414 research outputs found
Inversive Meadows and Divisive Meadows
Inversive meadows are commutative rings with a multiplicative identity
element and a total multiplicative inverse operation whose value at 0 is 0.
Divisive meadows are inversive meadows with the multiplicative inverse
operation replaced by a division operation. We give finite equational
specifications of the class of all inversive meadows and the class of all
divisive meadows. It depends on the angle from which they are viewed whether
inversive meadows or divisive meadows must be considered more basic. We show
that inversive and divisive meadows of rational numbers can be obtained as
initial algebras of finite equational specifications. In the spirit of
Peacock's arithmetical algebra, we study variants of inversive and divisive
meadows without an additive identity element and/or an additive inverse
operation. We propose simple constructions of variants of inversive and
divisive meadows with a partial multiplicative inverse or division operation
from inversive and divisive meadows. Divisive meadows are more basic if these
variants are considered as well. We give a simple account of how mathematicians
deal with 1 / 0, in which meadows and a customary convention among
mathematicians play prominent parts, and we make plausible that a convincing
account, starting from the popular computer science viewpoint that 1 / 0 is
undefined, by means of some logic of partial functions is not attainable.Comment: 18 pages; error corrected; 29 pages, combined with arXiv:0909.2088
[math.RA] and arXiv:0909.5271 [math.RA
How functional programming mattered
In 1989 when functional programming was still considered a niche topic, Hughes wrote a visionary paper arguing convincingly ‘why functional programming matters’. More than two decades have passed. Has functional programming really mattered? Our answer is a resounding ‘Yes!’. Functional programming is now at the forefront of a new generation of programming technologies, and enjoying increasing popularity and influence. In this paper, we review the impact of functional programming, focusing on how it has changed the way we may construct programs, the way we may verify programs, and fundamentally the way we may think about programs
Safe abstractions of data encodings in formal security protocol models
When using formal methods, security protocols are usually modeled at a high level of abstraction. In particular, data encoding and decoding transformations are often abstracted away. However, if no assumptions at all are made on the behavior of such transformations, they could trivially lead to security faults, for example leaking secrets or breaking freshness by collapsing nonces into constants. In order to address this issue, this paper formally states sufficient conditions, checkable on sequential code, such that if an abstract protocol model is secure under a Dolev-Yao adversary, then a refined model, which takes into account a wide class of possible implementations of the encoding/decoding operations, is implied to be secure too under the same adversary model. The paper also indicates possible exploitations of this result in the context of methods based on formal model extraction from implementation code and of methods based on automated code generation from formally verified model
Meta SOS - A Maude Based SOS Meta-Theory Framework
Meta SOS is a software framework designed to integrate the results from the
meta-theory of structural operational semantics (SOS). These results include
deriving semantic properties of language constructs just by syntactically
analyzing their rule-based definition, as well as automatically deriving sound
and ground-complete axiomatizations for languages, when considering a notion of
behavioural equivalence. This paper describes the Meta SOS framework by
blending aspects from the meta-theory of SOS, details on their implementation
in Maude, and running examples.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690
Congruence from the Operator's Point of View: Compositionality Requirements on Process Semantics
One of the basic sanity properties of a behavioural semantics is that it
constitutes a congruence with respect to standard process operators. This issue
has been traditionally addressed by the development of rule formats for
transition system specifications that define process algebras. In this paper we
suggest a novel, orthogonal approach. Namely, we focus on a number of process
operators, and for each of them attempt to find the widest possible class of
congruences. To this end, we impose restrictions on sublanguages of
Hennessy-Milner logic, so that a semantics whose modal characterization
satisfies a given criterion is guaranteed to be a congruence with respect to
the operator in question. We investigate action prefix, alternative
composition, two restriction operators, and parallel composition.Comment: In Proceedings SOS 2010, arXiv:1008.190
Two Decades of Maude
This paper is a tribute to José Meseguer, from the rest of us in the Maude team, reviewing the past, the present, and the future of the language and system with which we have been working for around two decades under his leadership. After reviewing the origins and the language's main features, we present the latest additions to the language and some features currently under development. This paper is not an introduction to Maude, and some familiarity with it and with rewriting logic are indeed assumed.Universidad de Málaga. Campus de Excelencia Internacional AndalucÃa Tech
Probability functions in the context of signed involutive meadows
The Kolmogorov axioms for probability functions are placed in the context of
signed meadows. A completeness theorem is stated and proven for the resulting
equational theory of probability calculus. Elementary definitions of
probability theory are restated in this framework.Comment: 20 pages, 6 tables, some minor errors are correcte
On the Invariance of G\"odel's Second Theorem with regard to Numberings
The prevalent interpretation of G\"odel's Second Theorem states that a
sufficiently adequate and consistent theory does not prove its consistency. It
is however not entirely clear how to justify this informal reading, as the
formulation of the underlying mathematical theorem depends on several arbitrary
formalisation choices. In this paper I examine the theorem's dependency
regarding G\"odel numberings. I introduce deviant numberings, yielding
provability predicates satisfying L\"ob's conditions, which result in provable
consistency sentences. According to the main result of this paper however,
these "counterexamples" do not refute the theorem's prevalent interpretation,
since once a natural class of admissible numberings is singled out, invariance
is maintained.Comment: Forthcoming in The Review of Symbolic Logi
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