On the application of the combinatorial theory of solvability to the analysis of chemographs: Part 2. Local completeness of invariants of chemographs in view of the combinatorial theory of solvability

AbstractThe paper proposes an extension of the definition of a canonical proof, central to proof-theoretic semantics, to a definition of a canonical derivation from open assumptions. The impact of the extension on the definition of (reified) proof-theoretic meaning of logical constants is discussed. The extended definition also sheds light on a puzzle regarding the definition of local-completeness of a natural-deduction proof-system, underlying its harmony.

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Aubert Daigneault. Introduction. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 1–5. - William Craig. Unification and abstraction in algebraic logic. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 6–57. - J. Donald Monk. Connections between combinatorial theory and algebraic logic. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 58–91. - Helena Rasiowa. Post algebras as a semantic foundation of m-valued logics. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 92–142. - Gonzalo E. Reyes. From sheaves to logic. Studies in algebraic logic, edited by Aubert Daigneault, Studies in mathematics, vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, pp. 143–204.

Journal of Symbolic Logic ◽

10.2307/2271959 ◽

1978 ◽

Vol 43 (1) ◽

pp. 145-147

Author(s):

Anne Preller

Keyword(s):

Algebraic Logic ◽

Combinatorial Theory ◽

Mathematical Association

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A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-078-4 ◽

2012 ◽

Vol 64 (4) ◽

pp. 822-844 ◽

Cited By ~ 20

Author(s):

J. Haglund ◽

J. Morse ◽

M. Zabrocki

Keyword(s):

Building Blocks ◽

Main Diagonal ◽

Combinatorial Theory ◽

Dyck Path ◽

Dyck Paths ◽

Littlewood Polynomials ◽

Diagonal Harmonics ◽

Littlewood Polynomial ◽

Theory Operator ◽

Shuffle Conjecture

Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

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Projective planes of infinite but isolic order

Journal of Symbolic Logic ◽

10.1017/s0022481200051458 ◽

1976 ◽

Vol 41 (2) ◽

pp. 391-404 ◽

Cited By ~ 1

Author(s):

J. C. E. Dekker

Keyword(s):

Projective Plane ◽

Finite Projective Plane ◽

Projective Planes ◽

Latin Squares ◽

Combinatorial Theory ◽

Ternary Ring ◽

Orthogonal Latin Squares ◽

Mutually Orthogonal Latin Squares ◽

Planar Ternary Ring ◽

Partial Recursive Functions

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

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A Combinatorial Theory of Higher-Dimensional Permutation Arrays

Advances in Applied Mathematics ◽

10.1006/aama.1999.0693 ◽

2000 ◽

Vol 25 (2) ◽

pp. 194-211 ◽

Cited By ~ 4

Author(s):

Kimmo Eriksson ◽

Svante Linusson

Keyword(s):

Combinatorial Theory ◽

Permutation Arrays ◽

Higher Dimensional

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On the Foundation of Combinatorial Theory. X. A Categorical Setting for Symmetric Functions

Studies in Applied Mathematics ◽

10.1002/sapm19928611 ◽

1992 ◽

Vol 86 (1) ◽

pp. 1-29 ◽

Cited By ~ 7

Author(s):

Flavio Bonetti ◽

Gian-Carlo Rota ◽

Domenico Senato ◽

Antonietta M. Venezia

Keyword(s):

Symmetric Functions ◽

Combinatorial Theory ◽

Categorical Setting ◽

Theory X

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Elementary combinatorial theory of complexes

Topology - Colloquium Publications ◽

10.1090/coll/012/02 ◽

1930 ◽

pp. 7-71

Keyword(s):

Combinatorial Theory

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Quantum algebras and parity-dependent spectra

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.0003 ◽

2007 ◽

Vol 463 (2086) ◽

pp. 2415-2427 ◽

Cited By ~ 5

Author(s):

C.V Sukumar ◽

Andrew Hodges

Keyword(s):

Quantum Optics ◽

Harmonic Oscillator ◽

Combinatorial Theory ◽

Squeezed States ◽

Quantum Algebras ◽

Matrix Elements ◽

Euler Numbers ◽

Simple Harmonic Oscillator ◽

Non Gaussian ◽

Special Case

We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.

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