Type systems certify program properties in a compositional way. From a bigger program one can abstract out a part and certify the properties of the resulting abstract program by just using the type of the part that was abstracted away. Termination and productivity are non-trivial yet desired program properties, and several type systems have been put forward that guarantee termination, compositionally. These type systems are intimately connected to the definition of least and greatest fixed-points by ordinal iteration. While most type systems use conventional iteration, we consider inflationary iteration in this article. We demonstrate how this leads to a more principled type system, with recursion based on well-founded induction. The type system has a prototypical implementation, MiniAgda, and we show in particular how it certifies productivity of corecursive and mixed recursive-corecursive functions.
In the calculus of dependent lambda eliminations (CDLE), it is possible to define inductive datatypes via lambda encodings that feature constant-time destructors and a course-of-values induction scheme. This paper begins to address the missing derivations for the dual, coinductive types. Our derivation utilizes new methods within CDLE, as there are seemingly fundamental difficulties in adapting previous known approaches for deriving inductive types. The lambda encodings we present implementing coinductive types feature constant-time constructors and a course-of-values corecursion scheme. Coind…
We introduce a generalized logic programming paradigm where programs, consisting of facts and rules with the usual syntax, can be enriched by co-facts, which syntactically resemble facts but have a special meaning. As in coinductive logic programming, interpretations are subsets of the complete Herbrand basis, including infinite terms. However, the intended meaning (declarative semantics) of a program is a fixed point which is not necessarily the least, nor the greatest one, but is determined by co-facts. In this way, it is possible to express predicates on non well-founded structures, such as…
We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly expressive. For example, all well-known basic types and type formers that are needed for using this type theory as a logic are definable: propositional connectives, like falsity, conjunction, disjunction, and function space, dependent function space, existential quantification, equality, natural numbers, vectors etc. The reduction relation on terms consists solely…
We formulate a framework for describing behaviour of effectful higher-order recursive programs. Examples of effects are implemented using effect operations, and include: execution cost, nondeterminism, global store and interaction with a user. The denotational semantics of a program is given by a coinductive tree in a monad, which combines potential return values of the program in terms of effect operations. Using a simple test logic, we construct two sorts of predicate liftings, which lift predicates on a result type to predicates on computations that produce results of that type, each captur…
Guarded recursion is a powerful modal approach to recursion that can be seen as an abstract form of step-indexing. It is currently used extensively in separation logic to model programming languages with advanced features by solving domain equations also with negative occurrences. In its multi-clocked version, guarded recursion can also be used to program with and reason about coinductive types, encoding the productivity condition required for recursive definitions in types. This paper presents the first type theory combining multi-clocked guarded recursion with the features of Cubical Type Th…
Theorem provers are tools that help users to write machine readable proofs. Some of this tools are also interactive. The need of such softwares is increasing since they provide proofs that are more certified than the hand written ones. Agda is based on type theory and on the propositions-as-types correspondence and has a Haskell-like syntax. This means that a proof of a statement is turned into a function. Inference systems are a way of defining inductive and coinductive predicates and induction and coinduction principles are provided to help proving their correctness with respect to a given s…
Recently we presented a concise survey of the formulation of the induction and coinduction principles, and some concepts related to them, in programming languages type theory and four other mathematical disciplines. The presentation in type theory involved the separate formulation of these concepts, first, in the theory of types of functional programming languages and, next, in the theory of types of object-oriented programming languages. In this article we show that separating these two formulations helps demonstrate some of the fundamental differences between structural subtyping, predominan…
Essential tasks for the verification of probabilistic programs include bounding expected outcomes and proving termination in finite expected runtime. We contribute a simple yet effective inductive synthesis approach for proving such quantitative reachability properties by generating inductive invariants on source-code level. Our implementation shows promise: It finds invariants for (in)finite-state programs, can beat state-of-the-art probabilistic model checkers, and is competitive with modern tools dedicated to invariant synthesis and expected runtime reasoning.
We present a rich type system with subtyping for an extension of System F. Our type constructors include sum and product types, universal and existential quantifiers, inductive and coinductive types. The latter two size annotations allowing the preservation of size invariants. For example it is possible to derive the termination of the quicksort by showing that partitioning a list does not increase its size. The system deals with complex programs involving mixed induction and coinduction, or even mixed (co-)induction and polymorphism (as for Scott-encoded datatypes). One of the key ideas is to…
This work exploits the logical foundation of session types to determine what kind of type discipline for the pi-calculus can exactly capture, and is captured by, lambda-calculus behaviours. Leveraging the proof theoretic content of the soundness and completeness of sequent calculus and natural deduction presentations of linear logic, we develop the first mutually inverse and fully abstract processes-as-functions and functions-as-processes encodings between a polymorphic session pi-calculus and a linear formulation of System F. We are then able to derive results of the session calculus from the…
This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of inductive-inductive definitions involving strictly positive occurrences of Hofmann-style quotient types, and Abel's size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definitions of the elimination and computation proper…
In this survey article (which hitherto is an ongoing work-in-progress) we present the formulation of the induction and coinduction principles using the language and conventions of each of order theory, set theory, programming languages' type theory, first-order logic, and category theory, for the purpose of examining some of the similarities and, more significantly, the dissimilarities between these various mathematical disciplines, and hence shed some light on the precise relation between these disciplines. Towards that end, in this article we discuss plenty of related concepts, such as fixed…
