31 research outputs found
Disproving in First-Order Logic with Definitions, Arithmetic and Finite Domains
This thesis explores several methods which enable a first-order
reasoner to conclude satisfiability of a formula modulo an
arithmetic theory. The most general method requires restricting
certain quantifiers to range over finite sets; such assumptions
are common in the software verification setting. In addition, the
use of first-order reasoning allows for an implicit
representation of those finite sets, which can avoid
scalability problems that affect other quantified reasoning
methods. These new techniques form a useful complement to
existing methods that are primarily aimed at proving validity.
The Superposition calculus for hierarchic theory combinations
provides a basis for reasoning modulo theories in a first-order
setting. The recent account of âweak abstractionâ and related
improvements make an mplementation of the calculus practical.
Also, for several logical theories of interest Superposition is
an effective decision procedure for the quantifier free fragment.
The first contribution is an implementation of that calculus
(Beagle), including an optimized implementation of Cooperâs
algorithm for quantifier elimination in the theory of linear
integer arithmetic. This includes a novel means of extracting
values
for quantified variables in satisfiable integer problems. Beagle
won an efficiency award at CADE Automated theorem prover System
Competition (CASC)-J7, and won the arithmetic non-theorem
category at CASC-25. This implementation is the start point for
solving the âdisproving with theoriesâ problem.
Some hypotheses can be disproved by showing that, together with
axioms the hypothesis is unsatisfiable. Often this is relative to
other axioms that enrich a base theory by defining new functions.
In that case, the disproof is contingent on the satisfiability of
the enrichment.
Satisfiability in this context is undecidable. Instead, general
characterizations of definition formulas, which do not alter the
satisfiability status of the main axioms, are given. These
general criteria apply to recursive definitions, definitions over
lists, and to arrays. This allows proving some non-theorems which
are otherwise intractable, and justifies similar disproofs of
non-linear arithmetic formulas.
When the hypothesis is contingently true, disproof requires
proving existence of
a model. If the Superposition calculus saturates a clause set,
then a model exists,
but only when the clause set satisfies a completeness criterion.
This requires each
instance of an uninterpreted, theory-sorted term to have a
definition in terms of
theory symbols.
The second contribution is a procedure that creates such
definitions, given that a subset of quantifiers range over finite
sets. Definitions are produced in a counter-example driven way
via a sequence of over and under approximations to the clause
set. Two descriptions of the method are given: the first uses the
component solver modularly, but has an inefficient
counter-example heuristic. The second is more general, correcting
many of the inefficiencies of the first, yet it requires tracking
clauses through a proof. This latter method is shown to apply
also to lists and to problems with unbounded quantifiers.
Together, these tools give new ways for applying successful
first-order reasoning methods to problems involving interpreted
theories
Convalescent plasma in patients admitted to hospital with COVID-19 (RECOVERY): a randomised controlled, open-label, platform trial
SummaryBackground Azithromycin has been proposed as a treatment for COVID-19 on the basis of its immunomodulatoryactions. We aimed to evaluate the safety and efficacy of azithromycin in patients admitted to hospital with COVID-19.Methods In this randomised, controlled, open-label, adaptive platform trial (Randomised Evaluation of COVID-19Therapy [RECOVERY]), several possible treatments were compared with usual care in patients admitted to hospitalwith COVID-19 in the UK. The trial is underway at 176 hospitals in the UK. Eligible and consenting patients wererandomly allocated to either usual standard of care alone or usual standard of care plus azithromycin 500 mg once perday by mouth or intravenously for 10 days or until discharge (or allocation to one of the other RECOVERY treatmentgroups). Patients were assigned via web-based simple (unstratified) randomisation with allocation concealment andwere twice as likely to be randomly assigned to usual care than to any of the active treatment groups. Participants andlocal study staff were not masked to the allocated treatment, but all others involved in the trial were masked to theoutcome data during the trial. The primary outcome was 28-day all-cause mortality, assessed in the intention-to-treatpopulation. The trial is registered with ISRCTN, 50189673, and ClinicalTrials.gov, NCT04381936.Findings Between April 7 and Nov 27, 2020, of 16 442 patients enrolled in the RECOVERY trial, 9433 (57%) wereeligible and 7763 were included in the assessment of azithromycin. The mean age of these study participants was65·3 years (SD 15·7) and approximately a third were women (2944 [38%] of 7763). 2582 patients were randomlyallocated to receive azithromycin and 5181 patients were randomly allocated to usual care alone. Overall,561 (22%) patients allocated to azithromycin and 1162 (22%) patients allocated to usual care died within 28 days(rate ratio 0·97, 95% CI 0·87â1·07; p=0·50). No significant difference was seen in duration of hospital stay (median10 days [IQR 5 to >28] vs 11 days [5 to >28]) or the proportion of patients discharged from hospital alive within 28 days(rate ratio 1·04, 95% CI 0·98â1·10; p=0·19). Among those not on invasive mechanical ventilation at baseline, nosignificant difference was seen in the proportion meeting the composite endpoint of invasive mechanical ventilationor death (risk ratio 0·95, 95% CI 0·87â1·03; p=0·24).Interpretation In patients admitted to hospital with COVID-19, azithromycin did not improve survival or otherprespecified clinical outcomes. Azithromycin use in patients admitted to hospital with COVID-19 should be restrictedto patients in whom there is a clear antimicrobial indication
Proving Infinite Satisfiability
We consider the problem of automatically disproving invalid conjectures over data structures such as lists and arrays over integers, in the presence of additional hypotheses over these data structures. We investigate a simple approach based on refutation
Finite Quantification in Hierarchic Theorem Proving
Abstract. Many applications of automated deduction require reasoning in firstorder logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are âreasonably complete â even in the presence of free function symbols ranging into a background theory sort. In this paper we consider the case when all variables occurring below such function symbols are quantified over a finite subset of their domains. We present a non-naive decision procedure for background theories extended this way on top of black-box decision procedures for the EA-fragment of the background theory. In its core, it employs a model-guided instantiation strategy for obtaining pure background formulas that are equi-satisfiable with the original formula. Unlike traditional finite model finders, it avoids exhaustive instantiation and, hence, is expected to scale better with the size of the domains. Our main results in this paper are a correctness proof and first experimental results.
Structural basis of hAT transposon end recognition by hermes, an octameric dna transposase from musca domestica
Hermes is a member of the hAT transposon superfamily that has active representatives, including McClintock\u27s archetypal Ac mobile genetic element, in many eukaryotic species. The crystal structure of the Hermes transposase-DNA complex reveals that Hermes forms an octameric ring organized as a tetramer of dimers. Although isolated dimers are active in vitro for all the chemical steps of transposition, only octamers are active in vivo. The octamer can provide not only multiple specific DNA-binding domains to recognize repeated subterminal sequences within the transposon ends, which are important for activity, but also multiple nonspecific DNA binding surfaces for target capture. The unusual assembly explains the basis of bipartite DNA recognition at hAT transposon ends, provides a rationale for transposon end asymmetry, and suggests how the avidity provided by multiple sites of interaction could allow a transposase to locate its transposon ends amidst a sea of chromosomal DNA