An Open Challenge Problem Repository for Systems Supporting Binders

A variety of logical frameworks support the use of higher-order abstract syntax in representing formal systems; however, each system has its own set of benchmarks. Even worse, general proof assistants that provide special libraries for dealing with binders offer a very limited evaluation of such libraries, and the examples given often do not exercise and stress-test key aspects that arise in the presence of binders. In this paper we design an open repository ORBI (Open challenge problem Repository for systems supporting reasoning with BInders). We believe the field of reasoning about languages with binders has matured, and a common set of benchmarks provides an important basis for evaluation and qualitative comparison of different systems and libraries that support binders, and it will help to advance the field.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

Automated Deduction in the B Set Theory using Typed Proof Search and Deduction Modulo

International audienceWe introduce an encoding of the set theory of the B method using polymorphic types and deduction modulo, which is used for the automated verication of proof obligations in the framework of theBWare project. Deduction modulo is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewrite rules. We also present the associated automated theorem prover Zenon Modulo, an extension of Zenon to polymorphic types and deduction modulo, along with its backend to the Dedukti universal proof checker, which also relies on types and deduction modulo, and which allows us to verify the proofs produced by Zenon Modulo. Finally, we assess our approach over the proof obligation benchmark of BWare

Extensional Higher-Order Paramodulation in Leo-III

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling e.g. proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics.Comment: 34 pages, 7 Figures, 1 Table; submitted articl

Semi Automated Partial Credit Grading of Programming Assignments

The grading of student programs is a time consuming process. As class sizes continue to grow, especially in entry level courses, manually grading student programs has become an even more daunting challenge. Increasing the difficulty of grading is the needs of graphical and interactive programs such as those used as part of the UNH Computer Science curriculum (and various textbooks). There are existing tools that support the grading of introductory programming assignments (TAME and Web-CAT). There are also frameworks that can be used to test student code (JUnit, Tester, and TestNG). While these programs and frameworks are helpful, they have little or no no support for programs that use real data structures or that have interactive or graphical features. In addition, the automated tests in all these tools provide only “all or nothing” evaluation. This is a significant limitation in many circumstances. Moreover, there is little or no support for dynamic alteration of grading criteria, which means that refactoring of test classes after deployment is not easily done. Our goal is to create a framework that can address these weaknesses. This framework needs to: 1. Support assignments that have interactive and graphical components. 2. Handle data structures in student programs such as lists, stacks, trees, and hash tables. 3. Be able to assign partial credit automatically when the instructor can predict errors in advance. 4. Provide additional answer clustering information to help graders identify and assign consistent partial credit for incorrect output that was not predefined. Most importantly, these tools, collectively called RPM (short for Rapid Program Management), should interface effectively with our current grading support framework without requiring large amounts of rewriting or refactoring of test code

No value restriction is needed for algebraic effects and handlers

We present a straightforward, sound Hindley-Milner polymorphic type system for algebraic effects and handlers in a call-by-value calculus, which allows type variable generalisation of arbitrary computations, not just values. This result is surprising. On the one hand, the soundness of unrestricted call-by-value Hindley-Milner polymorphism is known to fail in the presence of computational effects such as reference cells and continuations. On the other hand, many programming examples can be recast to use effect handlers instead of these effects. Analysing the expressive power of effect handlers with respect to state effects, we claim handlers cannot express reference cells, and show they can simulate dynamically scoped state

Efficient Data Structures for Automated Theorem Proving in Expressive Higher-Order Logics

Church's Simple Theory of Types (STT), also referred to as classical higher-order logik, is an elegant and expressive formal system built on top of the simply typed λ-calculus. Its mechanisms of explicit binding and quantification over arbitrary sets and functions allow the representation of complex mathematical concepts and formulae in a concise and unambiguous manner. Higher-order automated theorem proving (ATP) has recently made major progress and several sophisticated ATP systems for higher-order logic have been developed, including Satallax, Osabelle/HOL and LEO-II. Still, higher-order theorem proving is not as mature as its first-order counterpart, and robust implementation techniques for efficient data structures are scarce. In this thesis, a higher-order term representation based upon the polymorphically typed λ-calculus is presented. This term representation employs spine notation, explicit substitutions and perfect term sharing for efficient term traversal, fast β-normalization and reuse of already constructed terms, respectively. An evaluation of the term representation is performed on the basis of a heterogeneous benchmark set. It shows that while the presented term data structure performs quite well in general, the normalization results indicate that a context dependent choice of reduction strategies is beneficial. A term indexing data structure for fast term retrieval based on various low-level criteria is presented and discussed. It supports symbol-based term retrieval, indexing of terms via structural properties, and subterm indexing

Combining Algebraic and Set-Theoretic Specifications (Extended Version)

Specification frameworks such as B and Z provide power sets and cartesianproducts as built-in type constructors, and employ a rich notation fordefining (among other things) abstract data types using formulae of predicatelogic and lambda-notation. In contrast, the so-called algebraic specification frameworks often limit the type structure to sort constants andfirst-order functionalities, and restrict formulae to (conditional) equations.Here, we propose an intermediate framework where algebraic specificationsare enriched with a set-theoretic type structure, but formulae remain in thelogic of equational Horn clauses. This combines an expressive yet modestspecification notation with simple semantics and tractable proof theory