Logical Step-Indexed Logical Relations

Appel and McAllester's "step-indexed" logical relations have proven to be a simple and effective technique for reasoning about programs in languages with semantically interesting types, such as general recursive types and general reference types. However, proofs using step-indexed models typically involve tedious, error-prone, and proof-obscuring step-index arithmetic, so it is important to develop clean, high-level, equational proof principles that avoid mention of step indices. In this paper, we show how to reason about binary step-indexed logical relations in an abstract and elegant way. Specifically, we define a logic LSLR, which is inspired by Plotkin and Abadi's logic for parametricity, but also supports recursively defined relations by means of the modal "later" operator from Appel, Melli\`es, Richards, and Vouillon's "very modal model" paper. We encode in LSLR a logical relation for reasoning relationally about programs in call-by-value System F extended with general recursive types. Using this logical relation, we derive a set of useful rules with which we can prove contextual equivalence and approximation results without counting steps

Spin-Transfer-Torque Driven Magneto-Logic OR, AND and NOT Gates

We show that current induced magneto-logic gates like AND, OR and NOT can be designed with the simple architecture involving a single nano spin-valve pillar, as an extension of our recent work on spin-torque-driven magneto-logic universal gates, NAND and NOR. Here the logical operation is induced by spin-polarized currents which also form the logical inputs. The operation is facilitated by the simultaneous presence of a constant controlling magnetic field, in the absence of which the same element operates as a magnetoresistive memory element. We construct the relevant phase space diagrams for the free layer magnetization dynamics in the monodomain approximation and show the rationale and functioning of the proposed gates. The flipping time for the logical states of these non-universal gates is estimated to be within nano seconds, just like their universal counter parts.Comment: 9 pages,7 figure

An analysis of the equational properties of the well-founded fixed point

Well-founded fixed points have been used in several areas of knowledge representation and reasoning and to give semantics to logic programs involving negation. They are an important ingredient of approximation fixed point theory. We study the logical properties of the (parametric) well-founded fixed point operation. We show that the operation satisfies several, but not all of the equational properties of fixed point operations described by the axioms of iteration theories

On the Harmonic approximation for large Josephson junction coupling charge qubits

We revisit the harmonic approximation (HA) for a large Josephson junction interacting with some charge qubits through the variational approach for the quantum dynamics of the junction-qubit coupling system. By making use of numerical calculation and analytical treatment, the conditions under which HA works well can be precisely presented to control the parameters implementing the two-qubit quantum logical gate through the couplings to the large junction with harmonic oscillator (HO) Hamiltonian.Comment: 7 pages, 3 figure