Recursively Defined Research Articles

Bisimulation as path type for guarded recursive types

In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, allowing this to be encoded in types. Indeed, coinductive types can be encoded using a combination of guarded recursion and universal quantification over clocks. This paper studies the notion of bisimilarity for guarded recursive types in Ticked Cubical Type Theory, an extension of Cubical Type Theory with guarded recursion. We prove that, for any functor, an abstract, category theoretic notion of bisimilarity for the final guarded coalgebra is equivalent (in the sense of homotopy type theory) to path equality (the primitive notion of equality in cubical type theory). As a worked example we study a guarded notion of labelled transition systems, and show that, as a special case of the general theorem, path equality coincides with an adaptation of the usual notion of bisimulation for processes. In particular, this implies that guarded recursion can be used to give simple equational reasoning proofs of bisimilarity. This work should be seen as a step towards obtaining bisimilarity as path equality for coinductive types using the encodings mentioned above.

A relational logic for higher-order programs

AbstractRelational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security, or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called relational higher-order logic (RHOL), for proving relational properties of a simply typed λ-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavor of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic, and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule, and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.

Constructive uniformities of pseudometrics and Bishop topologies

We develop the first steps of a constructive theory of uniformities given by pseudometrics and study its relation to the constructive theory of Bishop topologies. Both these concepts are constructive, function-theoretic alternatives to the notion of a topology of open sets. After motivating the constructive study of uniformities of pseudometrics we present their basic theory and we prove a Stone-Cech theorem for them. We introduce the f-uniform spaces and we prove a Tychonoff embedding theorem for them. We study the uniformity of pseudometrics generated by some Bishop topology and the pseudo-compact Bishop topology generated by some uniformity of pseudometrics. Defining the large uniformity on reals we prove a ``large'' version of the Tychonoff embedding theorem for f-uniform spaces and we show that the notion of morphism between uniform spaces captures Bishop continuity. We work within Bishop's informal system of constructive mathematics BISH extended with inductive definitions with rules of countably many premisses.

DefunT: A Tool for Automating Termination Proofs by Using the Community Books (Extended Abstract)

We present a tool that automates termination proofs for recursive definitions by mining existing termination theorems.

Validating Back-links of FOLID Cyclic Pre-proofs

Cyclic pre-proofs can be represented as sets of finite tree derivations with back-links. In the frame of the first-order logic with inductive definitions, the nodes of the tree derivations are labelled by sequents and the back-links connect particular terminal nodes, referred to as buds, to other nodes labelled by a same sequent. However, only some back-links can constitute sound pre-proofs. Previously, it has been shown that special ordering and derivability conditions, defined along the minimal cycles of the digraph representing a particular normal form of the cyclic pre-proof, are sufficient for validating the back-links. In that approach, a same constraint could be checked several times when processing different minimal cycles, hence one may require additional recording mechanisms to avoid redundant computation in order to downgrade the time complexity to polynomial. We present a new approach that does not need to process minimal cycles. It based on a normal form that allows to define the validation conditions by taking into account only the root-bud paths from the non-singleton strongly connected components of its digraph.

Categoricity Results and Large Model Constructions for Second-Order ZF in Dependent Type Theory

We formalise second-order ZF set theory in the dependent type theory of Coq. Assuming excluded middle, we prove Zermelo’s embedding theorem for models, categoricity in all cardinalities, and the categoricity of extended axiomatisations fixing the number of Grothendieck universes. These results are based on an inductive definition of the cumulative hierarchy eliminating the need for ordinals and set-theoretic transfinite recursion. Following Aczel’s sets-as-trees interpretation, we give a concise construction of an intensional model of second-order ZF with a weakened replacement axiom. Whereas this construction depends on no additional logical axioms, we obtain intensional and extensional models with full replacement assuming a description operator for trees and a weak form of proof irrelevance. In fact, these assumptions yield large models with n Grothendieck universes for every number n.

