80,873 research outputs found
Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting
At the core of the quest for a logic for Ptime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to tackle this problem is witnessed symmetric choice. It allows for choices from definable orbits certified by definable witnessing automorphisms.
We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. When similarly extending pure fixed-point logic (IFP), IFP+WSC+I simulates counting which IFP+WSC fails to do. For IFPC+WSC, it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting.
In this paper, we separate IFPC+WSC from IFPC+WSC+I by showing that IFPC+WSC is not closed under FO-interpretations. By the same argument, we answer an open question of Dawar and Richerby regarding non-witnessed symmetric choice in IFP. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances
Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting
At the core of the quest for a logic for PTime is a mismatch between
algorithms making arbitrary choices and isomorphism-invariant logics. One
approach to overcome this problem is witnessed symmetric choice. It allows for
choices from definable orbits which are certified by definable witnessing
automorphisms.
We consider the extension of fixed-point logic with counting (IFPC) with
witnessed symmetric choice (IFPC+WSC) and a further extension with an
interpretation operator (IFPC+WSC+I). The latter operator evaluates a
subformula in the structure defined by an interpretation. This structure
possibly has other automorphisms exploitable by the WSC-operator. For similar
extensions of pure fixed-point logic (IFP) it is known that IFP+WSCI simulates
counting which IFP+WSC fails to do. For IFPC+WSC it is unknown whether the
interpretation operator increases expressiveness and thus allows studying the
relation between WSC and interpretations beyond counting.
We separate IFPC+WSC from IFPC+WSCI by showing that IFPC+WSC is not closed
under FO-interpretations. By the same argument, we answer an open question of
Dawar and Richerby regarding non-witnessed symmetric choice in IFP.
Additionally, we prove that nesting WSC-operators increases the expressiveness
using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a
particular class of base graphs, then it also canonizes the corresponding CFI
graphs. This differs from various other logics, where CFI graphs provide
difficult instances.Comment: 46 pages, 5 figures, [v2] and [v3] Corrected minor mistakes and added
figure
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
Subspace-Invariant AC Formulas
We consider the action of a linear subspace of on the set of
AC formulas with inputs labeled by literals in the set , where an element acts on formulas by
transposing the th pair of literals for all such that . A
formula is {\em -invariant} if it is fixed by this action. For example,
there is a well-known recursive construction of depth formulas of size
computing the -variable PARITY function; these
formulas are easily seen to be -invariant where is the subspace of
even-weight elements of . In this paper we establish a nearly
matching lower bound on the -invariant depth
formula size of PARITY. Quantitatively this improves the best known
lower bound for {\em unrestricted} depth
formulas, while avoiding the use of the switching lemma. More generally,
for any linear subspaces , we show that if a Boolean function is
-invariant and non-constant over , then its -invariant depth
formula size is at least where is the minimum Hamming
weight of a vector in
CZF does not have the Existence Property
Constructive theories usually have interesting metamathematical properties
where explicit witnesses can be extracted from proofs of existential sentences.
For relational theories, probably the most natural of these is the existence
property, EP, sometimes referred to as the set existence property. This states
that whenever (\exists x)\phi(x) is provable, there is a formula \chi(x) such
that (\exists ! x)\phi(x) \wedge \chi(x) is provable. It has been known since
the 80's that EP holds for some intuitionistic set theories and yet fails for
IZF. Despite this, it has remained open until now whether EP holds for the most
well known constructive set theory, CZF. In this paper we show that EP fails
for CZF
Inversion, Iteration, and the Art of Dual Wielding
The humble ("dagger") is used to denote two different operations in
category theory: Taking the adjoint of a morphism (in dagger categories) and
finding the least fixed point of a functional (in categories enriched in
domains). While these two operations are usually considered separately from one
another, the emergence of reversible notions of computation shows the need to
consider how the two ought to interact. In the present paper, we wield both of
these daggers at once and consider dagger categories enriched in domains. We
develop a notion of a monotone dagger structure as a dagger structure that is
well behaved with respect to the enrichment, and show that such a structure
leads to pleasant inversion properties of the fixed points that arise as a
result. Notably, such a structure guarantees the existence of fixed point
adjoints, which we show are intimately related to the conjugates arising from a
canonical involutive monoidal structure in the enrichment. Finally, we relate
the results to applications in the design and semantics of reversible
programming languages.Comment: Accepted for RC 201
Dependent choice, properness, and generic absoluteness
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory
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