Removing Algebraic Data Types from Constrained Horn Clauses Using Difference Predicates
Emanuele De Angelis, Fabio Fioravanti, Alberto Pettorossi, Maurizio Proietti
备注：
10th International Joint Conference on Automated Reasoning (IJCAR 2020) - version with appendix; added DOI of the final authenticated Springer publication; minor corrections
发表信息：
Lecture Notes in Computer Science, vol 12166. Springer, Cham, 2020, pp. 83-102
We address the problem of proving the satisfiability of Constrained Horn Clauses (CHCs) with Algebraic Data Types (ADTs), such as lists and trees. We propose a new technique for transforming CHCs with ADTs into CHCs where predicates are defined over basic types, such as integers and booleans, only. Thus, our technique avoids the explicit use of inductive proof rules during satisfiability proofs. The main extension over previous techniques for ADT removal is a new transformation rule, called differential replacement, which allows us to introduce auxiliary predicates corresponding to the lemmas that are often needed when making inductive proofs. We present an algorithm that uses the new rule, together with the traditional folding/unfolding transformation rules, for the automatic removal of ADTs. We prove that if the set of the transformed clauses is satisfiable, then so is the set of the original clauses. By an experimental evaluation, we show that the use of the differential replacement rule significantly improves the effectiveness of ADT removal, and we show that our transformation-based approach is competitive with respect to a well-established technique that extends the CVC4 solver with induction.
In this paper we show that checking satisfiability of a set of non-linear Horn clauses (also called a non-linear Horn clause program) can be achieved using a solver for linear Horn clauses. We achieve this by interleaving a program transformation with a satisfiability checker for linear Horn clauses (also called a solver for linear Horn clauses). The program transformation is based on the notion of tree dimension, which we apply to a set of non-linear clauses, yielding a set whose derivation trees have bounded dimension. Such a set of clauses can be linearised. The main algorithm then proceeds…
This volume contains the proceedings of HCVS 2016, the Third Workshop on Horn Clauses for Verification and Synthesis which was held on April 3, 2016 in Eindhoven, The Netherlands as a satellite event of the European Joint Conferences on Theory and Practice of Software (ETAPS 2016). Many program verification and synthesis problems of interest can be modeled directly using Horn clauses and many recent advances in the CLP and CAV communities have centered around efficiently solving problems presented as Horn clauses. The Third Workshop on Horn Clauses for Verification and Synthesis was organised …
The Workshop on Coalgebra, Horn Clause Logic Programming and Types was held on the 28-29 November 2016 in Edinburgh. The workshop marked the end of the EPSRC Grant Coalgebraic Logic Programming for Type Inference, by E. Komendantskaya, Heriot-Watt University and J. Power, University of Bath, UK.
We present a method for verifying the correctness of imperative programs which is based on the automated transformation of their specifications. Given a program prog, we consider a partial correctness specification of the form $\{φ\}$ prog $\{ψ\}$, where the assertions $φ$ and $ψ$ are predicates defined by a set Spec of possibly recursive Horn clauses with linear arithmetic (LA) constraints in their premise (also called constrained Horn clauses). The verification method consists in constructing a set PC of constrained Horn clauses whose satisfiability implies that $\{φ\}$ prog $\{ψ\}$ is valid…
Many Program Verification and Synthesis problems of interest can be modeled directly using Horn clauses and many recent advances in the CLP and CAV communities have centered around efficiently solving problems presented as Horn clauses. The HCVS series of workshops aims to bring together researchers working in the two communities of Constraint/Logic Programming (e.g., ICLP and CP), Program Verification (e.g., CAV, TACAS, and VMCAI), and Automated Deduction (e.g., CADE, IJCAR), on the topic of Horn clause based analysis, verification, and synthesis. Horn clauses for verification and synthesis h…
Automatically verifying safety properties of programs is hard, and it is even harder if the program acts upon arrays or other forms of maps. Many approaches exist for verifying programs operating upon Boolean and integer values (e.g. abstract interpretation, counterexample-guided abstraction refinement using interpolants), but transposing them to array properties has been fraught with difficulties.In contrast to most preceding approaches, we do not introduce a new abstract domain or a new interpolation procedure for arrays. Instead, we generate an abstraction as a scalar problem and feed it to…
Motivated by applications in automated verification of higher-order functional programs, we develop a notion of constrained Horn clauses in higher-order logic and a decision problem concerning their satisfiability. We show that, although satisfiable systems of higher-order clauses do not generally have least models, there is a notion of canonical model obtained through a reduction to a problem concerning a kind of monotone logic program. Following work in higher-order program verification, we develop a refinement type system in order to reason about and automate the search for models. This pro…
Alternation of forward and backward analyses is a standard technique in abstract interpretation of programs, which is in particular useful when we wish to prove unreachability of some undesired program states. The current state-of-the-art technique for combining forward (bottom-up, in logic programming terms) and backward (top-down) abstract interpretation of Horn clauses is query-answer transformation. It transforms a system of Horn clauses, such that standard forward analysis can propagate constraints both forward, and backward from a goal. Query-answer transformation is effective, but has i…
