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This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly infinitary operators and equations. We prove that QWI types can be derived from quotient types and inductive types in the type theory of toposes with natural number object and universes, provided those universes satisfy the Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types as colimits of a family of approximations to them defined by well-founded recursion over a suitable notion of size, whose definition involves the WISC axiom. We developed the proof and checked it using the Agda theorem prover.

Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence. Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting - a natural operation which gives a new type for any binary relation on a type and, in order to be well-behaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty. We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1-types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean., v2: essentially the version published in the proceedings of Logic in Computer Science, numbering identical

Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative conception of set has been widely recognized as insufficient to establish replacement and recursion, its supplementation by considerations pertaining to algorithms suggests a new and philosophically well-motivated reason to believe in such principles.

Hay and, then, Johnson extended the classic Rice and Rice-Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work (with some motivations presented) to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, . Other cases are done for all transfinite notations in a very natural, proper subsystem of , where has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in .

Labelled transition systems are generally recognised as an appropriate model for nondeterministic computations. This paper presents a fundamental theory of well-founded relations and relational systems in Schröder categories.

Transfinite electrical networks of ranks larger than 1 have previously been defined by arbitrarily joining together various infinite extremities through transfinite nodes that are independent of the networks' resistance values. Thus, some or all of those transfinite nodes may remain ineffective in transmitting current "through infinity." In this paper, transfinite nodes are defined in terms of the paths that permit currents to "reach infinity." This is accomplished by defining a suitable metricd v on the node setN S v of eachv-sectionS v, av-section being a maximal subnetwork whose nodes are connected by two-ended paths of ranks no larger thanv. Upon taking the completion ofN S v under that metricd v, we identify those extremities (now calledv-terminals) that are accessible to current flows. These are used to define transfinite nodes that combine such extremities. The construction is recursive and is carried out through all the natural number ranks, and then through the first arrow rank � and the first limit-ordinal rank �. The recursion can be carried still further. All this provides a more natural development of transfinite networks and indeed simplifies the theory of electrical behavior for such networks.

If a and /3 are ordinals, a < /3, and / a ? a, then a + 1 < P. The first result of this paper shows that the restriction of this statement to countable well orderings is provably equivalent to ACAO, a subsystem of second order arithmetic introduced by Friedman. The proof of the equivalence is reminiscent of Dekker's construction of a hypersimple set. An application of the theorem yields the equivalence of the set comprehension scheme ACAo and an arithmetical transfinite induction scheme. Almost every theorem of ordinal arithmetic that has been analyzed in the framework of reverse mathematics is either provable in RCAo or equivalent to ATRo. One notable exception is Girard's proof that ACAo is equivalent to the assertion that ordinal exponentiation preserves well orderings. (See [6] or [7] for a proof.) As shown below, ACAo is also necessary and sufficient to prove the equivalence of two natural definitions of strict inequality for ordinals. Furthermore, this result leads to a precise analysis of the logical strength of an arithmetical transfinite induction scheme. This paper uses some axiom systems introduced by Friedman [4]. RCAo, ACAo, and ATRo are subsystems of second order arithmetic in which induction is restricted to Y? formulas with parameters, and the only set existence principles are taken to be recursive (AO) comprehension, arithmetic comprehension, and a form of arithmetical transfinite recursion, respectively. In these theories, a countable well ordering is defined to be a set R coding a linear order of N, such that every nonempty subset of the field of R has an R-least element. In RCAO, this last statement is equivalent to the assertion that there are no infinite R-descending sequences. For a short introduction to the systems and the program of reverse mathematics, see Simpson [9]. More details on the encoding of well orderings and formalizations of arithmetical operations on them can be found in [5], [7], or [10]. Although technically our discussion is limited to arithmetical operations on countable well orderings, we use the typical notation for ordinal arithmetic. ?1. Two notions of strict inequality. If a and fi are well orderings, and there is an order preserving bijection between a and an initial segment of Pi, then we write Received April 7, 1997. 1991 Mathematics Subject Classification. 03F35.