PROOF-THEORETIC STRENGTHS OF WEAK THEORIES FOR POSITIVE INDUCTIVE DEFINITIONS

AbstractIn this article the lightface ${\rm{\Pi }}_1^1$-Comprehension axiom is shown to be proof-theoretically strong even over ${\rm{RCA}}_0^{\rm{*}}$, and we calibrate the proof-theoretic ordinals of weak fragments of the theory ${\rm{I}}{{\rm{D}}_1}$ of positive inductive definitions over natural numbers. Conjunctions of negative and positive formulas in the transfinite induction axiom of ${\rm{I}}{{\rm{D}}_1}$ are shown to be weak, and disjunctions are strong. Thus we draw a boundary line between predicatively reducible and impredicative fragments of ${\rm{I}}{{\rm{D}}_1}$.

Aggregates in Answer Set Programming

Aggregates are among the most important linguistic extensions of Answer Set Programming (ASP), allowing for compact representations of properties and inductive definitions involving sets of propositions. Common use cases of aggregates in ASP are reported in this paper, which mainly focus on the semantics implemented by mainstream solvers, namely the F-stable model semantics. Other well-established semantics are also briefly discussed, providing a historical perspective on the foundation of logic programs with aggregates.

ABSTRACT INDUCTIVE AND CO-INDUCTIVE DEFINITIONS

AbstractIn [G. Curi, On Tarski’s fixed point theorem. Proc. Amer. Math. Soc., 143 (2015), pp. 4439–4455], a notion of abstract inductive definition is formulated to extend Aczel’s theory of inductive definitions to the setting of complete lattices. In this article, after discussing a further extension of the theory to structures of much larger size than complete lattices, as the class of all sets or the class of ordinals, a similar generalization is carried out for the theory of co-inductive definitions on a set. As a corollary, a constructive version of the general form of Tarski’s fixed point theorem is derived.

Encountering the muse: An exploration of the relationship between inspiration and information in the museum context

How are information and inspiration connected? Answering this question can help information professionals facilitate the pathways to inspiration. Inspiration has previously been conceptualized as a goal or mode of information seeking, but this says little about the nature of inspiration or how it is experienced. In this study, we explore the connection between information and inspiration through a qualitative approach, using the museum as our setting; specifically, the researchers’ own visits to three separate museums. We use collaborative auto-hermeneutics, a methodology specifically suited to such a reflexive exploration, to document and analyze three individual museum visits. The following research questions were the main driver for this exploratory study: What is inspiration, and How are inspiration and information related? In answer, we present an inductive definition of inspiration as a kind of information, and we discuss how this definition fits in with the information science literature as well as offer some practical applications.

Vector Programming Using Structural Recursion

Vector programming is an important topic in many Introduction to Computer Science courses. Despite the importance of vectors, learning vector programming is a source for frustration to many students given that they feel left adrift when it comes to resolving vector indexing errors. Even though the size of a vector is a natural number, there have been no efforts to define a useful recursive data definition to help beginners design vector processing functions. This article defines the concept of a vector interval and describes how to exploit its recursive structure to design vector processing functions. The described methodology provides a context beginners can use to reason about proper vector indexing instead of leaving them adrift with this responsibility. A key feature of properly using the described methodology is that if students process the correct vector interval then vector indexing errors can not arise. The classroom deployment of this approach is described in detail. Students, to date, have found vector intervals helpful in avoiding out-of-bounds indexing errors when all the vector elements of the interval are processed.

Recursive approach for non-Markovian time-convolutionless master equations

We consider a general open system dynamics and we provide a recursive method to derive the associated non-Markovian master equation in a perturbative series. The approach relies on a momenta expansion of the open system evolution. Unlike previous perturbative approaches of this kind, the method presented in this paper provides a recursive definition of each perturbative term. Furthermore, we give an intuitive diagrammatic description of each term of the series, which pro- vides an useful analytical tool to build them and to derive their structure in terms of commutators and anticommutators. We eventually apply our formalism to the evolution of the observables of the reduced system, by showing how the method can be applied to the adjoint master equation, and by developing a diagrammatic description of the associated series.

