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Register a new userGeneric Fibrational Induction
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Added by
Neil Ghani
Publication date
2012
fields
Informatics Engineering
and research's language is
English
Authors
Neil Ghani
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This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, a sound induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of a particular syntactic form. We establish the soundness of our generic induction rule by reducing induction to iteration. We then show how our generic induction rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The first of these lies outside the scope of Hermida and Jacobs work because it is not polynomial, and as far as we are aware, no induction rules have been known to exist for the second and third in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set.
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User defined recursive types are a fundamental feature of modern functional programming languages like Haskell, Clean, and the ML family of languages. Properties of programs defined by recursion on the structure of recursive types are generally proved by structural induction on the type. It is well known in the theorem proving community how to generate structural induction principles from data type declarations. These methods deserve to be better know in the functional programming community. Existing functional programming textbooks gloss over this material. And yet, if functional programmers do not know how to write down the structural induction principle for a new type - how are they supposed to reason about it? In this paper we describe an algorithm to generate structural induction principles from data type declarations. We also discuss how these methods are taught in the functional programming course at the University of Wyoming. A Haskell implementation of the algorithm is included in an appendix.
Context: The algorithms for generating a safe fluent API are actively studied these years. A safe fluent API is the fluent API that reports incorrect chaining of the API methods as a type error to the API users. Although such a safe property improves the productivity of its users, the construction of a safe fluent API is too complicated for the developers. The generation algorithms are studied to reduce the development cost of a safe fluent API. The study on the generation would benefit a number of programmers since a fluent API is a popular design in the real world. Inquiry: The generation of a generic fluent API has been left untackled. A generic fluent API refers to the fluent API that provides generic methods (methods that contain type parameters in their definitions). The Stream API in Java is an example of such a generic API. The recent research on the safe fluent API generation rather focuses on the grammar class that the algorithm can deal with for syntax checking. The key idea of the previous study is to use nested generics to represent a stack structure for the parser built on top of the type system. In that idea, the role of a type parameter was limited to internally representing a stack element of that parser on the type system. The library developers could not use type parameters to include a generic method in their API so that the semantic constraints for their API would be statically checked, for example, the type constraint on the items passed through a stream. Approach: We propose an algorithm to generate a generic fluent API. Our translation algorithm is modeled as the construction of deterministic finite automaton (DFA) with type parameter information. Each state of the DFA holds information about which type parameters are already bound in that state. This information is used to identify whether a method invocation in a chain newly binds a type to a type parameter, or refers to a previously bound type. The identification is required since a type parameter in a chain is bound at a particular method invocation, and that bound type is referred to in the following method invocations. Our algorithm constructs the DFA by analyzing the binding time of type parameters and their propagation among the states in a DFA that is naively constructed from the given grammar. Knowledge and Importance: Our algorithm helps library developers to develop a generic fluent API. The ability to generate a generic fluent API is essential to bring the safe fluent API generation to the real world since the use of type parameters is a common technique in the library API design. By our algorithm, the generation of a safe fluent API will be ready for practical use. Grounding: We implemented a generator named Protocool to demonstrate our algorithm. We also generated several libraries using Protocool to show the ability and the limitations of our algorithm.
We introduce a universe of regular datatypes with variable binding information, for which we define generic formation and elimination (i.e. induction /recursion) operators. We then define a generic alpha-equivalence relation over the types of the universe based on name-swapping, and derive iteration and induction principles which work modulo alpha-conversion capturing Barendregts Variable Convention. We instantiate the resulting framework so as to obtain the Lambda Calculus and System F, for which we derive substitution operations and substitution lemmas for alpha-conversion and substitution composition. The whole work is carried out in Constructive Type Theory and machine-checked by the system Agda.
We study lax families of adjoints from a fibrational viewpoint, obtaining a version of the mate correspondence for (op)lax natural transformations of functors from an $infty$-category to the $(infty,2)$-category of $infty$-categories. We apply this to show that the left adjoint of a lax symmetric monoidal functor is oplax symmetric monoidal and that the internal Hom in a closed symmetric monoidal $infty$-category is lax symmetric monoidal in both variables. We also consider units and counits of such families of adjoints, and use them to derive the full (twisted) naturality of passing to the dual.
The principle of strong induction, also known as k-induction is one of the first techniques for unbounded SAT-based Model Checking (SMC). While elegant and simple to apply, properties as such are rarely k-inductive and when they can be strengthened, there is no effective strategy to guess the depth of induction. It has been mostly displaced by techniques that compute inductive strengthenings based on interpolation and property directed reachability (Pdr). In this paper, we present kAvy, an SMC algorithm that effectively uses k-induction to guide interpolation and Pdr-style inductive generalization. Unlike pure k-induction, kAvy uses Pdr-style generalization to compute and strengthen an inductive trace. Unlike pure Pdr, kAvy uses relative k-induction to construct an inductive invariant. The depth of induction is adjusted dynamically by minimizing a proof of unsatisfiability. We have implemented kAvy within the Avy Model Checker and evaluated it on HWMCC instances. Our results show that kAvy is more effective than both Avy and Pdr, and that using k-induction leads to faster running time and solving more instances. Further, on a class of benchmarks, called shift, kAvy is orders of magnitude faster than Avy, Pdr and k-induction.
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