We address the problem of verifying the satisfiability of Constrained Horn Clauses (CHCs) based on theories of inductively defined data structures, such as lists and trees. We propose a transformation technique whose objective is the removal of these data structures from CHCs, hence reducing their satisfiability to a satisfiability problem for CHCs on integers and booleans. We propose a transformation algorithm and identify a class of clauses where it always succeeds. We also consider an extension of that algorithm, which combines clause transformation with reasoning on integer constraints. Vi…
This paper surveys recent work on applying analysis and transformation techniques that originate in the field of constraint logic programming (CLP) to the problem of verifying software systems. We present specialisation-based techniques for translating verification problems for different programming languages, and in general software systems, into satisfiability problems for constrained Horn clauses (CHCs), a term that has become popular in the verification field to refer to CLP programs. Then, we describe static analysis techniques for CHCs that may be used for inferring relevant program prop…
Big-step and small-step Horn clause derivations applied to operational semantics
Roskilde University（罗斯基勒大学）；IMDEA Software Institute（IMDEA软件研究所）；Universidad Politécnica de Madrid (UPM)（马德里理工大学）；Spanish Council for Scientific Research (CSIC)（西班牙国家研究委员会）
Verification conditions (VCs) are logical formulas whose satisfiability guarantees program correctness. We consider VCs in the form of constrained Horn clauses (CHC) which are automatically generated from the encoding of (an interpreter of) the operational semantics of the programming language. VCs are derived through program specialization based on the unfold/fold transformation rules and, as it often happens when specializing interpreters, they contain unnecessary variables, that is, variables which are not required for the correctness proofs of the programs under verification. In this paper…
In model based development, embedded systems are modeled using a mix of dataflow formalism, that capture the flow of computation, and hierarchical state machines, that capture the modal behavior of the system. For safety analysis, existing approaches rely on a compilation scheme that transform the original model (dataflow and state machines) into a pure dataflow formalism. Such compilation often result in loss of important structural information that capture the modal behaviour of the system. In previous work we have developed a compilation technique from a dataflow formalism into modular Horn…
We address the problem of checking the satisfiability of Constrained Horn Clauses (CHCs) defined on Algebraic Data Types (ADTs), such as lists and trees. We propose a new technique for transforming CHCs defined on ADTs into CHCs where the arguments of the predicates have only basic types, such as integers and booleans. Thus, our technique avoids, during satisfiability checking, the explicit use of proof rules based on induction over the ADTs. The main extension over previous techniques for ADT removal is a new transformation rule, called differential replacement, which allows us to introduce a…
Higher-order constrained Horn clauses (HoCHC) are a semantically-invariant system of higher-order logic modulo theories. With semi-decidable unsolvability over a semi-decidable background theory, HoCHC is suitable for safety verification. Less is known about its relation to larger classes of higher-order verification problems. Motivated by program equivalence, we introduce a coinductive version of HoCHC that enjoys a greatest model property. We define an encoding of higher-order recursion schemes (HoRS) into HoCHC logic programs. Correctness of this encoding reduces decidability of the open Ho…
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…
Building on the successes of satisfiability modulo theories (SMT), Bjørner et al. initiated a research programme advocating Horn constraints as a suitable basis for automatic program verification. The notion of first-order constrained Horn clauses has recently been extended to higher-order logic by Cathcart Burn et al. To exploit the remarkable efficiency of SMT solving, a natural approach to solve systems of higher-order Horn constraints is to reduce them to systems of first-order Horn constraints. This paper presents a defunctionalization algorithm to achieve the reduction. Given a well-sort…
The proceedings consist of a keynote paper by Alberto followed by 6 invited papers written by Lorenzo Clemente (U. Warsaw), Alain Finkel (U. Paris-Saclay), John Gallagher (Roskilde U. and IMDEA Software Institute) et al., Neil Jones (U. Copenhagen) et al., Michael Leuschel (Heinrich-Heine U.) and Maurizio Proietti (IASI-CNR) et al.. These invited papers are followed by 4 regular papers accepted at VPT 2020 and the papers of HCVS 2020 which consist of three contributed papers and an invited paper on the third competition of solvers for Constrained Horn Clauses. In addition, the abstracts (in HT…
These proceedings include selected papers presented at the 9th Workshop on Horn Clauses for Verification and Synthesis and the Tenth International Workshop on Verification and Program Transformation, both affiliated with ETAPS 2022. Many Program Verification and Synthesis problems of interest can be modeled directly using Horn clauses and many recent advances in the CLP and CAV communities have centered around efficiently solving problems presented as Horn clauses. The HCVS series of workshops aims to bring together researchers working in the communities of Constraint/Logic Programming (e.g., …
Many applications require conformance with specifications that constrain the use of resources, such as execution time, energy, bandwidth, etc. We have presented a configurable framework for static resource usage verification where specifications can include lower and upper bound, data size-dependent resource usage functions. To statically check such specifications, our framework infers the same type of resource usage functions, which safely approximate the actual resource usage of the program, and compares them against the specification. We review how this framework supports several languages …
