59 research outputs found

    Degrees of extensionality in the theory of B\"ohm trees and Sall\'e's conjecture

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    The main observational equivalences of the untyped lambda-calculus have been characterized in terms of extensional equalities between B\"ohm trees. It is well known that the lambda-theory H*, arising by taking as observables the head normal forms, equates two lambda-terms whenever their B\"ohm trees are equal up to countably many possibly infinite eta-expansions. Similarly, two lambda-terms are equal in Morris's original observational theory H+, generated by considering as observable the beta-normal forms, whenever their B\"ohm trees are equal up to countably many finite eta-expansions. The lambda-calculus also possesses a strong notion of extensionality called "the omega-rule", which has been the subject of many investigations. It is a longstanding open problem whether the equivalence B-omega obtained by closing the theory of B\"ohm trees under the omega-rule is strictly included in H+, as conjectured by Sall\'e in the seventies. In this paper we demonstrate that the two aforementioned theories actually coincide, thus disproving Sall\'e's conjecture. The proof technique we develop for proving the latter inclusion is general enough to provide as a byproduct a new characterization, based on bounded eta-expansions, of the least extensional equality between B\"ohm trees. Together, these results provide a taxonomy of the different degrees of extensionality in the theory of B\"ohm trees

    Addressing Machines as models of lambda-calculus

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    Turing machines and register machines have been used for decades in theoretical computer science as abstract models of computation. Also the \lambda-calculus has played a central role in this domain as it allows to focus on the notion of functional computation, based on the substitution mechanism, while abstracting away from implementation details. The present article starts from the observation that the equivalence between these formalisms is based on the Church-Turing Thesis rather than an actual encoding of \lambda-terms into Turing (or register) machines. The reason is that these machines are not well-suited for modelling \lam-calculus programs. We study a class of abstract machines that we call \emph{addressing machine} since they are only able to manipulate memory addresses of other machines. The operations performed by these machines are very elementary: load an address in a register, apply a machine to another one via their addresses, and call the address of another machine. We endow addressing machines with an operational semantics based on leftmost reduction and study their behaviour. The set of addresses of these machines can be easily turned into a combinatory algebra. In order to obtain a model of the full untyped \lambda-calculus, we need to introduce a rule that bares similarities with the \omega-rule and the rule \zeta_\beta from combinatory logic

    Solution of a Problem of Barendregt on Sensible lambda-Theories

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    H is the theory extending β-conversion by identifying all closed unsolvables. Hω is the closure of this theory under the ω-rule (and β-conversion). A long-standing conjecture of H. Barendregt states that the provable equations of Hω form Π11-complete set. Here we prove that conjecture.Comment: 17 page

    two factor authentication for e government services using hardware like one time password generators

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    A safe and accessible authentication technique is a prerequisite for any modern e-government application. Two-factor authentication is currently widely adopted, since it alleviates many vulnerabilities of password-based authentication. The majority of e-government systems currently make use of text messages to deliver the second authentication factor, but these messages do not constitute an adequate (secure and reliable) solution. In this paper we show how to use One-Time Passwords (OTP) generated by a per-user, ad-hoc built application installed on a smartphone to support a two-factor authentication scheme specifically targeted to e-government tasks. In particular, we develop a process for the request, generation and distribution of such an application that achieves the same security of OTP hardware devices but avoids the related distribution and management costs, requiring no dedicated hardware and relying on the pre-existing administrative infrastructure. The process is designed to be accessible by any citizen who is able to perform very basic operations on a smartphone

    On the commutative equivalence of bounded context-free and regular languages: The code case

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    This is the first paper of a group of three where we prove the following result. Let A be an alphabet of t letters and let psi : A* -> N-t be the corresponding Parikh morphism. Given two languages L-1, L-2 subset of A*, we say that L1 is commutatively equivalent to L-2 if there exists a bijection f : L-1 -> L-2 from L-1 onto L-2 such that, for every u is an element of L-1, psi (u) = psi (f (u). Then every bounded context-free language is commutatively equivalent to a regular language. (C) 2014 Elsevier B.V. All rights reserved

    A note on coding techniques in bounded arithmetic

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    Technical Report n. 231, Dipartimento di Matematica, Via del Capitano 15, Sien

    On the commutative equivalence of semi-linear sets of N^k

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    Abstract Given two subsets S1,S2 of Nk, we say that S1 is commutatively equivalent to S2 if there exists a bijection f:S1鉄禨2 from S1 onto S2 such that, for every v鈭圫1, |v|=|f(v)|, where |v| denotes the sum of the components of v. We prove that every semi-linear set of Nk is commutatively equivalent to a recognizable subset of Nk