Towards an Intelligent Tutor for Mathematical Proofs

Computer-supported learning is an increasingly important form of study since it allows for independent learning and individualized instruction. In this paper, we discuss a novel approach to developing an intelligent tutoring system for teaching textbook-style mathematical proofs. We characterize the particularities of the domain and discuss common ITS design models. Our approach is motivated by phenomena found in a corpus of tutorial dialogs that were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor for textbook-style mathematical proofs can be built on top of an adapted assertion-level proof assistant by reusing representations and proof search strategies originally developed for automated and interactive theorem proving. The resulting prototype was successfully evaluated on a corpus of tutorial dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453

Deriving Proved Equality Tests in Coq-Elpi: Stronger Induction Principles for Containers in Coq

We describe a procedure to derive equality tests and their correctness proofs from inductive type declarations in Coq. Programs and proofs are derived compositionally, reusing code and proofs derived previously. The key steps are two. First, we design appropriate induction principles for data types defined using parametric containers. Second, we develop a technique to work around the modularity limitations imposed by the purely syntactic termination check Coq performs on recursive proofs. The unary parametricity translation of inductive data types turns out to be the key to both steps. Last but not least, we provide an implementation of the procedure for the Coq proof assistant based on the Elpi [Dunchev et al., 2015] extension language

Application of Decomposition and Generic Instantiation

It is believed that reusability in formal development should reduce the time and cost of formal modelling within a production environment. Event-B is a formal method that allows modelling and refinement of systems. Generic instantiation and decomposition are techniques that simplify formal developments by reusing existing models and avoiding re-proofs. We apply these techniques in Event-B for the development of a metro system case study based on safety properties. This work aims to be give some guidelines of a practical way to develop large systems by instantiation of generic models and (shared event) decompose components into smaller sub-components

Do mathematical explanations have instrumental value?

Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental value by promoting reuse of the resources they contain? In general, I argue that the answer to this question is "no" and demonstrate this in detail for the theory of mathematical explanation developed by Marc Lange

Towards correct-by-construction product variants of a software product line: GFML, a formal language for feature modules

Software Product Line Engineering (SPLE) is a software engineering paradigm that focuses on reuse and variability. Although feature-oriented programming (FOP) can implement software product line efficiently, we still need a method to generate and prove correctness of all product variants more efficiently and automatically. In this context, we propose to manipulate feature modules which contain three kinds of artifacts: specification, code and correctness proof. We depict a methodology and a platform that help the user to automatically produce correct-by-construction product variants from the related feature modules. As a first step of this project, we begin by proposing a language, GFML, allowing the developer to write such feature modules. This language is designed so that the artifacts can be easily reused and composed. GFML files contain the different artifacts mentioned above.The idea is to compile them into FoCaLiZe, a language for specification, implementation and formal proof with some object-oriented flavor. In this paper, we define and illustrate this language. We also introduce a way to compose the feature modules on some examples.Comment: In Proceedings FMSPLE 2015, arXiv:1504.0301

Reusing constraint proofs in symbolic analysis

Symbolic analysis is an important element of program verification and automatic testing. Symbolic analysis techniques abstract program properties as expressions of symbolic input values to characterise the program logical constraints, and rely on Satisfiability Modulo Theories (SMT) solvers to both validate the satisfiability of the constraint expression and verify the corresponding program properties. Despite the impressive improvements of constraint solving and the availability of mature solvers, constraint solving still represents a main bottleneck towards efficient and scalable symbolic program analysis. The work on the SMT bottleneck proceeds along two main research lines: (i) optimisation approaches that assist and complement the solvers in the context of the program analysis in various ways, and (ii) reuse approaches that reduce the invocation of constraint solvers, by reusing proofs while solving constraints during symbolic analysis. This thesis contributes to the research in reuse approaches, with REusing-Constraint- proofs-in-symbolic-AnaLysis (ReCal), a new approach for reusing proofs across constraints that recur during analysis. ReCal advances over state-of-the-art approaches for reusing constraints by (i) proposing a novel canonical form to efficiently store and retrieve equivalent and related-by- implication constraints, and (ii) defining a parallel framework for GPU-based platforms to optimise the storage and retrieval of constraints and reusable proofs. Equivalent constraints vary widely due to the program specific details. This thesis defines a canonical form of constraints in the context of symbolic analysis, and develops an original canonicalisation algorithm to generate the canonical form. The canonical form turns the complex problem of deciding the equivalence of two constraints to the simple problem of comparing for equality their canonical forms, thus enabling efficient catching recurring constraints during symbolic analysis. Constraints can become extremely large when analysing complex systems, and handling large constraints may introduce a heavy overhead, thus harming the scalability of proof-reusing approaches. The ReCal parallel framework largely improves both the performance and scalability of reusing proofs by benefitting from Graphics Processing Units (GPU) platforms that provide thousands of computing units working in parallel. The parallel ReCal framework ReCal-gpu achieves a 10- times speeding up in constraint solving during symbolic execution of various programs

Partial Quantifier Elimination By Certificate Clauses

We study partial quantifier elimination (PQE) for propositional CNF formulas. In contrast to full quantifier elimination, in PQE, one can limit the set of clauses taken out of the scope of quantifiers to a small subset of target clauses. The appeal of PQE is twofold. First, PQE can be dramatically simpler than full quantifier elimination. Second, it provides a language for performing incremental computations. Many verification problems (e.g. equivalence checking and model checking) are inherently incremental and so can be solved in terms of PQE. Our approach is based on deriving clauses depending only on unquantified variables that make the target clauses $\mathit{redundant}$. Proving redundancy of a target clause is done by construction of a ``certificate'' clause implying the former. We describe a PQE algorithm called $\mathit{START}$ that employs the approach above. We apply $\mathit{START}$ to generating properties of a design implementation that are not implied by specification. The existence of an $\mathit{unwanted}$ property means that this implementation is buggy. Our experiments with HWMCC-13 benchmarks suggest that $\mathit{START}$ can be used for generating properties of real-life designs