547 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
A map of dependencies among three-valued logics
International audienceThree-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value
Admissibility in Finitely Generated Quasivarieties
Checking the admissibility of quasiequations in a finitely generated (i.e.,
generated by a finite set of finite algebras) quasivariety Q amounts to
checking validity in a suitable finite free algebra of the quasivariety, and is
therefore decidable. However, since free algebras may be large even for small
sets of small algebras and very few generators, this naive method for checking
admissibility in \Q is not computationally feasible. In this paper,
algorithms are introduced that generate a minimal (with respect to a multiset
well-ordering on their cardinalities) finite set of algebras such that the
validity of a quasiequation in this set corresponds to admissibility of the
quasiequation in Q. In particular, structural completeness (validity and
admissibility coincide) and almost structural completeness (validity and
admissibility coincide for quasiequations with unifiable premises) can be
checked. The algorithms are illustrated with a selection of well-known finitely
generated quasivarieties, and adapted to handle also admissibility of rules in
finite-valued logics
Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories
In flowchart languages, predicates play an interesting double role. In the
textual representation, they are often presented as conditions, i.e.,
expressions which are easily combined with other conditions (often via Boolean
combinators) to form new conditions, though they only play a supporting role in
aiding branching statements choose a branch to follow. On the other hand, in
the graphical representation they are typically presented as decisions,
intrinsically capable of directing control flow yet mostly oblivious to Boolean
combination. While categorical treatments of flowchart languages are abundant,
none of them provide a treatment of this dual nature of predicates. In the
present paper, we argue that extensive restriction categories are precisely
categories that capture such a condition/decision duality, by means of
morphisms which, coincidentally, are also called decisions. Further, we show
that having these categorical decisions amounts to having an internal logic:
Analogous to how subobjects of an object in a topos form a Heyting algebra, we
show that decisions on an object in an extensive restriction category form a De
Morgan quasilattice, the algebraic structure associated with the (three-valued)
weak Kleene logic . Full classical propositional logic can be
recovered by restricting to total decisions, yielding extensive categories in
the usual sense, and confirming (from a different direction) a result from
effectus theory that predicates on objects in extensive categories form Boolean
algebras. As an application, since (categorical) decisions are partial
isomorphisms, this approach provides naturally reversible models of classical
propositional logic and weak Kleene logic.Comment: 19 pages, including 6 page appendix of proofs. Accepted for MFPS XXX
Implications of Foundational Crisis in Mathematics: A Case Study in Interdisciplinary Legal Research
As a result of a sequence of so-called foundational crises, mathematicians have come to realize that foundational inquiries are difficult and perhaps never ending. Accounts of the last of these crises have appeared with increasing frequency in the legal literature, and one piece of this Article examines these invocations with a critical eye. The other piece introduces a framework for thinking about law as a discipline. On the one hand, the disciplinary framework helps explain how esoteric mathematical topics made their way into the legal literature. On the other hand, the mathematics can be used to examine some aspects of interdisciplinary legal research
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