytically the resulting partial differential equation. The final purely analytical solution is compared to the well known steady state solution. The two match exactly as time approaches infinity. Furthermore it is shown that the steady-state and dynamic characteristics of the fin are directly tied to the Biot number. Copyright © 2009 by ASME.
Embedding a logical Theory of Constructions in agda
(2009-01-01) Bove, A.; Dybjer, P.; Sicard-Ra?irez, A.; Bove, A.; Dybjer, P.; Sicard-Ra?irez, A.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
We propose a new way to reason about general recursive functional in the dependently typed programming language Agda,is based on Martin-Löf's intuitionistic type theory. We show to embed an external programming logic, Aczel's Logical Theory Constructions (LTC) inside Agda. To this end we postulate existence of a domain of untyped functional programs and the rules for these programs. Furthermore, we represent the notions in LTC (intuitionistic predicate logic with equality,totality predicates) as inductive notions in Agda. To illustrate approach we specify an LTC-style logic for PCF, and show to prove the termination and correctness of a general recursive for computing the greatest common divisor of two numbers. © 2009.

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