CPP 2016

The 5th ACM SIGPLAN Conference on

Certified Programs and Proofs

in cooperation with ACM SIGLOG

Saint Petersburg, Florida, USA - January 18-19, 2016

# CPP 2016 Accepted Papers with Abstracts

Refinement Based Verification of Imperative Data Structures

**Abstract:**In this paper we present a stepwise refinement based top-down approach to

verified imperative data structures.

Our approach is modular in the sense that already verified data structures

can be used for construction of more complex data structures.

Moreover, our data structures can be used to verify real algorithms.

Our tool chain supports refinement down to executable code in various

programming languages, and is fully implemented

in Isabelle/HOL, such that its trusted code base is only the inference kernel and

code generator of Isabelle/HOL.

As a case study, we verify an indexed heap data structure, and use

it to generate an efficient verified implementation of Dijkstra's algorithm.

Formal proofs of transcendence for e and pi as an application of multivariate and symmetric polynomials

**Abstract:**We describe the formalisation in Coq of a proof that

the numbers e and pi are transcendental. This is the

first formalized proof of transcendance for pi.

This proof lies at

the interface of two domains of mathematics that are often considered

separately: calculus (real and elementary complex analysis) and

algebra. For the work on calculus, we rely on the Coquelicot library

and for the work on algebra, we rely on the Mathematical Components

library. Moreover, some of the elements of formalized proof originate

in the more ancient library for real numbers included in the Coq

distribution. The case of pi relied extensively on properties

of multivariate polynomials and this experiment was also an occasion

to put to test a newly developed library for these multivariate

polynomials. In the paper,

we describe the various original aspects of the work: orchestrating a

development based on several libraries, designing a new library for

multivariate polynomials, and specialized developments performed

solely for this proof. Apart from achieving a landmark result, this

work also teaches lessons about the evolution of modern formalized

mathematics.

Planning for Change in a Formal Verification of the Raft Consensus Protocol

**Abstract:**We present the first formal verification of the Raft consensus protocol,

a critical component of many distributed systems. We proved that our

implementation is a linearizable replicated state machine, which

required iteratively discovering and proving nSysInv system invariants.

Our verified implementation can be extracted and run on real networks.

The primary challenge we faced during the verification process was proof

maintenance, since proving one invariant often required strengthening

and updating other parts of our proof. To address this challenge, we

propose a methodology of planning for change during verification. Our

methodology adapts classical information hiding techniques to the

context of proof assistants, factors out common invariant strengthening

patterns into custom induction principles, proves higher-order lemmas

that show any property proved about a particular component imply

analogous properties about related components, and makes proofs robust

to change using structural tactics. We also discuss how our methodology

may be applied to systems verification more broadly.

A Modular, Efficient Formalisation of Real Algebraic Numbers

**Abstract:**This paper shows a construction of real algebraic numbers with executable arithmetic operations in Isabelle/HOL. Instead of verified resultants, arithmetic operations on real algebraic numbers are based on a decision procedure to decide the sign of a bivariate polynomial at real algebraic point. The modular design allows the safe use of fast external code. This work can be the basis for decision procedures that rely on real algebraic numbers.

Formalization of a Newton series representation of polynomials

**Abstract:**We formalize an algorithm to change the representation of a polynomial to a Newton power series. This provides a way to compute efficiently polynomials which roots are the sums or products of roots of other polynomials, and hence provide a base component of efficient computation for algebraic numbers. In order to achieve this, we formalize a notion of partial power series and develop an abstract theory of poles of fractions.

Axiomatic Semantics for Compiler Verification

**Abstract:**Based on constructive type theory, we study two idealized imperative languages GC and IC and verify the correctness of a compiler from GC to IC. GC is a guarded command language with underspecified execution order defined with an axiomatic semantics. IC is a deterministic low-level language with linear sequential composition and lexically scoped gotos defined with a small-step semantics. We characterize IC with an axiomatic semantics and prove that the compiler from GC to IC preserves specifications. The axiomatic semantics we consider model total correctness and map programs to continuous predicate transformers. We define the axiomatic semantics of GC and IC with elementary inductive predicates and show that the predicate transformer described by a program can be obtained compositionally by recursion on the syntax of the program using a fixed point operator for loops and continuations. We also show that two IC programs are contextually equivalent if and only if their predicate transformers are equivalent.

