5,224 research outputs found
User-friendly Support for Common Concepts in a Lightweight Verifier
Machine verification of formal arguments can only increase our confidence in the correctness of those arguments, but the costs of employing machine verification still outweigh the benefits for some common kinds of formal reasoning activities. As a result, usability is becoming increasingly important in the design of formal verification tools. We describe the "aartifact" lightweight verification system, designed for processing formal arguments involving basic, ubiquitous mathematical concepts. The system is a prototype for investigating potential techniques for improving the usability of formal verification systems. It leverages techniques drawn both from existing work and from our own efforts. In addition to a parser for a familiar concrete syntax and a mechanism for automated syntax lookup, the system integrates (1) a basic logical inference algorithm, (2) a database of propositions governing common mathematical concepts, and (3) a data structure that computes congruence closures of expressions involving relations found in this database. Together, these components allow the system to better accommodate the expectations of users interested in verifying formal arguments involving algebraic and logical manipulations of numbers, sets, vectors, and related operators and predicates. We demonstrate the reasonable performance of this system on typical formal arguments and briefly discuss how the system's design contributed to its usability in two case studies
Space-time extensions II
The global extendibility of smooth causal geodesically incomplete spacetimes
is investigated. Denote by one of the incomplete non-extendible causal
geodesics of a causal geodesically incomplete spacetime . First, it
is shown that it is always possible to select a synchronised family of causal
geodesics and an open neighbourhood of a final segment
of in such that is comprised by members of ,
and suitable local coordinates can be defined everywhere on
provided that does not terminate either on a tidal force tensor
singularity or on a topological singularity. It is also shown that if, in
addition, the spacetime, , is globally hyperbolic, and the
components of the curvature tensor, and its covariant derivatives up to order
are bounded on , and also the line integrals of the
components of the -order covariant derivatives are finite along the
members of ---where all the components are meant to be registered with
respect to a synchronised frame field on ---then there exists a
extension so that for each , which
is inextendible in , the image, , is
extendible in . Finally, it is also proved that
whenever does terminate on a topological singularity
cannot be generic.Comment: 42 pages, no figures, small changes to match the published versio
Deciding Confluence and Normal Form Properties of Ground Term Rewrite Systems Efficiently
It is known that the first-order theory of rewriting is decidable for ground
term rewrite systems, but the general technique uses tree automata and often
takes exponential time. For many properties, including confluence (CR),
uniqueness of normal forms with respect to reductions (UNR) and with respect to
conversions (UNC), polynomial time decision procedures are known for ground
term rewrite systems. However, this is not the case for the normal form
property (NFP). In this work, we present a cubic time algorithm for NFP, an
almost cubic time algorithm for UNR, and an almost linear time algorithm for
UNC, improving previous bounds. We also present a cubic time algorithm for CR
The Braid Shelf
The braids of can be equipped with a selfdistributive operation
enjoying a number of deep properties. This text is a
survey of known properties and open questions involving this structure, its
quotients, and its extensions
Certified Context-Free Parsing: A formalisation of Valiant's Algorithm in Agda
Valiant (1975) has developed an algorithm for recognition of context free
languages. As of today, it remains the algorithm with the best asymptotic
complexity for this purpose. In this paper, we present an algebraic
specification, implementation, and proof of correctness of a generalisation of
Valiant's algorithm. The generalisation can be used for recognition, parsing or
generic calculation of the transitive closure of upper triangular matrices. The
proof is certified by the Agda proof assistant. The certification is
representative of state-of-the-art methods for specification and proofs in
proof assistants based on type-theory. As such, this paper can be read as a
tutorial for the Agda system
- …