Verification problems of programs written in various paradigms (such as imperative, logic, concurrent, functional, and object-oriented ones) can be reduced to problems of solving Horn clause constraints on predicate variables that represent unknown inductive invariants. This paper presents a novel Horn constraint solving method based on inductive theorem proving: the method reduces Horn constraint solving to validity checking of first-order formulas with inductively defined predicates, which are then checked by induction on the derivation of the predicates. To automate inductive proofs, we int…
Several techniques and tools have been developed for verification of properties expressed as Horn clauses with constraints over a background theory (CHC). Current CHC verification tools implement intricate algorithms and are often limited to certain subclasses of CHC problems. Our aim in this work is to investigate the use of a combination of off-the-shelf techniques from the literature in analysis and transformation of Constraint Logic Programs (CLPs) to solve challenging CHC verification problems. We find that many problems can be solved using a combination of tools based on well-known techn…
We present a compositional SMT-based algorithm for safety of procedural C programs that takes the heap into consideration as well. Existing SMT-based approaches are either largely restricted to handling linear arithmetic operations and properties, or are non-compositional. We use Constrained Horn Clauses (CHCs) to represent the verification conditions where the memory operations are modeled using the extensional theory of arrays (ARR). First, we describe an exponential time quantifier elimination (QE) algorithm for ARR which can introduce new quantifiers of the index and value sorts. Second, w…
The functional properties of a program are often specified by providing a contract for each of its functions. A contract of a function consists of a pair of formulas, called a precondition and a postcondition, which, respectively, should hold before and after execution of that function. It might be the case that the contracts supplied by the programmer are not adequate to allow a verification system to prove program correctness, that is, to show that for every function, if the precondition holds and the execution of the function terminates, then the postcondition holds. We address this problem…
It is known that the verification of imperative, functional, and logic programs can be reduced to the satisfiability of constrained Horn clauses (CHCs), and this satisfiability check can be performed by using CHC solvers, such as Eldarica and Z3. These solvers perform well when they act on simple constraint theories, such as Linear Integer Arithmetic and the theory of Booleans, but their efficacy is very much reduced when the clauses refer to constraints on inductively defined structures, such as lists or trees. Recently, we have presented a transformation technique for eliminating those induc…
Proceedings 18th International Workshop on Logical and Semantic Frameworks, with Applications and 10th Workshop on Horn Clauses for Verification and Synthesis
The proof of a program property can be reduced to the proof of satisfiability of a set of constrained Horn clauses (CHCs) which can be automatically generated from the program and the property. In this paper we have conducted a case study in Horn clause verification by considering several sorting programs with the aim of exploring the effectiveness of a transformation technique which allows us to eliminate inductive data structures such as lists or trees. If this technique is successful, we derive a set of CHCs with constraints over the integers and booleans only, and the satisfiability check …
Many recent analyses for conventional imperative programs begin by transforming programs into logic programs, capitalising on existing LP analyses and simple LP semantics. We propose using logic programs as an intermediate program representation throughout the compilation process. With restrictions ensuring determinism and single-modedness, a logic program can easily be transformed to machine language or other low-level language, while maintaining the simple semantics that makes it suitable as a language for program analysis and transformation. We present a simple LP language that enforces det…
First-order logic is a natural way of expressing the properties of computation, traditionally used in various program logics for expressing the correctness properties and certificates. Subsequently, modern methods in the automated inference of program invariants progress towards the construction of first-order definable invariants. Although the first-order representations are very expressive for some theories, they fail to express many interesting properties of algebraic data types (ADTs). Thus we propose to represent program invariants regularly with tree automata. We show how to automaticall…
In recent years they have been numerous works that aim to automate relational verification. Meanwhile, although Constrained Horn Clauses (CHCs) empower a wide range of verification techniques and tools, they lack the ability to express hyperproperties beyond $k$-safety such as generalized non-interference and co-termination. This paper describes a novel and fully automated constraint-based approach to relational verification. We first introduce a new class of predicate Constraint Satisfaction Problems called pfwCSP where constraints are represented as clauses modulo first-order theories over p…
We investigate representations of imperative programs as constrained Horn clauses. Starting from operational semantics transition rules, we proceed by writing interpreters as constrained Horn clause programs directly encoding the rules. We then specialise an interpreter with respect to a given source program to achieve a compilation of the source language to Horn clauses (an instance of the first Futamura projection). The process is described in detail for an interpreter for a subset of C, directly encoding the rules of big-step operational semantics for C. A similar translation based on small…
