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This is for the Af-calculus; for the An-calculus, the required structure is an extensional combinatory model, which is like a combinatory model without e which satisfies: if, for all dE D, d):d=d,-d, then ... The author shows that every lambda model is a lambda algebra and, given any lambda algebra, the free combinatory algebra generated by an infinite set of variables over the lambda algebra is a lambda model ...
Combinatory Logic is an elegant and powerful logical theory that is used in computer science as a theoretical model for computation. ... A Neural Algebra, modeling the way we think, constitutes an interesting model of Combinatory Logic. There are other models, also based on the Graph Model (Engeler 1981), such as software testing. ... Graph Model that he did. ...doi:10.30958/ajs.9-1-3 fatcat:ecnzrflfc5hupag5fjvzxykc6i
Lecture Notes in Computer Science
In this paper we give an outline of recent algebraic results concerning theories and models of the untyped lambda calculus. ... Of course, an unorderable model can still arise from an order-theoretic construction, for instance as a subalgebra of some orderable model. ... A function f : C → C is representable in a combinatory algebra C if there exists an element c ∈ C such that cz = f (z) for all z ∈ C. ...doi:10.1007/978-3-642-32589-2_3 fatcat:jqadnhzb5jcudfucqdvxbirasa
algebras; (iii) a proof that no effective lambda-model can have lambda-beta or lambda-beta-eta as its equational theory (this can be seen as a partial answer to an open problem introduced by Honsell-Ronchi ... The main research achievements include: (i) a general construction of lambda-models from reflexive objects in (possibly non-well-pointed) categories; (ii) a Stone-style representation theorem for combinatory ... [35, Thm. 4.3.7 ] (Stone's representation theorem for combinatory algebras) Every combinatory algebra C can be represented as a weak Boolean product of indecomposable combinatory algebras C x (for x ...arXiv:0904.4756v1 fatcat:7j47i72zebaoxkltk6fqul5bsi
“In Section 5 we consider a problem in which the combinatorial structure (flows) is generalized in an algebraic setting. ... Thus a second direction of further research is the discussion of reasonable algebraic generalizations of combinatorial structures. For example, the dual of an algebraic linear program (cf. ...
We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. ... The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. ... Acknowledgments We thank Hayat Cheballah for useful discussions. ...arXiv:0904.1506v1 fatcat:vfs2s5wclzc47agmrdrxqxkvf4
Many combinatorial optimization problems assume, under such reformulation, the appearance of problems of linear algebra over an ordered system of scalars. ... This application is made relevant by the fact that many optimization questions depend essentially on the presence of two features: an algebraic language within which a system can be modelled and an algorithm ...doi:10.1090/s0273-0979-1985-15325-x fatcat:lfnlvbmtjfgu3gpi7ituv6yqni
., Extension of combinatory logic to a theory of combinatory representation, Theoretical Computer Science 97 (1992) 157-173. ... On one hand, a solution (h, f;-c,YI) of r' in a combinatory model M should render an inner Y-algebra M[! ... By adding this property to a semi-universal combinatory model, the resulting universal combinatory model can now deal with the entire representation problem. ...doi:10.1016/0304-3975(92)90392-s fatcat:iogen6fsz5gs5eurzfsfp6mjr4
Algebraic generalization is an important prerequisite to prove mathematically and to solve combinatorial problem at higher levels, i.e., using expressions and formulas, therefore a special focus should ... mathematical problems Specific relations between these three abilities, manifested in the solving of an open-ended ill-structured problem aimed at mathematical modeling, were investigated. ... As stated by Ericson and Lockwood  in order to prove a combinatorial identity various cognitive models are needed in order to understand the algebraic notations. ...doi:10.3390/math8122257 fatcat:tk2k2xo45jabtb3vy424zxp2ry
We survey three problem representations that are popularly applied in combinatorial optimization: algebraic modeling languages, constraint logic programming languages, and network diagrams. ... We focus especially on the possibility that general-purpose system designs, which are highly successful in other areas, might be extended to combinatorial optimization. ... For the user who has a difficult combinatorial optim ization problem, algebraic modeling languages offer an appealing environment in which to devise and test new formulations that may provide tighter bounds ...doi:10.1109/hicss.1996.495425 dblp:conf/hicss/CoullardF96 fatcat:63vhji73snbnbomgngy47ag7eq
In particular, it solves the problem of the notorious ξ-rule, which asserts that equations should be preserved under binders, and which fails to be sound for the usual interpretation. ... This paper serves as a self-contained, tutorial introduction to combinatory models of the untyped lambda calculus. We focus particularly on the interpretation of free variables. ... Acknowledgments I would like to thank the three anonymous referees for their valuable suggestions. ...doi:10.1017/s0956796801004294 fatcat:n34nmya5tfbbjddyepeomcwgjq
We focus on the basic strand algebra -combinatorial strand algebra, which is equivalent to the place-transition Petri nets, and on the version of the model driver of the Insertion Modeling System, based ... Letichevsky in 1987. and on the way of implementation of strand algebras -a process algebra for DNA computing devised by Luca Cardelli in order to compile other formal systems into the algebra, and compilation ... Some solution of this problem, based on the combinatorial DNA algebra, was given by Cardelli in paper  . ...dblp:conf/icteri/Klionov12 fatcat:l5xhot75cja6lfeqsdcvrtxemy
In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. ... , and 26 open problems. ... In an algebraic setting the problem of the order-incompleteness can be expressed as follows: (P19) Is there an n-permutable variety of LAAs for some n ≥ 2 (see  for the definition of n-permutability ...doi:10.1093/logcom/exn085 fatcat:muo26iplxbe2diepxtlnnutdxi
Functional lambda abstraction algebras arise as the ‘coordinatizations’ of environment models or lambda models, the natural combinatory models of the lambda calculus. ... to combinatory algebras in this regard. ...
We show in this paper, through a series of examples, how an algebraic modeling language might be extended to help with a greater variety of combinatorial optimization problems. ... Algebraic languages are at the heart of many successful optimization modeling systems, yet they have been used with only limited success for combinatorial (or discrete) optimization. ... Decision sets in algebraic modeling languages Our examples from previous sections suggest that algebraic modeling languages have several strengths for the specification of combinatorial optimization problems ...doi:10.1007/bf00248011 fatcat:pkxo4vvsjnhadim7jlmgv52dsm
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