Convex polyhedral abstractions, specialisation and property-based predicate splitting in Horn clause verification

We present an approach to constrained Horn clause (CHC) verification combining three techniques: abstract interpretation over a domain of convex polyhedra, specialisation of the constraints in CHCs using abstract interpretation of query-answer transformed clauses, and refinement by splitting predicates. The purpose of the work is to investigate how analysis and transformation tools developed for constraint logic programs (CLP) can be applied to the Horn clause verification problem. Abstract interpretation over convex polyhedra is capable of deriving sophisticated invariants and when used in conjunction with specialisation for propagating constraints it can frequently solve challenging verification problems. This is a contribution in itself, but refinement is needed when it fails, and the question of how to refine convex polyhedral analyses has not been studied much. We present a refinement technique based on interpolants derived from a counterexample trace; these are used to drive a property-based specialisation that splits predicates, leading in turn to more precise convex polyhedral analyses. The process of specialisation, analysis and splitting can be repeated, in a manner similar to the CEGAR and iterative specialisation approaches.Comment: In Proceedings HCVS 2014, arXiv:1412.082

Learning programs by learning from failures

We describe an inductive logic programming (ILP) approach called learning from failures. In this approach, an ILP system (the learner) decomposes the learning problem into three separate stages: generate, test, and constrain. In the generate stage, the learner generates a hypothesis (a logic program) that satisfies a set of hypothesis constraints (constraints on the syntactic form of hypotheses). In the test stage, the learner tests the hypothesis against training examples. A hypothesis fails when it does not entail all the positive examples or entails a negative example. If a hypothesis fails, then, in the constrain stage, the learner learns constraints from the failed hypothesis to prune the hypothesis space, i.e. to constrain subsequent hypothesis generation. For instance, if a hypothesis is too general (entails a negative example), the constraints prune generalisations of the hypothesis. If a hypothesis is too specific (does not entail all the positive examples), the constraints prune specialisations of the hypothesis. This loop repeats until either (i) the learner finds a hypothesis that entails all the positive and none of the negative examples, or (ii) there are no more hypotheses to test. We introduce Popper, an ILP system that implements this approach by combining answer set programming and Prolog. Popper supports infinite problem domains, reasoning about lists and numbers, learning textually minimal programs, and learning recursive programs. Our experimental results on three domains (toy game problems, robot strategies, and list transformations) show that (i) constraints drastically improve learning performance, and (ii) Popper can outperform existing ILP systems, both in terms of predictive accuracies and learning times.Comment: Accepted for the machine learning journa

An iterative approach to precondition inference using constrained Horn clauses

We present a method for automatic inference of conditions on the initial states of a program that guarantee that the safety assertions in the program are not violated. Constrained Horn clauses (CHCs) are used to model the program and assertions in a uniform way, and we use standard abstract interpretations to derive an over-approximation of the set of unsafe initial states. The precondition then is the constraint corresponding to the complement of that set, under-approximating the set of safe initial states. This idea of complementation is not new, but previous attempts to exploit it have suffered from the loss of precision. Here we develop an iterative specialisation algorithm to give more precise, and in some cases optimal safety conditions. The algorithm combines existing transformations, namely constraint specialisation, partial evaluation and a trace elimination transformation. The last two of these transformations perform polyvariant specialisation, leading to disjunctive constraints which improve precision. The algorithm is implemented and tested on a benchmark suite of programs from the literature in precondition inference and software verification competitions.Comment: Paper presented at the 34nd International Conference on Logic Programming (ICLP 2018), Oxford, UK, July 14 to July 17, 2018 18 pages, LaTe

Precondition Inference via Partitioning of Initial States

Precondition inference is a non-trivial task with several applications in program analysis and verification. We present a novel iterative method for automatically deriving sufficient preconditions for safety and unsafety of programs which introduces a new dimension of modularity. Each iteration maintains over-approximations of the set of \emph{safe} and \emph{unsafe} \emph{initial} states. Then we repeatedly use the current abstractions to partition the program's \emph{initial} states into those known to be safe, known to be unsafe and unknown, and construct a revised program focusing on those initial states that are not yet known to be safe or unsafe. An experimental evaluation of the method on a set of software verification benchmarks shows that it can solve problems which are not solvable using previous methods.Comment: 19 pages, 8 figure

Learning to Understand by Evolving Theories

In this paper, we describe an approach that enables an autonomous system to infer the semantics of a command (i.e. a symbol sequence representing an action) in terms of the relations between changes in the observations and the action instances. We present a method of how to induce a theory (i.e. a semantic description) of the meaning of a command in terms of a minimal set of background knowledge. The only thing we have is a sequence of observations from which we extract what kinds of effects were caused by performing the command. This way, we yield a description of the semantics of the action and, hence, a definition.Comment: KRR Workshop at ICLP 201

Constructive approaches to Program Induction

Search is a key technique in artificial intelligence, machine learning and Program Induction. No matter how efficient a search procedure, there exist spaces that are too large to search effectively and they include the search space of programs. In this dissertation we show that in the context of logic-program induction (Inductive Logic Programming, or ILP) it is not necessary to search for a correct program, because if one exists, there also exists a unique object that is the most general correct program, and that can be constructed directly, without a search, in polynomial time and from a polynomial number of examples. The existence of this unique object, that we term the Top Program because of its maximal generality, does not so much solve the problem of searching a large program search space, as it completely sidesteps it, thus improving the efficiency of the learning task by orders of magnitude commensurate with the complexity of a program space search. The existence of a unique Top Program and the ability to construct it given finite resources relies on the imposition, on the language of hypotheses, from which programs are constructed, of a strong inductive bias with relevance to the learning task. In common practice, in machine learning, Program Induction and ILP, such relevant inductive bias is selected, or created, manually, by the human user of a learning system, with intuition or knowledge of the problem domain, and in the form of various kinds of program templates. In this dissertation we show that by abandoning the reliance on such extra-logical devices as program templates, and instead defining inductive bias exclusively as First- and Higher-Order Logic formulae, it is possible to learn inductive bias itself from examples, automatically, and efficiently, by Higher-Order Top Program construction. In Chapter 4 we describe the Top Program in the context of the Meta-Interpretive Learning approach to ILP (MIL) and describe an algorithm for its construction, the Top Program Construction algorithm (TPC). We prove the efficiency and accuracy of TPC and describe its implementation in a new MIL system called Louise. We support theoretical results with experiments comparing Louise to the state-of-the-art, search-based MIL system, Metagol, and find that Louise improves Metagol’s efficiency and accuracy. In Chapter 5 we re-frame MIL as specialisation of metarules, Second-Order clauses used as inductive bias in MIL, and prove that problem-specific metarules can be derived by specialisation of maximally general metarules, by MIL. We describe a sub-system of Louise, called TOIL, that learns new metarules by MIL and demonstrate empirically that the metarules learned by TOIL match those selected manually, while maintaining the accuracy and efficiency of learning. iOpen Acces