Constraint specialisation in Horn clause verification

We present a method for specialising the constraints in constrained Horn clauses with respect to a goal. We use abstract interpretation to compute a model of a query-answer transformation of a given set of clauses and a goal. The effect is to propagate the constraints from the goal top-down and propagate answer constraints bottom-up. Our approach does not unfold the clauses at all; we use the constraints from the model to compute a specialised version of each clause in the program. The approach is independent of the abstract domain and the constraints theory underlying the clauses. Experimental results on verification problems show that this is an effective transformation, both in our own verification tools (convex polyhedra analyser) and as a pre-processor to other Horn clause verification tools.

We address the problem of verifying that the functions of a program meet their contracts, specified by pre/postconditions. We follow an approach based on constrained Horn clauses (CHCs) by which the verification problem is reduced to the problem of checking satisfiability of a set of clauses derived from the given program and contracts. We consider programs that manipulate algebraic data types (ADTs) and a class of contracts specified by catamorphisms, that is, functions defined by simple recursion schemata on the given ADTs. We show by several examples that state-of-the-art CHC satisfiability tools are not effective at solving the satisfiability problems obtained by direct translation of the contracts into CHCs. To overcome this difficulty, we propose a transformation technique that removes the ADT terms from CHCs and derives new sets of clauses that work on basic sorts only, such as integers and booleans. Thus, when using the derived CHCs there is no need for induction rules on ADTs. We prove that the transformation is sound, that is, if the derived set of CHCs is satisfiable, then so is the original set. We also prove that the transformation always terminates for the class of contracts specified by catamorphisms. Finally, we present the experimental results obtained by an implementation of our technique when verifying many non-trivial contracts for ADT manipulating programs.

This work addresses the problem of verifying imperative programs that manipulate data structures, e.g., Rust programs. Data structures are usually modeled by Algebraic Data Types (ADTs) in verification conditions. Inductive invariants of such programs often require recursively defined functions (RDFs) to represent abstractions of data structures. From the logic perspective, this reduces to solving Constrained Horn Clauses (CHCs) modulo both ADT and RDF. The underlying logic with RDFs is undecidable. Thus, even verifying a candidate inductive invariant is undecidable. Similarly, IC3-based algorithms for solving CHCs lose their progress guarantee: they may not find counterexamples when the program is unsafe. We propose a novel IC3-inspired algorithm Racer for solving CHCs modulo ADT and RDF (i.e., automatically synthesizing inductive invariants, as opposed to only verifying them as is done in deductive verification). Racer ensures progress despite the undecidability of the underlying theory, and is guaranteed to terminate with a counterexample for unsafe programs. It works with a general class of RDFs over ADTs called catamorphisms. The key idea is to represent catamorphisms as both CHCs, via relationification , and RDFs, using novel abstractions . Encoding catamorphisms as CHCs allows learning inductive properties of catamorphisms, as well as preserving unsatisfiabilty of the original CHCs despite the use of RDF abstractions, whereas encoding catamorphisms as RDFs allows unfolding the recursive definition, and relying on it in solutions. Abstractions ensure that the underlying theory remains decidable. We implement our approach in Z3 and show that it works well in practice.

We consider constrained Horn clause solving from the more general point of view of solving formula equations. Constrained Horn clauses correspond to the subclass of Horn formula equations. We state and prove a fixed-point theorem for Horn formula equations which is based on expressing the fixed-point computation of a minimal model of a set of Horn clauses on the object level as a formula in first-order logic with a least fixed point operator. We describe several corollaries of this fixed-point theorem, in particular concerning the logical foundations of program verification, and sketch how to generalise it to incorporate abstract interpretations.

Catamorphisms are functions that are recursively defined on list and trees and, in general, on algebraic data types (ADTs), and are often used to compute suitable abstractions of programs that manipulate ADTs. Examples of catamorphisms include functions that compute size of lists, orderedness of lists, and height of trees. It is well known that program properties specified through catamorphisms can be proved by showing the satisfiability of suitable sets of constrained Horn clauses (CHCs). We address the problem of checking the satisfiability of those sets of CHCs, and we propose a method for transforming sets of CHCs into equisatisfiable sets where catamorphisms are no longer present. As a consequence, clauses with catamorphisms can be handled without extending the satisfiability algorithms used by existing CHC solvers. Through an experimental evaluation on a nontrivial benchmark consisting of many list and tree processing algorithms expressed as sets of CHCs, we show that our technique is indeed effective and significantly enhances the performance of state-of-the-art CHC solvers.

We present Ultimate TreeAutomizer, a solver for satisfiability of sets of\nconstrained Horn clauses. Constrained Horn clauses (CHC) are a fragment of\nfirst order logic with attractive properties in terms of expressiveness and\naccessibility to algorithmic solving. Ultimate TreeAutomizer is based on the\ntechniques of trace abstraction, tree automata and tree interpolation. This\npaper serves as a tool description for TreeAutomizer in CHC-COMP 2019.\n

Quantifier elimination is used in various automated reasoning tasks, including quantified SMT solving, exists/forall solving, program synthesis, model checking, and constrained Horn clause (CHC) solving. Complete quantifier elimination, however, is computationally intractable for many theories. The recent algorithm QEL shows a promising approach to approximate quantifier elimination, which has resulted in improvements in solver performance. QEL performs partial quantifier elimination with a completeness guarantee that depends on a certain semantic property of the given formula. Considerably generalizing the previous approach, we identify a subclass of local theories in which partial quantifier elimination can be performed efficiently. We present $$\mathcal {T}$$ T -QEL a parametrized polynomial time algorithm that is a sound extension of QEL and is relatively complete for this class of theories. The algorithm utilizes the proof theoretic characterization of the theories, which is based on restricted derivations. Finally, we prove for $$\mathcal {T}$$ T -QEL, soundness in general, and relative completeness with respect to the identified class of theories.

