32,929 research outputs found
Proof Generation from Delta-Decisions
We show how to generate and validate logical proofs of unsatisfiability from
delta-complete decision procedures that rely on error-prone numerical
algorithms. Solving this problem is important for ensuring correctness of the
decision procedures. At the same time, it is a new approach for automated
theorem proving over real numbers. We design a first-order calculus, and
transform the computational steps of constraint solving into logic proofs,
which are then validated using proof-checking algorithms. As an application, we
demonstrate how proofs generated from our solver can establish many nonlinear
lemmas in the the formal proof of the Kepler Conjecture.Comment: Appeared in SYNASC'1
Verifying Safety Properties With the TLA+ Proof System
TLAPS, the TLA+ proof system, is a platform for the development and
mechanical verification of TLA+ proofs written in a declarative style requiring
little background beyond elementary mathematics. The language supports
hierarchical and non-linear proof construction and verification, and it is
independent of any verification tool or strategy. A Proof Manager uses backend
verifiers such as theorem provers, proof assistants, SMT solvers, and decision
procedures to check TLA+ proofs. This paper documents the first public release
of TLAPS, distributed with a BSD-like license. It handles almost all the
non-temporal part of TLA+ as well as the temporal reasoning needed to prove
standard safety properties, in particular invariance and step simulation, but
not liveness properties
A formally verified proof of the prime number theorem
The prime number theorem, established by Hadamard and de la Vall'ee Poussin
independently in 1896, asserts that the density of primes in the positive
integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of
the methods of complex analysis, elementary proofs were provided by Selberg and
Erd"os in 1948. We describe a formally verified version of Selberg's proof,
obtained using the Isabelle proof assistant.Comment: 23 page
Affine functions and series with co-inductive real numbers
We extend the work of A. Ciaffaglione and P. Di Gianantonio on mechanical
verification of algorithms for exact computation on real numbers, using
infinite streams of digits implemented as co-inductive types. Four aspects are
studied: the first aspect concerns the proof that digit streams can be related
to the axiomatized real numbers that are already axiomatized in the proof
system (axiomatized, but with no fixed representation). The second aspect
re-visits the definition of an addition function, looking at techniques to let
the proof search mechanism perform the effective construction of an algorithm
that is correct by construction. The third aspect concerns the definition of a
function to compute affine formulas with positive rational coefficients. This
should be understood as a testbed to describe a technique to combine
co-recursion and recursion to obtain a model for an algorithm that appears at
first sight to be outside the expressive power allowed by the proof system. The
fourth aspect concerns the definition of a function to compute series, with an
application on the series that is used to compute Euler's number e. All these
experiments should be reproducible in any proof system that supports
co-inductive types, co-recursion and general forms of terminating recursion,
but we performed with the Coq system [12, 3, 14]
A Survey of Satisfiability Modulo Theory
Satisfiability modulo theory (SMT) consists in testing the satisfiability of
first-order formulas over linear integer or real arithmetic, or other theories.
In this survey, we explain the combination of propositional satisfiability and
decision procedures for conjunctions known as DPLL(T), and the alternative
"natural domain" approaches. We also cover quantifiers, Craig interpolants,
polynomial arithmetic, and how SMT solvers are used in automated software
analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest,
Romania. 201
Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations
We present a formal tool for verification of multivariate nonlinear
inequalities. Our verification method is based on interval arithmetic with
Taylor approximations. Our tool is implemented in the HOL Light proof assistant
and it is capable to verify multivariate nonlinear polynomial and
non-polynomial inequalities on rectangular domains. One of the main features of
our work is an efficient implementation of the verification procedure which can
prove non-trivial high-dimensional inequalities in several seconds. We
developed the verification tool as a part of the Flyspeck project (a formal
proof of the Kepler conjecture). The Flyspeck project includes about 1000
nonlinear inequalities. We successfully tested our method on more than 100
Flyspeck inequalities and estimated that the formal verification procedure is
about 3000 times slower than an informal verification method implemented in
C++. We also describe future work and prospective optimizations for our method.Comment: 15 page
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