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### Page 2530 of Mathematical Reviews Vol. , Issue 84g [page]

1984 Mathematical Reviews
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  ...

### The Fixpoint Combinator in Combinatory Logic – A Step towards Autonomous Real-time Testing of Software?

Thomas Fehlmann, Eberhard Kranich
2022 Athens Journal of Sciences
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.  ...

### Scott Is Always Simple [chapter]

Antonino Salibra
2012 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.  ...

### Models and theories of lambda calculus [article]

Giulio Manzonetto
2009 arXiv   pre-print
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  ...

### Page 4360 of Mathematical Reviews Vol. , Issue 84j [page]

1984 Mathematical Reviews
“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.  ...

### Heisenberg-Weyl algebra revisited: Combinatorics of words and paths [article]

P. Blasiak, K. A. Penson Institut Galilee, University of Paris, France, LPTMC, University of Paris VI, France)
2009 arXiv   pre-print
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.  ...

### Book Review: Linear and combinatorial optimization in ordered algebraic structures

R. A. Cuninghame-Green
1985 Bulletin of the American Mathematical Society
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  ...

### Extension of combinatory logic to a theory of combinatory representation

Trudy Weibel
1992 Theoretical Computer Science
., 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.  ...

### Relations between Generalization, Reasoning and Combinatorial Thinking in Solving Mathematical Open-Ended Problems within Mathematical Contest

Janka Medová, Kristína Ovary Bulková, Soňa Čeretková
2020 Mathematics
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.  ...

### Algebraic, logical and network representations in the design of software for combinatorial optimization

C. Coullard, R. Fourer
1996 Proceedings of HICSS-29: 29th Hawaii International Conference on System Sciences
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  ...

### The lambda calculus is algebraic

PETER SELINGER
2002 Journal of functional programming
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.  ...

### Combinatorial Strand Algebra in Insertion Modeling System

Dmitriy M. Klionov
2012 International Conference on Information and Communication Technologies in Education, Research, and Industrial Applications
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  .  ...

### Applying Universal Algebra to Lambda Calculus

G. Manzonetto, A. Salibra
2008 Journal of Logic and Computation
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  ...

### Page 1477 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews
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.  ...

### New constructs for the description of combinatorial optimization problems in algebraic modeling languages

J. J. Bisschop, Robert Fourer
1996 Computational optimization and applications
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  ...
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