Pol (1996) and Arenas (1996) independently introduced transfinite extensions of finite order of mappings by the use of the length of a partially ordered set and Borst's order, respectively. By use of the transfinite order of mappings, Arenas introduced a transfinite dimension O-dim based on the Morita's theorem and proved that every countable-dimensional compact metric space has O-dim. Then he asked whether the converse is true. In the present note, we shall show that both the transfinite extensions given by Pol and Arenas are the same if we ignore the values, and give an affirmative answer to Arenas' question as follows: a metrizable space X has the order dimension O-dim X if and only if X has large transfinite dimension Ind X . Furthermore, we shall prove that if a metrizable space X has the order dimension O-dim, then Ind X ⩽ O -dim X and O -dim S α = α for every ordinal number α ω 1 , where S ga is Smirnov's compactum.

Four proof rules for recursive procedures in a Pascal-like language are presented. The main rule deals with total correctness and is based on results of Gries and Martin. The rule is easier to apply than Martin's. It is introduced as an extension of a specification format for Pascal-procedures, with its associated correctness and invocation rules. It uses well-founded recursion and is proved under the postulate that a procedure is semantically equal to its body. This rule for total correctness is compared with Hoare's rule for partial correctness of recursive procedures, in which no well-founded relation is needed. Both rules serve to prove correctness, i.e. sufficiency of certain preconditions. There are also two rules for proving necessity of preconditions. These rules can be used to give formal proofs of nontermination and refinement. They seem to be completely new.

Some new double analogues of induction and transfinite recursion are given which yields a relatively simple proof of a result of Robert Cowen, [2] which in turn is a strengthening of an earlier result of Smullyan [1], which in turn gives a unified approach to Zorn's Lemma, the transfinite recursion theorem and certain results about ordinal numbers.

We prove the strong normalization theorem for the natural deduction system for the constructive arithmetic TRDB (the system with Definition by Transfinite Recursion and Bar induction), which was introduced by Yasugi and Hayashi. We also establish the consistency of this system, applying the strong normalization theorem.

Four proof rules for recursive procedures in a Pascal-like language are presented. The main rule deals with total correctness and is based on results of Gries and Martin. The rule is easier to apply than Martin's. It is introduced as an extension of a specification format for Pascal-procedures, with its associated correctness and invocation rules. It uses well-founded recursion and is proved under the postulate that a procedure is semantically equal to its body. This rule for total correctness is compared with Hoare's rule for partial correctness of recursive procedures, in which no well-founded relation is needed. Both rules serve to prove correctness, i.e. sufficiency of certain preconditions. There are also two rules for proving necessity of preconditions. These rules can be used to give formal proofs of nontermination and refinement. They seem to be completely new.

Four proof rules for recursive procedures in a Pascal-like language are presented. The main rule deals with total correctness and is based on results of Gries and Martin. The rule is easier to apply than Martin's. It is introduced as an extension of a specification format for Pascal-procedures, with its associated correctness and invocation rules. It uses well-founded recursion and is proved under the postulate that a procedure is semantically equal to its body. This rule for total correctness is compared with Hoare's rule for partial correctness of recursive procedures, in which no well-founded relation is needed. Both rules serve to prove correctness, i.e. sufficiency of certain preconditions. There are also two rules for proving necessity of preconditions. These rules can be used to give formal proofs of nontermination and refinement. They seem to be completely new.

We define the notion of transfinite term rewriting: rewriting in which terms may be infinitely large and rewrite sequences may be of any ordinal length. For orthogonal rewrite systems, some fundamental properties known in the finite case are extended to the transfinite case. Among these are the Parallel Moves lemma and the Unique Normal Form property. The transfinite Church-Rosser property (CR∞) fails in general, even for orthogonal systems, including such well-known systems as combinatory logic. Syntactic characterisations are given of some classes of orthogonal TRSs which do satisfy CR∞. We also prove a weakening of CR∞ for all orthogonal systems, in which the property is only required to hold up to a certain equivalence relation on terms. Finally, we extend the theory of needed reduction from the finite to the transfinite case. The reduction strategy of needed reduction is normalising in the finite case, but not in the transfinite case. To obtain a normalising strategy, it is necessary and sufficient to add a requirement of fairness. Parallel outermost reduction is such a strategy.

