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Applying Universal Algebra to Lambda Calculus

G. Manzonetto, A. Salibra
2008 Journal of Logic and Computation  
From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (= equational extensions of untyped λ-calculus) and the models of lambda calculus via universal  ...  ., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics  ...  partially ordered model of lambda calculus.  ... 
doi:10.1093/logcom/exn085 fatcat:muo26iplxbe2diepxtlnnutdxi

Models and theories of lambda calculus [article]

Giulio Manzonetto
2009 arXiv   pre-print
In this paper we briefly summarize the contents of Manzonetto's PhD thesis which concerns denotational semantics and equational/order theories of the pure untyped lambda-calculus.  ...  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  ...  This is a natural generalization of Honsell-Ronchi Della Rocca's longstanding open problem about the existence of a continuous model having as equational theory exactly λβ or λβη (since λβ and λβη are  ... 
arXiv:0904.4756v1 fatcat:7j47i72zebaoxkltk6fqul5bsi

Graph easy sets of mute lambda terms

A. Bucciarelli, A. Carraro, G. Favro, A. Salibra
2016 Theoretical Computer Science  
In this paper, we define for each natural number n, an infinite and recursive set M n of mute terms, and show that it is graph-easy: for any closed term t of the lambda calculus there exists a graph model  ...  Alongside, we provide a brief survey of the notion of undefinedness in the lambda calculus.  ...  A total pair (D, p) generates a model of the λ-calculus of universe P(D), called graph λ-model.  ... 
doi:10.1016/j.tcs.2015.12.024 fatcat:4ronvkej6vfw7lpptda55gql4q

Reflexive Scott Domains are Not Complete for the Extensional Lambda Calculus

Alberto Carraro, Antonino Salibra
2009 2009 24th Annual IEEE Symposium on Logic In Computer Science  
In this paper we analyze the class of reflexive Scott domains, the models of λ-calculus living in the category of Scott domains (a full subcategory of Cpo).  ...  Based on a fine-grained study of an "effective" version of Scott's information systems, we have shown that there are equations not in λβ (resp. λβη) which hold in all (extensional) iweb-models.  ...  Based on a notion of an effective model of λ-calculus, in [5] it was shown that the order theory of an effective model cannot be r.e. and its equational theory is different from λβ (λβη).  ... 
doi:10.1109/lics.2009.22 dblp:conf/lics/CarraroS09 fatcat:vmupobwr7zb6bgu4bfiihko7qe

Some results on extensionality in lambda calculus

Benedetto Intrigila, Richard Statman
2005 Annals of Pure and Applied Logic  
The result of the application of a term of the form λx.M to Ω returns the same tree which is obtained replacing the "label" x with the "label" Ω in the tree of M, that is:  ...  The following lemma points out that the result of the application of a term of the form λx.M to Ω returns the same tree as one can obtain replacing the "label" x with the "label" Ω in the tree of M.  ...  Acknowledgements The first author would like to thank Alessandro Berarducci for useful conversations on the topics of the paper.  ... 
doi:10.1016/j.apal.2003.05.001 fatcat:kexbwstqizc6tnwmhsa4jm6mzu

Lambda theories of effective lambda models [article]

Chantal Berline, Antonio Salibra
2007 arXiv   pre-print
A longstanding open problem is whether there exists a non-syntactical model of untyped lambda-calculus whose theory is exactly the least equational lambda-theory (=Lb).  ...  both are the order/equ theories of an effective graph model.  ...  The authors do not know any example of unsolvable satisfying the above condition.  ... 
arXiv:math/0701684v2 fatcat:uo7ia755zjgzdf6t4hvnxn6p24

What is a categorical model of the differential and the resource lambda-calculi? [article]

Manzonetto Giulio
2010 arXiv   pre-print
In this paper we provide an abstract model theory for the untyped differential lambda-calculus and the resource calculus.  ...  Examples of models based on relations are provided.  ...  also a model of the resource calculus.  ... 
arXiv:1011.2307v1 fatcat:bjaj2rb3sjdq7bbs5lt3zadyl4

What is a categorical model of the differential and the resource λ-calculi?

