97 research outputs found
Building Decision Procedures in the Calculus of Inductive Constructions
It is commonly agreed that the success of future proof assistants will rely
on their ability to incorporate computations within deduction in order to mimic
the mathematician when replacing the proof of a proposition P by the proof of
an equivalent proposition P' obtained from P thanks to possibly complex
calculations. In this paper, we investigate a new version of the calculus of
inductive constructions which incorporates arbitrary decision procedures into
deduction via the conversion rule of the calculus. The novelty of the problem
in the context of the calculus of inductive constructions lies in the fact that
the computation mechanism varies along proof-checking: goals are sent to the
decision procedure together with the set of user hypotheses available from the
current context. Our main result shows that this extension of the calculus of
constructions does not compromise its main properties: confluence, subject
reduction, strong normalization and consistency are all preserved
Coq Modulo Theory - Short Paper
International audienceCoq Modulo Theory (CoqMT) is an extension of the Coq proof assistant incorporating, in its computational mechanism, validity entailment for user-defined first-order equational theories. Such a mechanism strictly enriches the system (more terms are typable), eases the use of dependent types and provides more automation during the development of proofs. CoqMT improves over the Calculus of Congruent Inductive Constructions by getting rid of various restrictions and simplifying the type-checking algorithm and the integration of first-order decision procedures
Certification of SAT Solvers in Coq
International audienceWe describe a fully portable, open source certifier for traces of SAT problems produced by zChaff. It can also be easily adapted for MiniSat, PicoSat or BooleForce, and we have done it for PicoSat. Our certifier has been developed with the proof assistant Coq. We give some figures based on the pigeon hole, comparing both PicoSat and zChaff on the one hand, and our certifier with another certifier also developed with Coq
Computer-aided proofs for multiparty computation with active security
Secure multi-party computation (MPC) is a general cryptographic technique
that allows distrusting parties to compute a function of their individual
inputs, while only revealing the output of the function. It has found
applications in areas such as auctioning, email filtering, and secure
teleconference. Given its importance, it is crucial that the protocols are
specified and implemented correctly. In the programming language community it
has become good practice to use computer proof assistants to verify correctness
proofs. In the field of cryptography, EasyCrypt is the state of the art proof
assistant. It provides an embedded language for probabilistic programming,
together with a specialized logic, embedded into an ambient general purpose
higher-order logic. It allows us to conveniently express cryptographic
properties. EasyCrypt has been used successfully on many applications,
including public-key encryption, signatures, garbled circuits and differential
privacy. Here we show for the first time that it can also be used to prove
security of MPC against a malicious adversary. We formalize additive and
replicated secret sharing schemes and apply them to Maurer's MPC protocol for
secure addition and multiplication. Our method extends to general polynomial
functions. We follow the insights from EasyCrypt that security proofs can be
often be reduced to proofs about program equivalence, a topic that is well
understood in the verification of programming languages. In particular, we show
that in the passive case the non-interference-based definition is equivalent to
a standard game-based security definition. For the active case we provide a new
NI definition, which we call input independence
From formal proofs to mathematical proofs: a safe, incremental way for building in first-order decision procedures
We investigate here a new version of the Calculus of Inductive Constructions
(CIC) on which the proof assistant Coq is based: the Calculus of Congruent
Inductive Constructions, which truly extends CIC by building in arbitrary
first-order decision procedures: deduction is still in charge of the CIC
kernel, while computation is outsourced to dedicated first-order decision
procedures that can be taken from the shelves provided they deliver a proof
certificate. The soundness of the whole system becomes an incremental property
following from the soundness of the certificate checkers and that of the
kernel. A detailed example shows that the resulting style of proofs becomes
closer to that of the working mathematician
A Relational Logic for Higher-Order Programs
Relational program verification is a variant of program verification where
one can reason about two programs and as a special case about two executions of
a single program on different inputs. Relational program verification can be
used for reasoning about a broad range of properties, including equivalence and
refinement, and specialized notions such as continuity, information flow
security or relative cost. In a higher-order setting, relational program
verification can be achieved using relational refinement type systems, a form
of refinement types where assertions have a relational interpretation.
Relational refinement type systems excel at relating structurally equivalent
terms but provide limited support for relating terms with very different
structures.
We present a logic, called Relational Higher Order Logic (RHOL), for proving
relational properties of a simply typed -calculus with inductive types
and recursive definitions. RHOL retains the type-directed flavour of relational
refinement type systems but achieves greater expressivity through rules which
simultaneously reason about the two terms as well as rules which only
contemplate one of the two terms. We show that RHOL has strong foundations, by
proving an equivalence with higher-order logic (HOL), and leverage this
equivalence to derive key meta-theoretical properties: subject reduction,
admissibility of a transitivity rule and set-theoretical soundness. Moreover,
we define sound embeddings for several existing relational type systems such as
relational refinement types and type systems for dependency analysis and
relative cost, and we verify examples that were out of reach of prior work.Comment: Submitted to ICFP 201
Proving uniformity and independence by self-composition and coupling
Proof by coupling is a classical proof technique for establishing
probabilistic properties of two probabilistic processes, like stochastic
dominance and rapid mixing of Markov chains. More recently, couplings have been
investigated as a useful abstraction for formal reasoning about relational
properties of probabilistic programs, in particular for modeling
reduction-based cryptographic proofs and for verifying differential privacy. In
this paper, we demonstrate that probabilistic couplings can be used for
verifying non-relational probabilistic properties. Specifically, we show that
the program logic pRHL---whose proofs are formal versions of proofs by
coupling---can be used for formalizing uniformity and probabilistic
independence. We formally verify our main examples using the EasyCrypt proof
assistant
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