Definition Of Duality Research Articles

Duality of Session Types: The Final Cut

Duality is a central concept in the theory of session types. Since a flaw was found in the original definition of duality for recursive types, several other definitions have been published. As their connection is not obvious, we compare the competing definitions, discuss tradeoffs, and prove some equivalences. Some of the results are mechanized in Agda.

Recovery operators, paraconsistency and duality

Abstract There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency (LFIs) and by the logics of formal undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core—in the case studied here, this common core is classical positive propositional logic. The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of LFI and undeterminedness, namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion one at a time. The last sections offer an algebraic account for such logics by adapting the swap structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite nondeterministic matrices.

Duality and geodesics for probabilistic frames

Probabilistic frames are a generalization of finite frames into the Wasserstein space of probability measures with finite second moment. We introduce new probabilistic definitions of duality, analysis, and synthesis and investigate their properties. In particular, we formulate a theory of transport duals for probabilistic frames and prove certain properties of this class. We also investigate paths of probabilistic frames, identifying conditions under which geodesic paths between two such measures are themselves probabilistic frames. In the discrete case this is related to ranks of convex combinations of matrices, while in the continuous case this is related to the continuity of the optimal transport plan.

Infinite linear programming and online searching with turn cost

We consider the problem of searching for a hidden target in an environment that consists of a set of concurrent rays. Every time the searcher turns direction, it incurs a fixed cost. The objective is to derive a search strategy for locating the target as efficiently as possible, and the performance of the strategy is evaluated by means of the well-established competitive ratio. In this paper we revisit an approach due to Demaine et al. [8] based on infinite linear-programming formulations of this problem. We first demonstrate that their definition of duality in infinite LPs can lead to erroneous results. We then provide a non-trivial correction which establishes the optimality of a certain round-robin search strategy.

Bilateralism in Proof-Theoretic Semantics

The paper suggests a revision of the notion of harmony, a major necessary condition in proof-theoretic semantics for a natural-deduction proof-system to qualify as meaning conferring, when moving to a bilateral proof-system. The latter considers both forces of assertion and denial as primitive, and is applied here to positive logics, lacking negation altogether. It is suggested that in addition to the balance between (positive) introduction and elimination rules traditionally imposed by harmony, a balance should be imposed also on: (i) negative introduction and elimination rules, and (ii) positive and negative introduction rules. The paper suggests a proof-theoretical definition of duality (not referring to truthtables), using which double harmony is defined. The paper proves that in a doubly-harmonious system, the coordination rule, typical to bilateral systems, is admissible.

Projective geometry from Poisson algebras

By analogy with the Poisson algebra of quadratic forms on the symplectic plane and with the concept of duality in the projective plane introduced by Arnold (2005) [1], where the concurrence of the triangle altitudes is deduced from the Jacobi identity, we consider the Poisson algebras of the first degree harmonics on the sphere, on the pseudo-sphere and on the hyperboloid, to obtain analogous duality concepts and similar results for spherical, pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra of quadratic forms, are isomorphic either to the Lie algebra of the vectors in R 3 , with the vector product, or to algebra s l 2 ( R ) . The Tomihisa identity, introduced in (Tomihisa, 2009) [3] for the algebra of quadratic forms, holds for all these Poisson algebras and has a geometrical interpretation. The relationships between the different definitions of duality in projective geometry inherited by these structures are shown here.

GF(2/sup m/) multiplication and division over the dual basis

In this paper an algorithm for GF(2/sup m/) multiplication/division is presented and a new, more generalized definition of duality is proposed. From these the bit-serial Berlekamp multiplier is derived and shown to be a specific case of a more general class of multipliers. Furthermore, it is shown that hardware efficient, bit-parallel dual basis multipliers can also be designed. These multipliers have a regular structure, are easily extended to different GF(2/sup m/) and hence suitable for VLSI implementations. As in the bit-serial case these bit-parallel multipliers can also be hardwired to carry out constant multiplication. These constant multipliers have reduced hardware requirements and are also simple to design. In addition, the multiplication/division algorithm also allows a bit-serial systolic finite field divider to be designed. This divider is modular, independent of the defining irreducible polynomial for the field, easily expanded to different GF(2/sup m/) and its longest delay path is independent of m.

Mesh duality and Legendre duality

We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.

Dual Pairs of Stopping Times for Random Walk

A definition of duality for pairs of stopping times of any random walk is motivated by the duality relation of ascending and descending ladder epochs $N, \bar{N}$ of random walk in $R^1$. Dual pairs share several of the properties of the pair $N, \bar{N}$.

Duality diagrams and the pomeron

A model for the pomeron is suggested, based upon an interpretation ofall planar quark diagrams. The model involves as a subclass those «legal» diagrams used to generalize the Veneziano model to supermultiplet scattering. In consequence the definition of duality is enlarged and more specifically the Harari-Freund identification of resonances with Regge poles and pomeron with background is modified.

An Unambiguous Definition of Duality for Unitary Amplitudes

A new definition of strong duality is suggested, namely that crossed-channel trajectories are built out of direct-channel trajectories. This definition is unambiguous and reduces to the usual statement of strong duality if the resonance amplitudes are chosen appropriately. The role of unitarity appears to be of some importance, and it is found that the nonunitary Veneziano model does not exhibit strong duality defined in this way.

A geometric representation for pairs of dual quadratic or linear programs

The relation between a linear program and its dual has been elegantly described in terms of a Lagrangian function, in which each set of variables appears as a set of Lagrange multipliers for the other [l]. Lagrangian functions have also been defined for quadratic programs [2], and in analogy with the linear case, the quadratic Lagrangian has been used as a stepping-stone to arrive at quadratic duals [3,4]. Pairs of dual quadratic programs so defined, however, are strangely unsymmetrical. The present paper is the result of an effort to understand this lack of symmetry. A special quadratic program brought in by an engineer turned out to have a perfectly symmetrical geometric representation and a perfectly symmetrical associated problem which differed slightly from the dual based on the Lagrangian. This suggested a geometric definition of duality. From the geometric point of view the midpoint between primal and dual appears to be not the Lagrangian but a set of relations called here a decomposition and equivalent to the conditions of Kuhn and Tucker [2]. In cases more general than the symmetrical one just mentioned, the geometric definition of duality leads to asymmetric duals which agree in essence with those previously defined. These can, however, be reduced to a simple, symmetrical canonical form1 by changing coordinates. For the general quadratic program the coordinate change is computationally nontrivial, but for the special class of quadratic programs with which the paper is mainly concerned the coordinate change is merely a translation and is computationally easy. A computation procedure for the special class of quadratic programs considered here is presented at the end of the paper. This procedure is referred to as the projection method.