2012 Mathematical Structures in Computer Science  
We prove that the extensional model D of λ-calculus we have recently built in MRel is linear, and therefore it is also an extensional model of the untyped differential λ-calculus.  ...  constitute sound and complete models of the untyped differential λ-calculus.  ...  a model of the resource calculus.  ... 
doi:10.1017/s0960129511000594 fatcat:oahkwj3ihndovlixd4tp5yflg4

On the completeness of order-theoretic models of the λ-calculus

Furio Honsell, Gordon Plotkin
2009 Information and Computation  
Domain theory Completeness Consistency Mal'cev operators Scott discovered his domain-theoretic models of the λ-calculus, isomorphic to their function space, in 1969.  ...  on the consistency of certain critical sentences asserting the existence of certain functions analogous to the generalized Mal'cev operators first considered in the context of the λ-calculus by Selinger  ...  Let T be a first-order theory, e.g., that for combinatory logic, for the λβ-calculus or for the λβη-calculus; let C be a class of models of T; and let F be a set of sentences.  ... 
doi:10.1016/j.ic.2008.03.027 fatcat:so6e3udqifhjbeootdpyrbyoii

Infinite λ-calculus and non-sensible models [chapter]

Alessandro Berarducci
2017 Logic and algebra  
We define a model of λβ-calculus which is similar to the model of Böhm trees, but it does not identify all the unsolvable lambda-terms.  ...  We set the foundations for such a calculus, which might turn out to be a useful tool for the study of non-sensible models of λ-calculus.  ...  During our attempts to solve the problem Intrigila suggested to expand the language of λ-calculus with a new constant δ together with the rule δ → δω 3 .  ... 
doi:10.1201/9780203748671-18 fatcat:pzzkf3iwdjbnpg2eueosqsmojy

A Fuzzy Weibull model with application in stocks optimization

Campean Emilia, Morar Liviu, Blaga Lucia
2011 Procedia - Social and Behavioral Sciences  
Therefore the whole numerical procedure applied in the followings using the (FAHP) and (FQFD) algorithms depends on the solved problem's data.  ...  In the fuzzy neural networks' applications the membership functions _A (t) are selected from a standardized sequence of fuzzy sets without any modification depending on the class of problems to be solved  ...  This family of Fuzzy numbers is no longer defuzzificated, it will result in validation of the models obtained by numerical algorithms (FAHP) and Neuro-Fuzzy networks that are attached.  ... 
doi:10.1016/j.sbspro.2011.09.122 fatcat:rfxfunmoc5cwxdtcjd5nu35pqi

The third parafermionic chiral algebra with the symmetry Z3

Vladimir S. Dotsenko, Raoul Santachiara
2005 Physics Letters B  
Acknowledgments: We are grateful to J.L.Jacobsen for his collaboration during the initial stages of this project.  ...  This algebra is of the following form: Ψ(z ′ )Ψ(z) = 1 (z ′ − z) ∆ Ψ {λΨ + (z) + (z ′ − z)λβ (1) ΨΨ,Ψ + ∂Ψ + (z) +(z ′ − z) 2 [λβ (11) ΨΨ,Ψ + ∂ 2 Ψ + (z) + λβ (2) ΨΨ,Ψ + L −2 Ψ + (z) + ζΨ(z)] + ...}  ...  The third solution, with ∆ Ψ = 5/2, is know as the chiral algebra of the W B 2 model [5] .  ... 
doi:10.1016/j.physletb.2005.01.085 fatcat:rltwe27penezliswy75z2kc5ea

On the Spontaneous Dynamics of Synaptic Weights in Stochastic Models with Pair-Based STDP [article]

Philippe Robert, Gaëtan Vignoud
2021 arXiv   pre-print
We show that the pairing model of presynaptic and postsynaptic spikes controls the synaptic weight dynamics for small external input, on an excitatory synapse.  ...  This result implies in particular that mean-field analysis of plasticity may miss some important properties of STDP.  ...  We show in particular that several interesting properties of the synaptic weight dynamics are lost when using classical models with numerous excitatory inputs, leading to an underestimation of the role  ... 
arXiv:2111.07919v1 fatcat:5m2yfupmijdirbihnfyy2lmgfe

Gravity induced from quantum spacetime

Edwin J Beggs, Shahn Majid
2014 Classical and quantum gravity  
We show that tensoriality constraints in noncommutative Riemannian geometry in the 2-dimensional bicrossproduct model quantum spacetime algebra [x,t]=λ x drastically reduce the moduli of possible metrics  ...  The classical geometry is conformally flat and its geodesics motivate new coordinates which we extend to the quantum case as a new description of the quantum spacetime model as a quadratic algebra.  ...  It is not assumed that 1-forms and functions commute, so this is more general than classical differential calculus. In the case of (1.2) there is only one standard calculus of dimension n.  ... 
doi:10.1088/0264-9381/31/3/035020 fatcat:hyiu27zgy5gnhgkf77v43zdgfq

Type-safe run-time polytypic programming

2006 Journal of functional programming  
Polytypic programming is a way of defining type-indexed operations, such as map, fold and zip, based on type information.  ...  Although it is natural to define polytypic programs as operating over first-class types, such a semantics suffers from a number of drawbacks.  ...  All equivalent terms have the same erasure, so we can argue that they model the same computation.  ... 
doi:10.1017/s0956796806005879 fatcat:565tbvem5ndabegfpvsxpov5cm
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