Motivated by applications in computable analysis, we study fixpoints of certain endofunctors over categories of containers. More specifically, we focus on fibred endofunctors over the fibrewise opposite of the codomain fibration that can be themselves be represented by families of polynomial endofunctors. In this setting, we show how to compute initial algebras, terminal coalgebras and another kind of fixpoint ζ. We then explore a number of examples of derived operators inspired by Weihrauch complexity and the usual construction of the free polynomial monad. We introduce ζ-expressions as the syntax of μ-bicomplete categories, extended with ζ-binders and parallel products, which thus have a natural denotation in containers. By interpreting certain ζ-expressions in a category of type-2 computable maps, we are able to capture a number of meaningful Weihrauch degrees, ranging from closed choice on {0,1} to determinacy of infinite parity games, via an "answerable part" operator.

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving two distinguished sets of relational symbols. One set of relations is for the "source data" I, and the other is for the "interface data" O. Such a specification is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a transformation (or "query" below) that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC). We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest, between NRC queries, interpretations, a standard mechanism for defining structure-to-structure translation in logic, and between interpretations and implicit to definability "up to unique isomorphism". The latter connection uses a variation of a result of Gaifman concerning "relatively categorical" theories. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving the source data I and the interface data O. It is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a query that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous effective implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be effectively converted to explicit definitions in the nested relational calculus (NRC). As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

Nested relations, built up from atomic types via product and set types, form a rich data model. Over the last decades the nested relational calculus, NRC, has emerged as a standard language for defining transformations on nested collections. NRC is a strongly-typed functional language which allows building up transformations using tupling and projections, a singleton-former, and a map operation that lifts transformations on tuples to transformations on sets. In this work we describe an alternative declarative method of describing transformations in logic. A formula with distinguished inputs and outputs gives an implicit definition if one can prove that for each input there is only one output that satisfies it. Our main result shows that one can synthesize transformations from proofs that a formula provides an implicit definition, where the proof is in an intuitionistic calculus that captures a natural style of reasoning about nested collections. Our polynomial time synthesis procedure is based on an analog of Craig's interpolation lemma, starting with a provable containment between terms representing nested collections and generating an NRC expression that interpolates between them. We further show that NRC expressions that implement an implicit definition can be found when there is a classical proof of functionality, not just when there is an intuitionistic one. That is, whenever a formula implicitly defines a transformation, there is an NRC expression that implements it.

This paper introduces a new automata-theoretic class of string-to-string functions with polynomial growth. Several equivalent definitions are provided: a machine model which is a restricted variant of pebble transducers, and a few inductive definitions that close the class of regular functions under certain operations. Our motivation for studying this class comes from another characterization, which we merely mention here but prove elsewhere, based on a λ-calculus with a linear type system. As their name suggests, these comparison-free polyregular functions form a subclass of polyregular functions; we prove that the inclusion is strict. We also show that they are incomparable with HDT0L transductions, closed under usual function composition - but not under a certain "map" combinator - and satisfy a comparison-free version of the pebble minimization theorem. On the broader topic of polynomial growth transductions, we also consider the recently introduced layered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that a function can be obtained by composing such transducers together if and only if it is polyregular, and that k-layered SSTs (or k-marble transducers) are closed under "map" and equivalent to a corresponding notion of (k+1)-layered HDT0L systems.

Abstract We study the Horn theories of Kleene algebras and star continuous Kleene algebras, from the complexity point of view. While their equational theories coincide and are PSpace-complete, their Horn theories differ and are undecidable. We characterise the Horn theory of star continuous Kleene algebras in terms of downward closed languages and we show that when restricting the shape of allowed hypotheses, the problems lie in various levels of the arithmetical or analytical hierarchy. We also answer a question posed by Cohen about hypotheses of the form $$1=S$$ where S is a sum of letters: we show that it is decidable.

Abstract We devise a variant of Dialectica interpretation of intuitionistic linear logic for "Equation missing", a linear logic-based version $$\mathsf {MSO}$$ over infinite words. "Equation missing" was known to be correct and complete w.r.t. Church’s synthesis, thanks to an automata-based realizability model. Invoking Büchi-Landweber Theorem and building on a complete axiomatization of $$\mathsf {MSO}$$ on infinite words, our interpretation provides us with a syntactic approach, without any further construction of automata on infinite words. Via Dialectica, as linear negation directly corresponds to switching players in games, we furthermore obtain a complete logic: either a closed formula or its linear negation is provable. This completely axiomatizes the theory of the realizability model of "Equation missing". Besides, this shows that in principle, one can solve Church’s synthesis for a given $$\forall \exists $$ -formula by only looking for proofs of either that formula or its linear negation.

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We study the strength of axioms needed to prove various results related to automata on infinite words and B\"uchi's theorem on the decidability of the MSO theory of $(N, {\le})$. We prove that the following are equivalent over the weak second-order arithmetic theory $RCA_0$: (1) the induction scheme for $\Sigma^0_2$ formulae of arithmetic, (2) a variant of Ramsey's Theorem for pairs restricted to so-called additive colourings, (3) B\"uchi's complementation theorem for nondeterministic automata on infinite words, (4) the decidability of the depth-$n$ fragment of the MSO theory of $(N, {\le})$, for each $n \ge 5$. Moreover, each of (1)-(4) implies McNaughton's determinisation theorem for automata on infinite words, as well as the "bounded-width" version of K\"onig's Lemma, often used in proofs of McNaughton's theorem.

Church's synthesis problem asks whether there exists a finite-state stream transducer satisfying a given input-output specification. For specifications written in Monadic Second-Order Logic (MSO) over infinite words, Church's synthesis can theoretically be solved algorithmically using automata and games. We revisit Church's synthesis via the Curry-Howard correspondence by introducing SMSO, an intuitionistic variant of MSO over infinite words, which is shown to be sound and complete w.r.t. synthesis thanks to an automata-based realizability model.

We propose LMSO, a proof system inspired from Linear Logic, as a proof-theoretical framework to extract finite-state stream transducers from linear-constructive proofs of omega-regular specifications. We advocate LMSO as a stepping stone toward semi-automatic approaches to Church's synthesis combining computer assisted proofs with automatic decisions procedures. LMSO is correct in the sense that it comes with an automata-based realizability model in which proofs are interpreted as finite-state stream transducers. It is moreover complete, in the sense that every solvable instance of Church's synthesis problem leads to a linear-constructive proof of the formula specifying the synthesis problem.

In this paper, we show how to integrate linear types with type dependency, by extending the linear/non-linear calculus of Benton to support type dependency. Next, we give an application of this calculus by giving a proof-theoretic account of imperative programming, which requires extending the calculus with computationally irrelevant quantification, proof irrelevance, and a monad of computations. We show the soundness of our theory by giving a realizability model in the style of Nuprl, which permits us to validate not only the beta-laws for each type, but also the eta-laws. These extensions permit us to decompose Hoare triples into a collection of simpler type-theoretic connectives, yielding a rich equational theory for dependently-typed higher-order imperative programs. Furthermore, both the type theory and its model are relatively simple, even when all of the extensions are considered.