120 research outputs found
Grazing incidence X-Ray scattering from the interfaces of thin film magnetic device materials
Thin film devices have found many applications in recently developed technology. With the need to increase data storage capacity and performance there are ever more demanding requirements of these devices. Gaining an understanding at the atomic scale of the growth and subsequent manufacturing treatments is fundamental to improving the device design. Grazing incidence x-ray scattering techniques have been used to study the interfaces in a sequence of samples, starting with repeated bi layers of single element material and sequentially working up to a realistic Magnetic Tunnnel junction (MTJ) structure. The width of the diffuse Bragg sheet from repeated bi-layers of Co/Pd and Co/Ru shows that the correlation length of the out-of-plane toughness is shorter for higher frequency roughness components than longer wavelength features. Scaling behaviour in the intensity profile demonstrates that the interfaces become more two-dimensional as more layers are deposited Reflectivity measurements with in-situ annealing reveal that the interfaces in CoFe/Ru repeated bi-layers are stable with temperature. The interfaces of amorphous CoFeB with ruthenium are also stable until the CoFeB crystallises. Similar measurements on repeated bi-layers of CoFeB/AlO, show sharpening of the interface during annealing. The diffuse scatter shows this to be a reduction in the intetdiffusion of the interface and not a change in topological roughness. The scatter from a single CoFeB/A10, interface on a realistic MTJ sub-structure also shows changes with annealing which are consistent with interface sharpening. This sharpening is matched to enhancements in the tunnel magneto- resistance of the MTJ. The changes occurring cannot be explained solely by sharpening of this particular interface and more sophisticated modelling has been attempted to identify the changes. Simulations show that changes in the manganese profile from an IrMn pinning layer in the MTJ should result in a significant change in the variable energy reflectivity recorded at a constant scattering vector
Negation-as-Failure in the Base-extension Semantics for Intuitionistic Propositional Logic
Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of them as provided by uniform proof-search—the proof-theoretic foundation of logic programming (LP)—to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Definite Formulae, Negation-as-Failure, and the Base-extension Semantics of Intuitionistic Propositional Logic
Proof-theoretic semantics (P-tS) is the paradigm of semantics in which
meaning in logic is based on proof (as opposed to truth). A particular instance
of P-tS for intuitionistic propositional logic (IPL) is its base-extension
semantics (B-eS). This semantics is given by a relation called support,
explaining the meaning of the logical constants, which is parameterized by
systems of rules called bases that provide the semantics of atomic
propositions. In this paper, we interpret bases as collections of definite
formulae and use the operational view of the latter as provided by uniform
proof-search -- the proof-theoretic foundation of logic programming (LP) -- to
establish the completeness of IPL for the B-eS. This perspective allows
negation, a subtle issue in P-tS, to be understood in terms of the
negation-as-failure protocol in LP. Specifically, while the denial of a
proposition is traditionally understood as the assertion of its negation, in
B-eS we may understand the denial of a proposition as the failure to find a
proof of it. In this way, assertion and denial are both prime concepts in P-tS.Comment: submitte
Proof-theoretic Semantics and Tactical Proof
The use of logical systems for problem-solving may be as diverse as in
proving theorems in mathematics or in figuring out how to meet up with a
friend. In either case, the problem solving activity is captured by the search
for an \emph{argument}, broadly conceived as a certificate for a solution to
the problem. Crucially, for such a certificate to be a solution, it has be
\emph{valid}, and what makes it valid is that they are well-constructed
according to a notion of inference for the underlying logical system. We
provide a general framework uniformly describing the use of logic as a
mathematics of reasoning in the above sense. We use proof-theoretic validity in
the Dummett-Prawitz tradition to define validity of arguments, and use the
theory of tactical proof to relate arguments, inference, and search.Comment: submitte
Provability in BI's Sequent Calculus is Decidable
The logic of Bunched Implications (BI) combines both additive and
multiplicative connectives, which include two primitive intuitionistic
implications. As a consequence, contexts in the sequent presentation are not
lists, nor multisets, but rather tree-like structures called bunches. This
additional complexity notwithstanding, the logic has a well-behaved metatheory
admitting all the familiar forms of semantics and proof systems. However, the
presentation of an effective proof-search procedure has been elusive since the
logic's debut. We show that one can reduce the proof-search space for any given
sequent to a primitive recursive set, the argument generalizing Gentzen's
decidability argument for classical propositional logic and combining key
features of Dyckhoff's contraction-elimination argument for intuitionistic
logic. An effective proof-search procedure, and hence decidability of
provability, follows as a corollary.Comment: Submitted to CADE-2
From Proof-theoretic Validity to Base-extension Semantics for Intuitionistic Propositional Logic
Proof-theoretic semantics (P-tS) is the approach to meaning in logic based on
\emph{proof} (as opposed to truth). There are two major approaches to P-tS:
proof-theoretic validity (P-tV) and base-extension semantics (B-eS). The former
is a semantics of arguments, and the latter is a semantics of logical constants
in a logic. This paper demonstrates that the B-eS for intuitionistic
propositional logic (IPL) encapsulates the declarative content of a basic
version of P-tV. Such relationships have been considered before yielding
incompleteness results. This paper diverges from these approaches by accounting
for the constructive, hypothetical setup of P-tV. It explicates how the B-eS
for IPL works
Defining Logical Systems via Algebraic Constraints on Proofs
We comprehensively present a program of decomposition of proof systems for
non-classical logics into proof systems for other logics, especially classical
logic, using an algebra of constraints. That is, one recovers a proof system
for a target logic by enriching a proof system for another, typically simpler,
logic with an algebra of constraints that act as correctness conditions on the
latter to capture the former; for example, one may use Boolean algebra to give
constraints in a sequent calculus for classical propositional logic to produce
a sequent calculus for intuitionistic propositional logic. The idea behind such
forms of reduction is to obtain a tool for uniform and modular treatment of
proof theory and provide a bridge between semantics logics and their proof
theory. The article discusses the theoretical background of the project and
provides several illustrations of its work in the field of intuitionistic and
modal logics. The results include the following: a uniform treatment of modular
and cut-free proof systems for a large class of propositional logics; a general
criterion for a novel approach to soundness and completeness of a logic with
respect to a model-theoretic semantics; and a case study deriving a
model-theoretic semantics from a proof-theoretic specification of a logic.Comment: submitte
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