59 research outputs found

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

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

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

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

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

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

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

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

Abstract Given two subsets S1,S2 of Nk, we say that S1 is commutatively equivalent to S2 if there exists a bijection f:S1⟶S2 from S1 onto S2 such that, for every v∈S1, |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

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