405 research outputs found
Extending Coq with Imperative Features and its Application to SAT Verification
This work was supported in part by the french ANR DECERT initiativeInternational audienceCoq has within its logic a programming language that can be used to replace many deduction steps into a single computation, this is the so-called reflection. In this paper, we present two extensions of the evaluation mechanism that preserve its correctness and make it possible to deal with cpu-intensive tasks such as proof checking of SAT traces
Extending SMTCoq, a Certified Checker for SMT (Extended Abstract)
This extended abstract reports on current progress of SMTCoq, a communication
tool between the Coq proof assistant and external SAT and SMT solvers. Based on
a checker for generic first-order certificates implemented and proved correct
in Coq, SMTCoq offers facilities both to check external SAT and SMT answers and
to improve Coq's automation using such solvers, in a safe way. Currently
supporting the SAT solver zChaff, and the SMT solver veriT for the combination
of the theories of congruence closure and linear integer arithmetic, SMTCoq is
meant to be extendable with a reasonable amount of effort: we present work in
progress to support the SMT solver CVC4 and the theory of bit vectors.Comment: In Proceedings HaTT 2016, arXiv:1606.0542
Applying Formal Methods to Networking: Theory, Techniques and Applications
Despite its great importance, modern network infrastructure is remarkable for
the lack of rigor in its engineering. The Internet which began as a research
experiment was never designed to handle the users and applications it hosts
today. The lack of formalization of the Internet architecture meant limited
abstractions and modularity, especially for the control and management planes,
thus requiring for every new need a new protocol built from scratch. This led
to an unwieldy ossified Internet architecture resistant to any attempts at
formal verification, and an Internet culture where expediency and pragmatism
are favored over formal correctness. Fortunately, recent work in the space of
clean slate Internet design---especially, the software defined networking (SDN)
paradigm---offers the Internet community another chance to develop the right
kind of architecture and abstractions. This has also led to a great resurgence
in interest of applying formal methods to specification, verification, and
synthesis of networking protocols and applications. In this paper, we present a
self-contained tutorial of the formidable amount of work that has been done in
formal methods, and present a survey of its applications to networking.Comment: 30 pages, submitted to IEEE Communications Surveys and Tutorial
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of by suprema of quadratic forms with a well
chosen curvature. Thus, we reduce the initial goal to a hierarchy of
semialgebraic optimization problems, solved by sums of squares relaxations. Our
implementation tool interleaves semialgebraic approximations with sums of
squares witnesses to form certificates. It is interfaced with Coq and thus
benefits from the trusted arithmetic available inside the proof assistant. This
feature is used to produce, from the certificates, both valid underestimators
and lower bounds for each approximated constituent. The application range for
such a tool is widespread; for instance Hales' proof of Kepler's conjecture
yields thousands of multivariate transcendental inequalities. We illustrate the
performance of our formal framework on some of these inequalities as well as on
examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table
Implementing and reasoning about hash-consed data structures in Coq
We report on four different approaches to implementing hash-consing in Coq
programs. The use cases include execution inside Coq, or execution of the
extracted OCaml code. We explore the different trade-offs between faithful use
of pristine extracted code, and code that is fine-tuned to make use of OCaml
programming constructs not available in Coq. We discuss the possible
consequences in terms of performances and guarantees. We use the running
example of binary decision diagrams and then demonstrate the generality of our
solutions by applying them to other examples of hash-consed data structures
From LCF to Isabelle/HOL
Interactive theorem provers have developed dramatically over the past four
decades, from primitive beginnings to today's powerful systems. Here, we focus
on Isabelle/HOL and its distinctive strengths. They include automatic proof
search, borrowing techniques from the world of first order theorem proving, but
also the automatic search for counterexamples. They include a highly readable
structured language of proofs and a unique interactive development environment
for editing live proof documents. Everything rests on the foundation conceived
by Robin Milner for Edinburgh LCF: a proof kernel, using abstract types to
ensure soundness and eliminate the need to store proofs. Compared with the
research prototypes of the 1970s, Isabelle is a practical and versatile tool.
It is used by system designers, mathematicians and many others
Itauto: An Extensible Intuitionistic SAT Solver
We present the design and implementation of itauto, a Coq reflexive tactic for intuitionistic propositional logic. The tactic inherits features found in modern SAT solvers: definitional conjunctive normal form; lazy unit propagation and conflict driven backjumping. Formulae are hash-consed using native integers thus enabling a fast equality test and a pervasive use of Patricia Trees. We also propose a hybrid proof by reflection scheme whereby the extracted solver calls user-defined tactics on the leaves of the propositional proof search thus enabling theory reasoning and the generation of conflict clauses. The solver has decent efficiency and is more scalable than existing tactics on synthetic benchmarks and preliminary experiments are encouraging for existing developments
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
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