31 research outputs found
Concrete Categorical Model of a Quantum Circuit Description Language with Measurement
In this paper, we introduce dynamic lifting to a quantum circuit-description language, following the Proto-Quipper language approach. Dynamic lifting allows programs to transfer the result of measuring quantum data - qubits - into classical data - booleans -. We propose a type system and an operational semantics for the language and we state safety properties. Next, we introduce a concrete categorical semantics for the proposed language, basing our approach on a recent model from Rios&Selinger for Proto-Quipper-M. Our approach is to construct on top of a concrete category of circuits with measurements a Kleisli category, capturing as a side effect the action of retrieving classical content out of a quantum memory. We then show a soundness result for this semantics
Variant-Based Satisfiability
Although different satisfiability decision procedures
can be combined by algorithms such as those of Nelson-Oppen or
Shostak, current tools typically can only support a finite number of
theories to use in such combinations. To make SMT solving more
widely applicable, generic satisfiability algorithms that can
allow a potentially infinite number of decidable theories to be
user-definable, instead of needing to be built in by the
implementers, are highly desirable. This work studies how
folding variant narrowing, a generic
unification algorithm that offers
good extensibility in unification theory, can be extended to
a generic variant-based satisfiability algorithm for the initial
algebras of its user-specified input theories when such theories
satisfy Comon-Delaune's finite variant property (FVP) and some
extra conditions. Several, increasingly larger infinite classes of
theories whose initial algebras enjoy decidable variant-based satisfiability
are identified, and a method based on descent maps to bring other theories
into these classes and to improve the generic
algorithm's efficiency is proposed and illustrated with examples.Partially supported by NSF Grant CNS 13-19109.Ope
A Constructor-Based Reachability Logic for Rewrite Theories
Reachability logic has been applied to K
rewrite-rule-based language definitions as a
language-generic logic of programs. It has been proved
successful in verifying a wide range of sophisticated programs in
conventional languages. Here we study how reachability logic
can be made not just language-generic, but
rewrite-theory-generic to make it available not just
for conventional program verification, but also to verify rewriting-logic-based
programs and distributed system designs. A theory-generic
reachability logic is presented and proved sound for a wide class of
rewrite theories. Particular attention is given to increasing the
logic's automation by means of constructor-based semantic
unification, matching, and satisfiability procedures. The
relationships to Hoare logic and LTL are discussed, new methods for
proving invariants of possibly never terminating distributed systems
are developed, and experiments with a prototype implementation
illustrating the new methods are presented.Partially supported by NSF Grants CNS 13-19109 and CNS 14-09416, and AFOSR Contract FA8750-11-2-0084.Ope
Algebraic Algorithm Design and Local Search
Formal, mathematically-based techniques promise to play an expanding role in the development and maintenance of the software on which our technological society depends. Algebraic techniques have been applied successfully to algorithm synthesis by the use of algorithm theories and design tactics, an approach pioneered in the Kestrel Interactive Development System (KIDS). An algorithm theory formally characterizes the essential components of a family of algorithms. A design tactic is a specialized procedure for recognizing in a problem specification the structures identified in an algorithm theory and then synthesizing a program. Design tactics are hard to write, however, and much of the knowledge they use is encoded procedurally in idiosyncratic ways. Algebraic methods promise a way to represent algorithm design knowledge declaratively and uniformly. We describe a general method for performing algorithm design that is more purely algebraic than that of KIDS. This method is then applied to local search. Local search is a large and diverse class of algorithms applicable to a wide range of problems; it is both intrinsically important and representative of algorithm design as a whole. A general theory of local search is formalized to describe the basic properties common to all local search algorithms, and applied to several variants of hill climbing and simulated annealing. The general theory is then specialized to describe some more advanced local search techniques, namely tabu search and the Kernighan-Lin heuristic
Generalized Rewrite Theories, Coherence Completion and Symbolic Methods
A new notion of generalized rewrite theory
suitable for symbolic reasoning and generalizing the standard notion
is motivated and defined.
Also, new requirements for symbolic executability
of generalized rewrite theories that extend those
for standard rewrite theories, including
a generalized notion of coherence, are given.
Symbolic executability, including coherence,
is both ensured and made available for
a wide class of such theories by
automatable theory transformations.
Using these foundations, several symbolic reasoning methods
using generalized rewrite theories are studied, including:
(i) symbolic description of sets of terms by
pattern predicates; (ii) reasoning about universal reachability properties
by generalized rewriting; (iii) reasoning about existential
reachability properties by constrained narrowing; and (iv) symbolic
verification of safety properties such
as invariants and stability properties.This work has been partially supported by NRL under contract number N00173-17-1-G002.Ope
Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation
AC-completion efficiently handles equality modulo associative and commutative
function symbols. When the input is ground, the procedure terminates and
provides a decision algorithm for the word problem. In this paper, we present a
modular extension of ground AC-completion for deciding formulas in the
combination of the theory of equality with user-defined AC symbols,
uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our
algorithm, called AC(X), is obtained by augmenting in a modular way ground
AC-completion with the canonizer and solver present for the theory X. This
integration rests on canonized rewriting, a new relation reminiscent to
normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is
proved sound, complete and terminating, and is implemented to extend the core
of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized
Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for
publication by LMCS (Logical Methods in Computer Science
Generalized Rewrite Theories and Coherence Completion
A new notion of generalized rewrite theory
suitable for symbolic reasoning and generalizing the standard notion
is motivated and defined.
Also, new requirements for symbolic executability
of generalized rewrite theories that extend those
for standard rewrite theories, including
a generalized notion of coherence, are given.
Finally, symbolic executability, including coherence,
is both ensured and made available for
a wide class of such theories by
automatable theory transformations.Partially supported by by NRL under contract number N00173-17-1-G002.Ope
The fundamental pro-groupoid of an affine 2-scheme
A natural question in the theory of Tannakian categories is: What if you
don't remember \Forget? Working over an arbitrary commutative ring , we
prove that an answer to this question is given by the functor represented by
the \'etale fundamental groupoid \pi_1(\spec(R)), i.e.\ the separable
absolute Galois group of when it is a field. This gives a new definition
for \'etale \pi_1(\spec(R)) in terms of the category of -modules rather
than the category of \'etale covers. More generally, we introduce a new notion
of "commutative 2-ring" that includes both Grothendieck topoi and symmetric
monoidal categories of modules, and define a notion of for the
corresponding "affine 2-schemes." These results help to simplify and clarify
some of the peculiarities of the \'etale fundamental group. For example,
\'etale fundamental groups are not "true" groups but only profinite groups, and
one cannot hope to recover more: the "Tannakian" functor represented by the
\'etale fundamental group of a scheme preserves finite products but not all
products.Comment: 46 pages + bibliography. Diagrams drawn in Tik