A predicative analysis of structural recursion

ACM SIGPLAN Notices, 2004

In the interest of designing a recursive module extension to ML that is as simple and general as possible, we propose a novel type system for general recursion over effectful expressions. The presence of effects seems to necessitate a backpatching semantics for recursion based on Scheme's. Our type system ensures statically that recursion is well-founded (that the body of a recursive expression will evaluate without attempting to access the undefined recursive variable), which avoids some unnecessary run-time costs associated with backpatching. To ensure well-founded recursion in the presence of multiple recursive variables and separate compilation, we track the usage of individual recursive variables, represented statically by "names". So that our type system may eventually be integrated smoothly into ML's, reasoning involving names is only required inside code that uses our recursive construct and does not need to infect existing ML code.

1996

We study recursive types from a syntactic perspective. In particular, we compare the formulations of recursive types that are used in programming languages and formal systems. Our main tool is a new syntactic explanation of type expressions as functors. We also introduce a simple logic for programs with recursive types in which we carry out our proofs. 1 Introduction Recursive types are common in both programming languages and formal systems. By now, there is a deep and well-developed semantic theory of recursive types. The syntactic aspects of recursive types are also well understood in some special cases. In particular, there is an important body of knowledge about covariant recursive types, which include datatypes like natural numbers, lists, and trees. Beyond the covariant case, however, the syntactic understanding of recursive types becomes rather spotty. Consequently, the relations between various alternative formulations of recursive types are generally unclear. Furthermore, ...

Information and Computation, 1997

2003

It is well known that there are problems associated with formal systems which attempt to combine higher order abstract syntax (HOAS) with principles of induction and recursion. We describe a formal system, called Bsyntax, which we have implemented in Isabelle HOL. Our contribution is to prove the existence of a combinator for primitive recursion with parameters over HOAS. The definition of the combinator is facilitated by the use of terms with infinite contexts. In particular, our work is purely definitional, and is thus consistent with classical logic and choice. An immediate payoff is that we obtain a primitive recursive definition of higher order substitution. We give a presheaf model of Bsyntax, providing additional semantic validation of Bsyntax's principles of recursion. We outline an application of our work to mechanized reasoning about the compiler intermediate language MIL-lite [2].

Acta Informatica, 1989

We develop the semantics of a language with arbitrary atomic statements, unbounded nondeterminacy, and mutual recursion. The semantics is expressed in weakest preconditions and weakest liberal preconditions. Individual states are not mentioned. The predicates on the state space are treated as elements of a distributive lattice. The semantics of recursion is constructed by means of the theorem of Knaster-Tarski. It is proved that the law of the excluded miracle can be preserved, if that is wanted. The universal conjunctivity of the weakest liberal precondition, and the connection between the weakest precondition and the weakest liberal precondition are proved to remain valid. Finally we treat Hoare-triple methods for proving correctness and conditional correctness of programs.

Lecture Notes in Computer Science, 2010

Final coalgebras for a functor serve as semantic domains for state based systems of various types. For example, formal languages, streams, nonwell-founded sets and behaviors of CCS processes form final coalgebras. We present a uniform account of the semantics of recursive definitions in final coalgebras by combining two ideas: (1) final coalgebras are also initial completely iterative algebras (cia); (2) additional algebraic operations on final coalgebras may be presented in terms of a distributive law λ. We first show that a distributive law leads to new extended cia structures on the final coalgebra. Then we formalize recursive function definitions involving operations given by λ as recursive program schemes for λ, and we prove that unique solutions exist in the extended cias. We illustrate our results by the four concrete final coalgebras mentioned above, e. g., a finite stream circuit defines a unique stream function and we show how to define new process combinators from given ones by sos rules involving recursion.

Journal of the ACM, 2006

The nominal approach to abstract syntax deals with the issues of bound names and α-equivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalization of common, but often technically incorrect uses of structural recursion and induction for abstract syntax modulo α-equivalence. At the heart of this approach is the notion of finitely supported mathematical objects. This article explains the idea in as concrete a way as possible and gives a new derivation within higher-order classical logic of principles of α- structural recursion and induction for α-equivalence classes from the ordinary versions of these principles for abstract syntax trees.

The expressive power of higher order logic makes it possible to define a wide variety of types within the logic and to prove theorems that state the properties of these types concisely and abstractly. This paper contains a tutorial introduction to the logical basis for such type definitions. Examples are given of the formal definitions in logic of several simple types. A method is then described for systematically defining any instance of a certain class of commonly-used recursive types. The automation of this method in HOL, an interactive system for generating proofs in higher order logic, is also discussed.

2015

The theory of recursive functions where the domain of a function is inductively defined at the same time as the function is called induction-recursion. This theory has been introduced in Martin-Löf type theory by Dybjer [37] and further explored in a series of papers by Dybjer and Setzer [38, 39, 40]. Important data types like universes closed under dependent type operators are instances of this theory. In this thesis we study the class of data types arising from inductive-recursive definitions, taking the seminal work of Dybjer and Setzer as our starting point. We show how the theories of inductive and indexed inductive types arise as sub-theories of induction-recursion, by revealing the role played by a notion of of size within the theory of induction-recursion. We then expand the expressive power of induction-recursion, showing how to extend the theory of induction-recursion in two different ways: in one direction we investigate the changes needed to obtain a more flexible semantics which gives rise to a more comprehensive elimination principle for inductive-recursive types. In another direction we generalize the theory of induction-recursion to a fibrational setting. In both extensions we provide a finite axiomatization of the theories introduced, we show applications and examples of these theories not previously covered by induction-recursion, and we justify the existence of data types built within these theories. v 7 Conclusion 7.

1991

Existing approaches to semantics of algebraically specified data types such as Initial Algebra Semantics and Final Algebra Semantics do not take into account the possibility of general recursion and hence nontermination in the ambient programming language. Any technical development of this problem needs to be in the setting of domain theory. In this paper we present extensions of initial and final algebra semantics to algebras with an underlying domain structure. Four possibilities for specification methodologies arise: two each in the Initial and Final algebra paradigms. We demonstrate that the initial/final objects (as appropriate) exist in all four situations. The final part of the paper attempts to explicate the notion of abstractness of ADT's by defining a notion of operational semantics for ADT's, and then studying the relationship between the various algebraic-semantics proposed and the operational semantics. Comments University of Pennsylvania Department of Computer ...