Martin-L��f's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and relations formed by inverse images, addition, multiplication, and exponentiation of other relations. The constructions are given in full detail to allow their use in theorem provers for Type Theory, such as Nuprl. The theory is compared with work in the field of ordinal recursion over higher types.

The following is a self-contained proof theoretic treatment of two of the principal axiom schemata of current intuitionistic analysis: the axiom of bar induction (Brouwer's bar theorem) and the axiom of continuity. The results are formulated in terms of formal derivability in elementary intuitionistic analysis H(§ 1), so the positive (i.e., derivability) results also apply to elementary classical analysis Z1 (Appendix 1). Both schemata contain the combination of quantifiers νfΛn, where f, g, … are intended to range over free choice sequences of suitable kinds of objects x, y, …; for example, natural numbers or sequences of natural numbers, and n, m, p, r, … over natural numbers (non-negative integers).

the reduced sequence corresponding to (W)n, modulo m. It is easily shown the (A)n is periodic; following Carmichael,: we shall call its smallest period, r, the characteristic number of (W)n modulo m. The object of this memoir is to attack the following fundamental distribution problem:? Given the numerical values of the integers P, Q, R, WO, W1, W2, m and r, to determine the distribution of the residues 0, 1, 2, m, m -1 among any r terms of the reduced sequence (A)n. There are really two distinct problems involved here: the determination of the particular place a given residue occurs in (A)n and the determination of the number of times a given residue occurs in any r terms of (A)n. Both

we shall call (A)n the reduced sequence corresponding to (W)n modulo m. If after s terms in the reduced sequence, a cycle of t terms keeps repeating itself indefinitely, (W)n will be said to admit the period t, modulo m. The least period that (W)n admits (modulo m) is called its characteristic number.t In this paper, I give a number of new results on the form of the characteristic number of a sequence. The principal result is the following

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are given in §2. The additional rules have the form of definition by transfinite recursion up to some ordinal ξ (where ξ is represented by a primitive recursive (p.r.) ordering). In §3 we discuss some elementary closure properties (under rules of inference and definition) of systems with recursion up to ξ. Let Rξ denote (temporarily) the system with recursion up to ξ. The main results of this paper are of two sorts:Sections 5–7 are concerned with less elementary closure properties of the systems Rξ. Namely, we show that certain classes of functional equations in Rη can be solved in Rη for some explicitly determined η < ε(η) (the least ε-number > ξ). The classes of functional equations considered all have roughly the form of definition by recursion on the partial ordering of unsecured sequences of a given functional F, or on some ordering which is obtained from this by simple ordinal operations. The key lemma (Theorem 1) needed for the reduction of these equations to transfinite recursion is simply a sharpening of the Brouwer-Kleene idea.

Publisher Summary This chapter addresses some technical questions related to the formal system ATR 0 of arithmetical transfinite recursion with quantifier free induction on the natural numbers. This system and all of the formal systems considered in this chapter are subsystems of second order arithmetic and use classical logic. A formula is said to be arithmetical if it contains no set quantifiers.

Publisher Summary This chapter discusses number theory and the Bachmann/Howard ordinal. The work of Gentzen and Hilbert/Bernays in the 1930s ɛ o is considered to be the characteristic ordinal of classical number theory. It is the least upper bound on the derivability of transfinite induction in number theory and the consistency of this theory can be proved by transfinite induction up to ɛ o included. Gentzen's technique of establishing the consistency of a formal system by transfinite induction up to a certain ordinal, and of showing transfinite induction to be derivable within the system for each smaller ordinal has been successfully applied to many other theories—especially to various subsystems of analysis. The characteristic ordinal found in this way is often interpreted as a measure of the proof-theoretical strength of the considered system. The proof of Girard's conjecture is based on the fact that the induction principles have recursion counterparts in fast- and slow-growing hierarchies of number-theoretic functions.