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Extensional Set Equality in the Calculus of Constructions

J. P. Seldin
2001 Journal of Logic and Computation  
The original representation of set theory in the calculus of constructions, has met an objection on the grounds that the axiom of exstensionality is incompatible with the assumptions for arithmetic.  ...  In this paper, it is shown that there is really no problem with the axiom of extensionality.  ...  Acknowledgements This work was supported in part by grant RGP-23391-98 from the Natural Sciences and Engineering Research Council of Canada, and was presented at the Festival Workshop on the Foundations  ... 
doi:10.1093/logcom/11.3.483 fatcat:vyjschlvc5c2dmjpsfjko5hgim

Internalizing Relational Parametricity in the Extensional Calculus of Constructions

Neelakantan R. Krishnaswami, Derek Dreyer, Marc Herbstritt
2013 Annual Conference for Computer Science Logic  
We give the first relationally parametric model of the extensional calculus of constructions.  ...  Our model remains as simple as traditional PER models of types, but unlike them, it additionally permits the relating of terms that implement abstract types in different ways.  ...  Almost all of the typing rules are standard for the calculus of constructions, and we discuss only our variations in detail. Our treatment of equality follows extensional type theory.  ... 
doi:10.4230/lipics.csl.2013.432 dblp:conf/csl/KrishnaswamiD13 fatcat:ekzodbdhxjd7xfpc2iylif3aia

Deduction System for TIL-2010

Marie Duzí, Marek Mensík, Lukás Vích
2012 Recent Advances in Slavonic Natural Languages Processing  
Then we describe the adjustments of the calculus so that it be applicable to hyperintensions within the ramified hierarchy of types.  ...  Tichý defined a sequent calculus for pre-1988 TIL, that is TIL based on the simple theory of types. Thus we first briefly recapitulate the rules of this simple-type calculus.  ...  of Semantics' and also by the internal grant agency of VSB-Technical University Ostrava, Project SP2012/26, "An utilization of artificial intelligence in knowledge mining from software processes".  ... 
dblp:conf/raslan/DuziMV12 fatcat:ma2bklnidrexvhocladriyelzy

Realizability Models for Type Theories

Bernhard Reus
1999 Electronical Notes in Theoretical Computer Science  
These range from simply typed -calculus over second-order polymorphic -calculus to the Calculus of Constructions as an example of dependent t ype theory.  ...  Finally, b y considering complete extensional pers, an approach to bridge the gap from type theory to constructive domain theory is discussed. Reus (see e.g. 7]).  ...  Thanks also go to Thorsten Altenkirch, Alexander Knapp, and several workshop participants who gave v aluable feedback on di erent preliminary drafts of this survey, and to the editors, in particular Pino  ... 
doi:10.1016/s1571-0661(04)00108-2 fatcat:7v24nb5g6nfifec3xmtg7swqum

Page 12 of Mathematical Reviews Vol. , Issue 81G [page]

1981 Mathematical Reviews  
The author then extends the construction of this model in two ways. The first is to obtain a model of the ASK-calculus without the w-rule.  ...  Any set-theoretic models of the A-calculus which are natural (in the above sense) have to be constructed in set theories without this axiom.  ... 

On a Question of H. Friedman

Gordon Plotkin
1996 Information and Computation  
In this paper we answer a question of Friedman, providing an ω-separable model M of the λβη-calculus. There therefore exists an α-separable model for any α ≥ 0.  ...  The open term model embeds in M: by way of contrast we provide a model which cannot embed in any non-trivial model separating all pairs of distinct elements.  ...  In Section 2 we answer Friedman's question positively, giving a term model construction of an ω-separable model of the λβη-calculus.  ... 
doi:10.1006/inco.1996.0035 fatcat:flysla72wzexrgsba56z6br4li

Towards an Implementation in LambdaProlog of the Two Level Minimalist Foundation (short paper)

Alberto Fiori, Claudio Sacerdoti Coen
2018 International Conference on Intelligent Computer Mathematics  
Linking the two level there is an interpretation of the extensional level in a dependent setoid model build upon the intensional level where extensionality is modeled by intensional propositions.  ...  In this work we will present an implementation of the Minimalist Foundation [MS05; Mai09] using the Prolog [NM88] implementation ELPI [Dun+15] .  ...  To reasonably implement the extensional calculus and the interpretation we need a way to construct intensional proofs from extensional proof and a way to contain the undecidability of the strong elimination  ... 
dblp:conf/mkm/FioriC18 fatcat:hm66eh7eqbby5eaclj5ve64oei

