2,119 research outputs found
Non-deterministic Semantics in Polynomial Format
AbstractThe method for automatic theorem proving proposed in [Carnielli, W. A., Polynomial ring calculus for many-valued logics, Proceedings of the 35th International Symposium on Multiple-Valued Logic, IEEE Computer Society. Calgary, Canada (2005), 20–25], called Polynomial Ring Calculus, is an algebraic proof mechanism based on handling polynomials over finite fields. Although useful in general domains, as in first-order logic, certain non-truth-functional logics and even in modal logics (see [Agudelo, J. C., Carnielli, W. A., Polynomial Ring Calculus for Modal Logics: a new semantics and proof method for modalities, The Review of Symbolic Logic. 4 (2011), 150–170, URL: doi:10.1017/S1755020310000213]), the method is particularly apt for deterministic and non-deterministic many-valued logics, as shown here. The aim of the present paper is to show how the method can be extended to any finite-valued non-deterministic semantics, and also to explore the computational character of the method through the development of a software capable of translating provability in deterministic and non-deterministic finite-valued logical systems into operations on polynomial rings
Tangent Categories from the Coalgebras of Differential Categories
Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science
An Embedding of the BSS Model of Computation in Light Affine Lambda-Calculus
This paper brings together two lines of research: implicit characterization
of complexity classes by Linear Logic (LL) on the one hand, and computation
over an arbitrary ring in the Blum-Shub-Smale (BSS) model on the other. Given a
fixed ring structure K we define an extension of Terui's light affine
lambda-calculus typed in LAL (Light Affine Logic) with a basic type for K. We
show that this calculus captures the polynomial time function class FP(K):
every typed term can be evaluated in polynomial time and conversely every
polynomial time BSS machine over K can be simulated in this calculus.Comment: 11 pages. A preliminary version appeared as Research Report IAC CNR
Roma, N.57 (11/2004), november 200
POLYNOMIAL RING CALCULUS FOR MODAL LOGICS: A NEW SEMANTICS AND PROOF METHOD FOR MODALITIES
Fundação de Amparo Ă Pesquisa do Estado de SĂŁo Paulo (FAPESP)Conselho Nacional de Desenvolvimento CientĂfico e TecnolĂłgico (CNPq)A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra-Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics.41150170Fundação de Amparo Ă Pesquisa do Estado de SĂŁo Paulo (FAPESP)Conselho Nacional de Desenvolvimento CientĂfico e TecnolĂłgico (CNPq)Fonds National de la Recherche LuxembourgFundação de Amparo Ă Pesquisa do Estado de SĂŁo Paulo (FAPESP)Conselho Nacional de Desenvolvimento CientĂfico e TecnolĂłgico (CNPq)FAPESP [2004/14107-2, 05/04123-3]CNPq [300702/2005-1
Modelling Coeffects in the Relational Semantics of Linear Logic
Various typing system have been recently introduced giving a parametric version of the exponential modality of linear logic. The parameters are taken from a semi-ring, and allow to express coeffects - i.e. specific requirements of a program with respect to the environment (availability of a resource, some prerequisite of the input, etc.).
We show that all these systems can be interpreted in the relational category (Rel) of sets and relations. This is possible because of the notion of multiplicity semi-ring and allowing a great variety of exponential comonads in Rel. The interpretation of a particular typing system corresponds then to give a suitable notion of stratification of the exponential comonad associated with the semi-ring parametrising the exponential modality
3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries
Recent advances in electron microscopy have enabled the imaging of single
cells in 3D at nanometer length scale resolutions. An uncharted frontier for in
silico biology is the ability to simulate cellular processes using these
observed geometries. Enabling such simulations requires watertight meshing of
electron micrograph images into 3D volume meshes, which can then form the basis
of computer simulations of such processes using numerical techniques such as
the Finite Element Method. In this paper, we describe the use of our recently
rewritten mesh processing software, GAMer 2, to bridge the gap between poorly
conditioned meshes generated from segmented micrographs and boundary marked
tetrahedral meshes which are compatible with simulation. We demonstrate the
application of a workflow using GAMer 2 to a series of electron micrographs of
neuronal dendrite morphology explored at three different length scales and show
that the resulting meshes are suitable for finite element simulations. This
work is an important step towards making physical simulations of biological
processes in realistic geometries routine. Innovations in algorithms to
reconstruct and simulate cellular length scale phenomena based on emerging
structural data will enable realistic physical models and advance discovery at
the interface of geometry and cellular processes. We posit that a new frontier
at the intersection of computational technologies and single cell biology is
now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies
available upon reques
Cognitive networks: brains, internet, and civilizations
In this short essay, we discuss some basic features of cognitive activity at
several different space-time scales: from neural networks in the brain to
civilizations. One motivation for such comparative study is its heuristic
value. Attempts to better understand the functioning of "wetware" involved in
cognitive activities of central nervous system by comparing it with a computing
device have a long tradition. We suggest that comparison with Internet might be
more adequate. We briefly touch upon such subjects as encoding, compression,
and Saussurean trichotomy langue/langage/parole in various environments.Comment: 16 page
A Deductive Approach towards Reasoning about Algebraic Transition Systems
Algebraic transition systems are extended from labeled transition systems by allowing transitions labeled by algebraic equations for modeling more complex systems in detail. We present a deductive approach for specifying and verifying algebraic transition systems. We modify the standard dynamic logic by introducing algebraic equations into modalities. Algebraic transition systems are embedded in modalities of logic formulas which specify properties of algebraic transition systems. The semantics of modalities and formulas is defined with solutions of algebraic equations. A proof system for this logic is constructed to verify properties of algebraic transition systems. The proof system combines with inference rules decision procedures on the theory of polynomial ideals to reduce a proof-search problem to an algebraic computation problem. The proof system proves to be sound but inherently incomplete. Finally, a typical example illustrates that reasoning about algebraic transition systems with our approach is feasible
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