Homeomorphic Embedding for Online Termination of Symbolic Methods

Well-quasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of techniques for program analysis, specialisation, transformation, and verification. In this paper we survey and discuss this use of homeomorphic embedding and clarify the advantages of such an approach over one using well-founded orders. We also discuss various extensions of the homeomorphic embedding relation. We conclude with a study of homeomorphic embedding in the context of metaprogramming, presenting some new (positive and negative) results and open problems

Sciduction: Combining Induction, Deduction, and Structure for Verification and Synthesis

Even with impressive advances in automated formal methods, certain problems in system verification and synthesis remain challenging. Examples include the verification of quantitative properties of software involving constraints on timing and energy consumption, and the automatic synthesis of systems from specifications. The major challenges include environment modeling, incompleteness in specifications, and the complexity of underlying decision problems. This position paper proposes sciduction, an approach to tackle these challenges by integrating inductive inference, deductive reasoning, and structure hypotheses. Deductive reasoning, which leads from general rules or concepts to conclusions about specific problem instances, includes techniques such as logical inference and constraint solving. Inductive inference, which generalizes from specific instances to yield a concept, includes algorithmic learning from examples. Structure hypotheses are used to define the class of artifacts, such as invariants or program fragments, generated during verification or synthesis. Sciduction constrains inductive and deductive reasoning using structure hypotheses, and actively combines inductive and deductive reasoning: for instance, deductive techniques generate examples for learning, and inductive reasoning is used to guide the deductive engines. We illustrate this approach with three applications: (i) timing analysis of software; (ii) synthesis of loop-free programs, and (iii) controller synthesis for hybrid systems. Some future applications are also discussed

A Universal Machine for Biform Theory Graphs

Broadly speaking, there are two kinds of semantics-aware assistant systems for mathematics: proof assistants express the semantic in logic and emphasize deduction, and computer algebra systems express the semantics in programming languages and emphasize computation. Combining the complementary strengths of both approaches while mending their complementary weaknesses has been an important goal of the mechanized mathematics community for some time. We pick up on the idea of biform theories and interpret it in the MMTt/OMDoc framework which introduced the foundations-as-theories approach, and can thus represent both logics and programming languages as theories. This yields a formal, modular framework of biform theory graphs which mixes specifications and implementations sharing the module system and typing information. We present automated knowledge management work flows that interface to existing specification/programming tools and enable an OpenMath Machine, that operationalizes biform theories, evaluating expressions by exhaustively applying the implementations of the respective operators. We evaluate the new biform framework by adding implementations to the OpenMath standard content dictionaries.Comment: Conferences on Intelligent Computer Mathematics, CICM 2013 The final publication is available at http://link.springer.com

Middle-Out Reasoning for Logic Program Synthesis

We propose a novel approach to automating the synthesis of logic programs: Logic programs are synthesized as a by-product of the planning of a verification proof. The approach is a two-level one: At the object level, we prove program verification conjectures in a sorted, first-order theory. The conjectures are of the form 8args \Gamma\Gamma\Gamma\Gamma! : prog(args \Gamma\Gamma\Gamma\Gamma! ) $ spec(args \Gamma\Gamma\Gamma\Gamma! ). At the meta-level, we plan the object-level verification with an unspecified program definition. The definition is represented with a (second-order) meta-level variable, which becomes instantiated in the course of the planning

A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the issues of defining such sequent calculi for Pure Type Systems (PTS, which were originally presented in natural deduction style) and then organizing their rules for effective proof-search. We introduce the idea of Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the syntax of a permutation-free sequent calculus for propositional logic due to Herbelin, which is strongly related to natural deduction and already well adapted to proof-search. The operational semantics is adapted from Herbelin's and is defined by a system of local rewrite rules as in cut-elimination, using explicit substitutions. We prove confluence for this system. Restricting our attention to PTSC, a type system for the ground terms of this system, we obtain the Subject Reduction property and show that each PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising iff the latter is. We show how to make the logical rules of PTSC into a syntax-directed system PS for proof-search, by incorporating the conversion rules as in syntax-directed presentations of the PTS rules for type-checking. Finally, we consider how to use the explicitly scoped meta-variables of PTSCalpha to represent partial proof-terms, and use them to analyse interactive proof construction. This sets up a framework PE in which we are able to study proof-search strategies, type inhabitant enumeration and (higher-order) unification