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Proving Noninterference by a Fully Complete Translation to the Simply Typed lambda-calculus  [PDF]
Naokata Shikuma,Atsushi Igarashi
Computer Science , 2008, DOI: 10.2168/LMCS-4(3:10)2008
Abstract: Tse and Zdancewic have formalized the notion of noninterference for Abadi et al.'s DCC in terms of logical relations and given a proof of noninterference by reduction to parametricity of System F. Unfortunately, their proof contains errors in a key lemma that their translation from DCC to System F preserves the logical relations defined for both calculi. In fact, we have found a counterexample for it. In this article, instead of DCC, we prove noninterference for sealing calculus, a new variant of DCC, by reduction to the basic lemma of a logical relation for the simply typed lambda-calculus, using a fully complete translation to the simply typed lambda-calculus. Full completeness plays an important role in showing preservation of the two logical relations through the translation. Also, we investigate relationship among sealing calculus, DCC, and an extension of DCC by Tse and Zdancewic and show that the first and the last of the three are equivalent.
A short proof of the strong normalization of the simply typed $λμ$-calculus  [PDF]
René David,Karim Nour
Mathematics , 2009,
Abstract: We give an elementary and purely arithmetical proof of the strong normalization of Parigot's simply typed $\lambda\mu$-calculus.
Definable functions in the simply typed lambda-calculus  [PDF]
Mateusz Zakrzewski
Computer Science , 2007,
Abstract: It is a common knowledge that the integer functions definable in simply typed lambda-calculus are exactly the extended polynomials. This is indeed the case when one interprets integers over the type (p->p)->p->p where p is a base type and/or equality is taken as beta-conversion. It is commonly believed that the same holds for beta-eta equality and for integers represented over any fixed type of the form (t->t)->t->t. In this paper we show that this opinion is not quite true. We prove that the class of functions strictly definable in simply typed lambda-calculus is considerably larger than the extended polynomials. Namely, we define F as the class of strictly definable functions and G as a class that contains extended polynomials and two additional functions, or more precisely, two function schemas, and is closed under composition. We prove that G is a subset of F. We conjecture that G exactly characterizes strictly definable functions, i.e. G=F, and we gather some evidence for this conjecture proving, for example, that every skewly representable finite range function is strictly representable over (t->t)->t->t for some t.
Simply Typed Lambda-Calculus Modulo Type Isomorphisms  [PDF]
Alejandro Díaz-Caro,Gilles Dowek
Computer Science , 2015,
Abstract: We define a simply typed, non-deterministic lambda-calculus where isomorphic types are equated. To this end, an equivalence relation is settled at the term level. We then provide a proof of strong normalisation modulo equivalence. Such a proof is a non-trivial adaptation of the reducibility method.
Strong Typed Boehm Theorem and Functional Completeness on the Linear Lambda Calculus  [PDF]
Satoshi Matsuoka
Computer Science , 2015,
Abstract: In this paper, we prove a version of the typed B\"{o}hm theorem on the linear lambda calculus, which says, for any given types $A$ and $B$, when $s_1$ and $s_2$ (respectively, $u_1$ and $u_2$) are different closed terms of $A$ (resp. $B$), there is a term $t$ such that \[ t \, s_1 =_{\beta \eta {\rm c}} u_1 \quad \mbox{and} \quad t \, s_2 =_{\beta \eta {\rm c}} u_2 \, . \] Several years ago, a weaker version of this theorem was proved, but the stronger version was open. As a corollary of this theorem, we prove that if $A$ has two different closed terms $s_1$ and $s_2$, then $A$ is functionally complete with regard to $s_1$ and $s_2$. So far, it was only known that a few types are functionally complete.
Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
Murdoch J. Gabbay,Dominic P. Mulligan
Electronic Proceedings in Theoretical Computer Science , 2011, DOI: 10.4204/eptcs.71.5
Abstract: We investigate a class of nominal algebraic Henkin-style models for the simply typed lambda-calculus in which variables map to names in the denotation and lambda-abstraction maps to a (non-functional) name-abstraction operation. The resulting denotations are smaller and better-behaved, in ways we make precise, than functional valuation-based models. Using these new models, we then develop a generalisation of λ-term syntax enriching them with existential meta-variables, thus yielding a theory of incomplete functions. This incompleteness is orthogonal to the usual notion of incompleteness given by function abstraction and application, and corresponds to holes and incomplete objects.
Simply-typed underdeterminism
Simply-typed Underdeterminism

Ewen Denney,

计算机科学技术学报 , 1998,
Abstract: An extension of the simply-typed lambda calculus with constructs for expressing a notioncalled underdeterminism is studied. This allows us to interpret notions of stub and skeletonused in top-down program development. We axiomatise a simple notion of program refinement,and give a semantics, for which the calculus is proved sound and complete.
Axioms for Definability and Full Completeness  [PDF]
Samson Abramsky
Computer Science , 2014,
Abstract: Axioms are presented which encapsulate the properties satisfied by categories of games which form the basis of results on full abstraction for PCF and other programming languages, and on full completeness for various logics and type theories. Axioms are presented on models of PCF from which full abstraction can be proved. These axioms have been distilled from recent results on definability and full abstraction of game semantics for a number of programming languages. Full completeness for pure simply-typed $\lambda$-calculus is also axiomatized.
A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of

Klaus Aehlig
Computer Science , 2007, DOI: 10.2168/LMCS-3(3:1)2007
Abstract: Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type hierarchy upon this interpretation of the base type gives a finite semantics for simply-typed lambda-trees. A calculus based on this semantics is proven sound and complete. In particular, for regular infinite lambda-trees it is decidable whether a given automaton has a run or not. As regular lambda-trees are precisely recursion schemes, this decidability result holds for arbitrary recursion schemes of arbitrary level, without any syntactical restriction.
Weak Typed Boehm Theorem on IMLL  [PDF]
Satoshi Matsuoka
Computer Science , 2004, DOI: 10.1016/j.apal.2006.06.001
Abstract: In the Boehm theorem workshop on Crete island, Zoran Petric called Statman's ``Typical Ambiguity theorem'' typed Boehm theorem. Moreover, he gave a new proof of the theorem based on set-theoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Boehm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment of intuitionistic linear logic without the multiplicative unit 1 (for short IMLL) weak typed Boehm theorem holds. The system IMLL exactly corresponds to the linear lambda calculus without exponentials, additives and logical constants. The system IMLL also exactly corresponds to the free symmetric monoidal closed category without the unit object. As far as we know, our separation result is the first one with regard to these systems in a purely syntactical manner.
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