A DIRECT PROOF OF SCHWICHTENBERG’S BAR RECURSION CLOSURE THEOREM

AbstractIn [12], Schwichtenberg showed that the System T definable functionals are closed under a rule-like version Spector’s bar recursion of lowest type levels 0 and 1. More precisely, if the functional Y which controls the stopping condition of Spector’s bar recursor is T-definable, then the corresponding bar recursion of type levels 0 and 1 is already T-definable. Schwichtenberg’s original proof, however, relies on a detour through Tait’s infinitary terms and the correspondence between ordinal recursion for $\alpha < {\varepsilon _0}$ and primitive recursion over finite types. This detour makes it hard to calculate on given concrete system T input, what the corresponding system T output would look like. In this paper we present an alternative (more direct) proof based on an explicit construction which we prove correct via a suitably defined logical relation. We show through an example how this gives a straightforward mechanism for converting bar recursive definitions into T-definitions under the conditions of Schwichtenberg’s theorem. Finally, with the explicit construction we can also easily state a sharper result: if Y is in the fragment Ti then terms built from $BR^{\mathbb{N},\sigma } $ for this particular Y are definable in the fragment ${T_{i + {\rm{max}}\left\{ {1,{\rm{level}}\left( \sigma \right)} \right\} + 2}}$.

Рекурсивные определения реляционных преобразований

Рекурсивные определения реляционных преобразований

Foundations for natural proofs and quantifier instantiation

We give foundational results that explain the efficacy of heuristics used for dealing with quantified formulas and recursive definitions. We develop a framework for first order logic (FOL) over an uninterpreted combination of background theories. Our central technical result is that systematic term instantiation is complete for a fragment of FOL that we call safe . Coupled with the fact that unfolding recursive definitions is essentially term instantiation and with the observation that heap verification engines generate verification conditions in the safe fragment explains the efficacy of verification engines like natural proofs that resort to such heuristics. Furthermore, we study recursive definitions with least fixpoint semantics and show that though they are not amenable to complete procedures, we can systematically introduce induction principles that in practice bridge the divide between FOL and FOL with recursive definitions.

Inductive definitions

Inductive definitions

SymMerge 알고리즘의 재귀적 정의

병합 알고리즘은 원소의 개수가 과 인 두 개의 배열된 인접수열을 원소의 개수가 인 배열된 하나의 수열로 만드는 알고리즘이다. 우리는 SymMerge라고 명명한 스테이블하고, 비트의 메모리를 추가적으로 필요로 하는 병합알고리즘을 소개하였고 비교횟수 그리고 덮어쓰기 (assignment)와 관련해 최악의 경우에 대한 복잡도 분석을 소개했다. 그리고 기존의 논문에서 우리는 SymMerge 알고리즘의 정의를 유사부호 (Pseudocode)를 이용해 묘사하였다.BR 이 논문에서 우리는 -재귀함수를 이용해 SymMerge 알고리즘이 정의 될 수 있음을 보이고자 한다. 추가로 이러한 재귀적 정의를 바탕으로 원소의 개수가 과 인 두 개의 입력수열을 병합하기 위해 요구되는 비교횟수가 재귀 함수에 의해 계산될 수 있음을 보인다.

A relational logic for higher-order programs

Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called Relational Higher Order Logic (RHOL), for proving relational properties of a simply typed λ-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavour of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic (HOL), and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.

One-particle many-body Green's function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms.

A thorough analytical and numerical characterization of the whole perturbation series of one-particle many-body Green's function (MBGF) theory is presented in a pedagogical manner. Three distinct but equivalent algebraic (first-quantized) recursive definitions of the perturbation series of the Green's function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green's function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 but converges at the full-configuration-interaction (FCI) limit at n=∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approximation. The diagrammatic linkedness and thus size-consistency of the one-particle Green's function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green's function to expose a self-energy diagram and the removal of reducible diagrams are also justified mathematically using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams.

Exchange process-based social mechanisms and social functions: an operational approach to the macro functional aspects of agent societies

This paper presents elements for an operational approach to the formal modeling of the macro functional aspects of agent societies. The concept of agent society used in the paper is summarized. The exchange process-based concept of elementary social function is reviewed and a corresponding concept of elementary social mechanism is introduced. Together, these concepts allow for the recursive definition of the concept of functional system, with which one can account for the general functions performed by the core organizational structure of agent societies. Two case studies are developed to illustrate the type of functional modeling of agent societies that is enabled by the concepts introduced in the paper. The first case study concerns the functional analysis of a simple motivating thought experiment. The second concerns the use of agent societies as formal models for natural societies: it sketches the formalization of Pierre Bourdieu’s functional analysis of the reproduction process of contemporary human societies.