This paper presents a new characterization of provably recursive functions of first-order arithmetic. We consider functions defined by sets of equations. The equations can be arbitrary, not necessarily defining primitive recursive, or even total, functions. The main result states that a function is provably recursive iff its totality is provable (using natural deduction) from the defining set o...

Wander types are a coinductive version of inductive-recursive definitions. They are defined by simultaneously specifying the constructors of the type and a function on the type itself. The types of the constructors can refer to the function component and the function itself is given by pattern matching on the constructors. Wander types are different from inductive-recursive types in two ways: t...

Wander types are a coinductive version of inductive-recursive definitions. They are defined by simultaneously specifying the constructors of the type and a function on the type itself. The types of the constructors can refer to the function component and the function itself is given by pattern matching on the constructors. Wander types are different from inductive-recursive types in two ways: t...

The class of recursive functions over the reals, denoted by REC(R), was introduced by Cristopher Moore in his seminal paper written in 1995. Since then many subsequent investigations brought new results: the class REC(R) was put in relation with the class of functions generated by the General Purpose Analogue Computer of Claude Shannon; classical digital computation was embedded in several ways...

1. Definition of recursive functions. In this paper, we shall consider certain reductions in the recursion scheme for defining primitive recursive functions. Hereafter, we shall refer to such functions simply as recursive functions. In §1, we define what is meant by a recursive function, and also define some recursive functions which will be used. The statement of the principal results of the p...

This manuscript contains technical results related to a particular approach for the design of Model Predictive Control (MPC) laws. The approach, named “generalized” terminal state constraint, induces the recursive feasibility of the underlying optimization problem and recursive satisfaction of state and input constraints, and it can be used for both tracking MPC (i.e. when the objective is to t...

Assume that we may prove in Arithmetic with Comprehension axiom that a primitive recursive binary relation R is well-founded, using the inductive definition of well-founded. In this paper we prove that the proof that R is well-founded may be made intuitionistic. Our result generalizes to any implication between such formulas. We conclude that if we are able to formulate any mathematical problem...

GSOS RULES AND A MODULAR TREATMENT OF RECURSIVE DEFINITIONS STEFAN MILIUS , LAWRENCE S. MOSS , AND DANIEL SCHWENCKE c a Lehrstuhl für Theoretische Informatik, Friedrich-Alexander Universität Erlangen-Nürnberg, Germany e-mail address: [email protected] b Department of Mathematics, Indiana University, Bloomington, IN, USA e-mail address: [email protected] c Institute of Transportation System...

Recursive inference rules arise in recursive definitions in logic programming systems and in database systems with recursive query languages. Let D be a recursive definition of a relation t. We say that D is minimal if for any predicate p in a recursive rule in D, p must appear in a recursive rule in any definition of t. We show that testing for minimality is in general undecidable. However, we...