Hot: A concurrent automated theorem prover based on higher-order tableaux [chapter]

Karsten Konrad
1998 Lecture Notes in Computer Science  
The rst part of this paper introduces an improved variant of the calculus which closely corresponds to the proof procedure implemented in Hot.  ...  Hot is an automated higher-order theorem prover based on HT E, an extensional higher-order tableaux calculus.  ...  o px ) py The following tableau construction rules complement ETAB with regard to extensionality in Henkin model semantics: Soundness and Completeness Soundness and completeness proofs for extensional  ... 
doi:10.1007/bfb0055140 fatcat:tuxfckgj5bao5awpbid6rhe2ha

A Calculus of Coroutines [chapter]

J. Laird
2004 Lecture Notes in Computer Science  
extensional in the presence of of errors; in particular adding errors to PCF allows the order of evaluation of functions to be observed.  ...  In particular, Laird showed that the bistable functions (a new notion) gave rise to a universal model of the λ-calculus with two constants, and that CPS-translations allowed this to be extended to programming  ...  Moreover, the extensional understanding of computation in games models afforded by this work allowed the construction of new models of nondeterminism: by completing the model with meets or joins, one obtains  ... 
doi:10.1007/978-3-540-27836-8_74 fatcat:hrcyftg22bcc3mw52qs3vbyr5y

Extensional Universal Types for Call-by-Value [chapter]

Kazuyuki Asada
2008 Lecture Notes in Computer Science  
To demonstrate validity of the calculus, we construct concrete models for the calculus in a generic manner, exploiting "relevant" parametricity.  ...  In the present paper we propose a second-order polymorphic call-by-value lambda calculus with extensional universal types.  ...  The λc-calculus λ c 2 η -calculus, to demonstrate how many effects are consistent with extensional universal types.  ... 
doi:10.1007/978-3-540-89330-1_9 fatcat:icvmldilfzgm7jmgpo3thtwxv4

Page 1375 of Mathematical Reviews Vol. , Issue 82d [page]

1982 Mathematical Reviews  
(The author’s construction of C; is in fact Cy, with M@ the closed term model of 7.) If there are enough arrows 1— U (1 is the terminal object), then I2(C, U) is (weakly) extensional.  ...  Given a domain U (and the retraction maps) in a Ccc C, one can extend the mentioned correspondence to type-free A-calculus and construct a A-theory T(C, U).  ... 

Programs as Data Structures in λSF-Calculus

Barry Jay
2016 Electronical Notes in Theoretical Computer Science  
In turn, all closed normal forms are data structures, in the sense that their internal structure is accessible through queries defined in the calculus, even to the point of constructing the Goedel number  ...  First, N is extensionally equivalent to M where extensional equivalence is defined in terms of eta-reduction.  ...  In conclusion, λSF -calculus adds intensionality to the extensional nature of λcalculus, so that one can query the internal structure of arbitrary closed normal forms, and treat programs as data structures  ... 
doi:10.1016/j.entcs.2016.09.040 fatcat:didnzadflzd2nbt5i6qfjstcti

Page 304 of The Philosophical Review Vol. 52, Issue 3 [page]

1943 The Philosophical Review  
For the intuitionistic rejection of certain principles of classical mathematics (e.g., the principle of excluded middle) refers, not to the abstract construction of a calculus, but to the construction  ...  of a calculus whose interpretation is at least partly given.  ... 

Extensional Models of Untyped Lambda-mu Calculus

Koji Nakazawa, Shin-ya Katsumata
2012 Electronic Proceedings in Theoretical Computer Science  
The other is called the stream combinatory algebra, which is an extension of the combinatory algebra, and it is proved that the extensional equality of the Lambda-mu calculus is equivalent to equality  ...  This paper proposes new mathematical models of the untyped Lambda-mu calculus.  ...  One natural direction of study is to analyze the local structure of the domain-theoretic extensional stream models constructed from the solutions of (2) in Section 3.2.  ... 
doi:10.4204/eptcs.97.3 fatcat:53yagybcbzaxtmvvmob65vmt24

Dr. Zilsel on the Concept of Physical Law

Mortimer Taube
1943 Philosophical Review  
For the intuitionistic rejection of certain principles of classical mathematics (e.g., the principle of excluded middle) refers, not to the abstract construction of a calculus, but to the construction  ...  of a calculus whose interpretation is at least partly given.  ... 
doi:10.2307/2180924 fatcat:rg24t7e4drbufn5uv3gxignt74
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