A Nominal Exploration of Intuitionism

**Abstract:**This papers extends the Nuprl proof assistant (a system representative

of the class of extensional type theories a la Martin-Lof) with

named exceptions and handlers, as well as a nominal

fresh operator. Using these new features, we prove a version

of Brouwer's Continuity Principle for numbers. We also provide a

simpler proof of a weaker version of this principle that only uses

diverging terms. We prove these two principles in Nuprl's meta-theory

using our formalization of Nuprl in Coq and show how we can reflect

these meta-theoretical results in the Nuprl theory as derivation

rules. We also show that these additions preserve Nuprl's key

meta-theoretical properties, in particular consistency and the

congruence of Howe's computational equivalence relation. Using

continuity and the fan theorem we prove important results of

Intuitionistic Mathematics: Brouwer's continuity theorem and bar

induction on monotone bars.

A Verified Algorithm for Detecting Conflicts in XACML Access Control Rules

**Abstract:**We describe the formalization of a correctness proof for a conflict detection algorithm for XACML (eXtensible Access Control Markup Language). XACML is a standardized declarative access control policy language that is increasingly used in industry. In practice it is common for rule sets to grow large, and contain unintended errors, often due to conflicting rules. A conflict occurs in a policy when one rule permits a request and another denies that same request. Such errors can lead to serious risks involving both allowing access to an unauthorized user as well as denying access to someone who needs it. Removing conflicts is thus an important aspect of debugging policies, and the use of a verified algorithm provides the highest assurance in a domain where security is important. In this paper, we focus on several complex XACML constructs, including time ranges and integer intervals, as well as ways to combine any number of functions using the boolean operators and, or, and not. The latter are the most complex, and add significant expressive power to the language. We propose an algorithm to find conflicts and then use the Coq Proof Assistant to prove the algorithm correct. We develop a library of tactics to help automate the proof.

A Unified Coq Framework for Verifying C Programs with Floating-Point Computations

**Abstract:**We provide concrete evidence that floating-point computations in C

programs can be verified in a homogeneous verification setting based on

Coq only, by evaluating the practicality of the combination of the

formal semantics of CompCert Clight and the Flocq formal specification

of IEEE 754 floating-point arithmetic for the verification of

properties of floating-point computations in C programs. To this end,

we develop a framework to automatically compute real-number

expressions of C floating-point computations with rounding error terms

along with their correctness proofs. We apply our framework to the

complete analysis of an energy-efficient C implementation of a radar

image processing algorithm, for which we provide a certified bound on

the total noise introduced by floating-point rounding errors and

energy-efficient approximations of square root and sine.

Formal Verification of Control-flow Graph Flattening

**Abstract:**Code obfuscation is emerging as a key asset in security by obscurity. It aims at hiding sensitive

information in programs so that they become more difficult to understand and reverse engineer.

Since the results on the impossibility of perfect and universal obfuscation, many obfuscation

techniques have been proposed in the literature, ranging from simple variable encoding to hiding

the control-flow of a program.

In this paper, we formally verify in Coq an advanced code obfuscation called control-flow graph

flattening, that is used in state-of-the-art program obfuscators. Our control-flow graph flattening

is a program transformation operating over C programs, that is integrated into the CompCert

formally verified compiler. The semantics preservation proof of our program obfuscator relies on

a simulation proof performed on a realistic language, the Clight language of CompCert.

The automatic extraction of our program obfuscator into OCaml yields a program with competitive

results.

Higher-order Representation Predicates in Separation Logic

**Abstract:**In Separation Logic, representation predicates are used to describe mutable data structures, by establishing a relationship between the entry point of the structure, the piece of heap over which this structure spans, and the logical model associated with the structure. When a data structure is polymorphic, such as in the case of a container, its representation predicate needs to be parameterized not just by the type of the items stored in the structure, but also by the representation predicates associated with these items. Such higher-order representation predicates can be used in particular to control whether containers should own their items. In this paper, we present, through a collection of practical examples, solutions to the challenges associated with reasoning about accesses into data structures that own their elements.

Bisimulation Up-to Techniques for Psi-calculi

**Abstract:**Psi-calculi is a parametric framework for process calculi similar to

popular pi-calculus extensions such as the explicit fusion calculus,

the applied pi-calculus and the spi calculus. Remarkably, machine-

checked proofs of standard algebraic and congruence properties of

bisimilarity apply to all calculi within the framework.

Bisimulation up-to techniques are methods for reducing the

size of relations needed in bisimulation proofs. In this paper, we

show how these bisimulation proof methods can be adapted to psi-

calculi. We formalise all our definitions and theorems in Nominal

Isabelle, and show examples where the use of up to-techniques

yields drastically simplified proofs of known results. We also prove

new structural laws about the replication operator.

