618 research outputs found
Strongly Universal Reversible Gate Sets
It is well-known that the Toffoli gate and the negation gate together yield a
universal gate set, in the sense that every permutation of can be
implemented as a composition of these gates. Since every bit operation that
does not use all of the bits performs an even permutation, we need to use at
least one auxiliary bit to perform every permutation, and it is known that one
bit is indeed enough. Without auxiliary bits, all even permutations can be
implemented. We generalize these results to non-binary logic: If is a
finite set of odd cardinality then a finite gate set can generate all
permutations of for all , without any auxiliary symbols. If the
cardinality of is even then, by the same argument as above, only even
permutations of can be implemented for large , and we show that indeed
all even permutations can be obtained from a finite universal gate set. We also
consider the conservative case, that is, those permutations of that
preserve the weight of the input word. The weight is the vector that records
how many times each symbol occurs in the word. It turns out that no finite
conservative gate set can, for all , implement all conservative even
permutations of without auxiliary bits. But we provide a finite gate set
that can implement all those conservative permutations that are even within
each weight class of .Comment: Submitted to Rev Comp 201
Applying quantitative semantics to higher-order quantum computing
Finding a denotational semantics for higher order quantum computation is a
long-standing problem in the semantics of quantum programming languages. Most
past approaches to this problem fell short in one way or another, either
limiting the language to an unusably small finitary fragment, or giving up
important features of quantum physics such as entanglement. In this paper, we
propose a denotational semantics for a quantum lambda calculus with recursion
and an infinite data type, using constructions from quantitative semantics of
linear logic
An Explicit Framework for Interaction Nets
Interaction nets are a graphical formalism inspired by Linear Logic
proof-nets often used for studying higher order rewriting e.g. \Beta-reduction.
Traditional presentations of interaction nets are based on graph theory and
rely on elementary properties of graph theory. We give here a more explicit
presentation based on notions borrowed from Girard's Geometry of Interaction:
interaction nets are presented as partial permutations and a composition of
nets, the gluing, is derived from the execution formula. We then define
contexts and reduction as the context closure of rules. We prove strong
confluence of the reduction within our framework and show how interaction nets
can be viewed as the quotient of some generalized proof-nets
A feasible algorithm for typing in Elementary Affine Logic
We give a new type inference algorithm for typing lambda-terms in Elementary
Affine Logic (EAL), which is motivated by applications to complexity and
optimal reduction. Following previous references on this topic, the variant of
EAL type system we consider (denoted EAL*) is a variant without sharing and
without polymorphism. Our algorithm improves over the ones already known in
that it offers a better complexity bound: if a simple type derivation for the
term t is given our algorithm performs EAL* type inference in polynomial time.Comment: 20 page
Quantitative Models and Implicit Complexity
We give new proofs of soundness (all representable functions on base types
lies in certain complexity classes) for Elementary Affine Logic, LFPL (a
language for polytime computation close to realistic functional programming
introduced by one of us), Light Affine Logic and Soft Affine Logic. The proofs
are based on a common semantical framework which is merely instantiated in four
different ways. The framework consists of an innovative modification of
realizability which allows us to use resource-bounded computations as realisers
as opposed to including all Turing computable functions as is usually the case
in realizability constructions. For example, all realisers in the model for
LFPL are polynomially bounded computations whence soundness holds by
construction of the model. The work then lies in being able to interpret all
the required constructs in the model. While being the first entirely semantical
proof of polytime soundness for light logi cs, our proof also provides a
notable simplification of the original already semantical proof of polytime
soundness for LFPL. A new result made possible by the semantic framework is the
addition of polymorphism and a modality to LFPL thus allowing for an internal
definition of inductive datatypes.Comment: 29 page
On paths-based criteria for polynomial time complexity in proof-nets
Girard's Light linear logic (LLL) characterized polynomial time in the
proof-as-program paradigm with a bound on cut elimination. This logic relied on
a stratification principle and a "one-door" principle which were generalized
later respectively in the systems L^4 and L^3a. Each system was brought with
its own complex proof of Ptime soundness.
In this paper we propose a broad sufficient criterion for Ptime soundness for
linear logic subsystems, based on the study of paths inside the proof-nets,
which factorizes proofs of soundness of existing systems and may be used for
future systems. As an additional gain, our bound stands for any reduction
strategy whereas most bounds in the literature only stand for a particular
strategy.Comment: Long version of a conference pape
A convenient category of locally preordered spaces
As a practical foundation for a homotopy theory of abstract spacetime, we
extend a category of certain compact partially ordered spaces to a convenient
category of locally preordered spaces. In particular, we show that our new
category is Cartesian closed and that the forgetful functor to the category of
compactly generated spaces creates all limits and colimits.Comment: 26 pages, 0 figures, partially presented at GETCO 2005; changes:
claim of Prop. 5.11 weakened to finite case and proof changed due to problems
with proof of Lemma 3.26, now removed; Eg. 2.7, statement before Lem. 2.11,
typos, and other minor problems corrected throughout; extensive rewording;
proof of Lem. 3.31, now 3.30, adde
- …