The problem of synthesizing specifications of undefined procedures has a broad range of applications, but the usefulness of the generated specifications depends on their quality. In this paper, we propose a technique for finding maximal and non-vacuous specifications. Maximality allows for more choices for implementations of undefined procedures, and non-vacuity ensures that safety assertions are reachable. To handle programs with complex control flow, our technique discovers not only specifications but also inductive invariants. Our iterative algorithm lazily generalizes non-vacuous specifications in a counterexample-guided loop. The key component of our technique is an effective non-vacuous specification synthesis algorithm. We have implemented the approach in a tool called HornSpec, taking as input systems of constrained Horn clauses. We have experimentally demonstrated the tool's effectiveness, efficiency, and the quality of generated specifications on a range of benchmarks.

We present a verification technique for program safety that combines Iterated Specialization and Interpolating Horn Clause Solving. Our new method composes together these two techniques in a modular way by exploiting the common Horn Clause representation of the verification problem. The Iterated Specialization verifier transforms an initial set of verification conditions by using unfold/fold equivalence preserving transformation rules. During transformation, program invariants are discovered by applying widening operators. Then the output set of specialized verification conditions is analyzed by an Interpolating Horn Clause solver, hence adding the effect of interpolation to the effect of widening. The specialization and interpolation phases can be iterated, and also combined with other transformations that change the direction of propagation of the constraints (forward from the program preconditions or backward from the error conditions). We have implemented our verification technique by integrating the VeriMAP verifier with the FTCLP Horn Clause solver, based on Iterated Specialization and Interpolation, respectively. Our experimental results show that the integrated verifier improves the precision of each of the individual components run separately.

The expressiveness of First Order Logic (FOL) languages is severely counterbalanced by the complexity of matching formulas on a universe. Matching is an instance of the class of Constraint Satisfaction Problems (CSP), which have shown to undergo a phase transition with respect to two order parameters: constraint density and constraint tightness. This paper analyzes the problem of satisfying FOL Horn clauses in the light of these recent results. By means of an extensive experimental analysis, we show how Horn clause verification exhibits a typical phase transition with respect to the number of binary (or greater arity) predicates, and with respect to the ratio between the number of constants in the universe and the cardinality of the basic predicates extension.

Smart contracts are tempting targets of attacks, as they often hold and manipulate significant financial assets, are immutable after deployment, and have publicly available source code, with assets estimated in the order of millions of dollars being lost in the past due to vulnerabilities. Formal verification is thus a necessity, but smart contracts challenge the existing highly efficient techniques routinely applied in the symbolic verification of software, due to specificities not present in general programming languages. A common feature of existing works in this area is the attempt to reuse off-the-shelf verification tools designed for general programming languages. This reuse can lead to inefficiency and potentially unsound results, as domain translation is required. In this article, we describe a carefully crafted approach that directly models the central aspects of smart contracts natively, going from the contract to its logical representation without intermediary steps. We use the expressive and highly automatable logic of constrained Horn clauses for modeling and instantiate our approach to the Solidity language. A tool implementing our approach, called Solicitous , was developed and integrated into the SMTChecker module of the Solidity compiler solc. We evaluated our approach on an extensive benchmark set containing 22,446 real-world smart contracts deployed on the Ethereum blockchain over a 27-month period. The results show that our approach is able to establish safety of significantly more contracts than comparable, publicly available verification tools, with an order of magnitude increase in the percentage of formally verified contracts.

Tamarin is a mature, state-of-the-art tool for cryptographic protocol verification. We survey some of the larger tour de force results achieved and show how Tamarin can formalize protocols, adversary models, and properties, and scale to substantial, real world, verification problems.

We develop a practical solution to the problem of automatic verification of the interface between device drivers and the operating system. Our solution relies on a combination of improved driver architecture and verification tools. Unlike previous proposals for verification-friendly drivers, our methodology supports drivers written in C and can be implemented in any existing OS. Our Linuxbased evaluation shows that this methodology amplifies the power of existing model checking tools in detecting driver bugs, making it possible to verify properties that are beyond the reach of traditional techniques.

The abstraction and over-approximation of protocols and web services by a set of Horn clauses is a very successful method in practice. It has however limitations for protocols and web services that are based on databases of keys, contracts, or even access rights, where revocation is possible, so that the set of true facts does not monotonically grow with state transitions. We extend the scope of these over-approximation methods by defining a new way of abstraction that can handle such databases, and we formally prove that the abstraction is sound. We realize a translator from a convenient specification language to standard Horn clauses and use the verifier ProVerif and the theorem prover SPASS to solve them. We show by a number of examples that this approach is practically feasible for wide variety of verification problems of security protocols and web services

In this paper we present InterHorn, a solver for recursion-free Horn clauses. The main application domain of InterHorn lies in solving interpolation problems arising in software verification. We show how a range of interpolation problems, including path, transition, nested, state/transition and well-founded interpolation can be handled directly by InterHorn. By detailing these interpolation problems and their Horn clause representations, we hope to encourage the emergence of a common back-end interpolation interface useful for diverse verification tools.