The Vampire and the FOOL

**Abstract:**This paper presents new features recently implemented in the theorem prover Vampire, namely support for first-order logic with a first class boolean sort (FOOL) and polymorphic arrays. In addition to having a first class boolean sort, FOOL also contains if-then-else and let-in expressions. We argue that presented extensions facilitate reasoning-based program analysis, both by increasing the expressivity of first-order reasoners and by gains in efficiency.

Improving automation in interactive theorem provers by efficient encoding of lambda-abstractions

**Abstract:**Hammers are tools for empolying external automatic theorem provers (ATPs) to improve automation in formal proof assistants. Strong automation can greatly ease the task of developing formal proofs. An essential component of any hammer is the translation of the logic of a proof assistant to the format of an ATP. Formalisms of state-of-the-art ATPs are usually first-order, so some method for eliminating lambda-abstractions is needed. We present an experimental comparison of several combinatory abstraction algorithms for HOL(y)Hammer -- a hammer for HOL Light. The algorithms are compared on the HOL Light standard library extended with a library for multivariate analysis. We succeeded in developing algorithms which outperform both lambda-lifting and the simple Schonfinkel's algorithm used in Sledgehammer for Isabelle/HOL. This visibly improves the ATPs' success rate on translated problems, thus enhancing automation in proof assistants.

Mizar Environment for Isabelle

**Abstract:**The Isabelle/Isar language has been heavily inspired by the Mizar style, however already from the beginning it has been different in many ways and it has been evolving in different directions than the syntax the Mizar language. These differences were mostly motivated by the particular needs of integration with Isabelle at a particular time, in particular in order to make various proof tactics and other techniques associated with the LCF style available.

In this paper we explore the possibility of emulating the Mizar environment as close as possible inside the Isabelle logical framework. We introduce adaptations to the Isabelle/FOL object logic that correspond to the meta-logic of Mizar, as well as Isar inner syntax notations that correspond to these of the Mizar language. We show how meta-level types can be used to differenciate between the syntactic categories of the Mizar language, such as sets and Mizar types including modes and attributes, and show how they interact with the basic constructs of the Tarski-Groethendieck set theory. We discuss Mizar definitions and provide simple abbreviations that allow the introduction of Mizar predicates, functions, attributes and modes using the Pure definition syntax. We finally consider the definite and indefinite description operators in Mizar and their use to introduce definitions by "means" and "equals". We demonstrate the usability of the environment on a sample Mizar-style formalization, with cluster inferences and "by" steps performed manually.

Formalizing Jordan Normal Forms in Isabelle/HOL

**Abstract:**In automated complexity analysis of term rewriting, estimating

the growth rate of the values in A n -- for a fixed matrix A and

increasing n -- is of fundamental interest. This growth rate can

be exactly characterized via A’s Jordan normal form (JNF). We

formalized this result in our library IsaFoR and our certifier CeTA,

and thereby improve the support for certifying polynomial bounds

derived by (untrusted) complexity analysis tools.

To this end, we develop a new library for matrices that admits

to conveniently work with block matrices. Besides the mentioned

complexity result, we formalize Gram-Schmidt’s orthogonalization

algorithm and Schur decomposition in order to prove existence of

JNFs. We also provide a uniqueness result for JNFs which allows

us to compute Jordan blocks for individual eigenvalues. In order to

determine eigenvalues automatically, we moreover formalize Yun’s

square-free factorization algorithm.

A Logic of Proofs for Differential Dynamic Logic

**Abstract:**Differential dynamic logic is a logic for specifying and verifying safety, liveness, and other properties about models of cyber-physical systems. Theorem provers based on differential dynamic logic have been used to verify safety properties for models of self-driving cars and collision avoidance protocols for aircraft. Unfortunately, these theorem provers do not have explicit proof terms, which makes the implementation of a number of important features unnecessarily complicated without soundness-critical and extra logical extensions to the theorem prover. Examples include: an unambiguous separation between proof checking and proof search, the ability to extract program traces corresponding to counter-examples, and synthesis of surely-live deterministic programs from liveness proofs for nondeterministic programs.

This paper presents a differential dynamic logic with such an explicit representation of proofs. The resulting logic extends both the syntax and semantics of differential dynamic logic with proof terms -- syntactic representations of logical deductions. To support axiomatic theorem proving, the logic allows equivalence rewriting deep within formulas and supports both uniform renaming and uniform substitutions.

Constructing the Propositional Truncation using Non-recursive HITs

**Abstract:**In homotopy type theory, we construct the propositional truncation as a colimit, using only non-recursive higher inductive types (HITs). This is a first step towards reducing recursive HITs to non-recursive HITs. This construction gives a characterization of functions from the propositional truncation to an arbitrary type, extending the universal property of the propositional truncation. We have fully formalized all results in a new proof assistant, Lean.