Recursively Defined Research Articles

Transforming orthogonal inductive definition sets into confluent term rewrite systems

In this paper, we transform an orthogonal inductive definition set, which is a set of productions for inductive predicates, into a confluent term rewrite system such that a quantifier-free sequent is valid w.r.t. the inductive definition set if and only if an equation representing the sequent is an inductive theorem of the term rewrite system. To this end, we first propose a transformation of an orthogonal inductive definition set into a confluent term rewrite system that is equivalent to the inductive definition set in the sense of evaluating ground formulas. Then, we show that termination of the inductive definition set is proved by the generalized subterm criterion if and only if termination of the transformed term rewrite system is so. Finally, we show that the transformed term rewrite system with some rewrite rules for sequents has the expected property. In addition, we show a termination criterion for the union of term rewrite systems whose termination is proved by the generalized subterm criterion.

AN AXIOMATIC APPROACH TO FORCING IN A GENERAL SETTING

AbstractThe technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which one still has control from the ground model, and has shown that these axiomatics necessarily lead to the usual concepts of genericity and of forcing extensions, and also that one can infer from them the usual recursive definition of forcing predicates. In this paper, we present a more general such approach, covering both class forcing and set forcing, over various base theories, and we provide additional details regarding the formal setting that was outlined in [3].

Visual Modeling of Rice Root Growth Based on B-Spline Curve

As a major food production crop in China, the growth and development of rice is an extremely complex systemic process, and the root system is the main organ for rice to obtain nutrients. Therefore, 3D modeling and visualization of the rice root system can help to further understand its morphology, structure and function, and provide an aid for scientific cultivation of rice and improving rice yield for decision making. In this paper, a mathematical model of the rice root system is established based on the B spline curve combined with the L-system approach, using mathematical knowledge based on the 3D morphological characteristics of the real rice root system. The B-Spline Curve is chosen to simulate this, and the recursive definition of B-Spline Curve and its formula are used to realize the modeling of the rice root system curve. Based on the mathematical method of rice root system integration, the bending effect of rice root system at different periods and different growth positions is realized. Finally, the L-system combined with B-Spline Curve is used to construct a rice root system model and realize the rice root system visualization simulation. The simulated image is closer to the real rice root system image in terms of morphological structure and has a strong sense of realism.

The Importance of Play in Middle School Mathematics

Using question 28 from the May Problems to Ponder in volume 114, the author and her seventh- and eighth-grade students launched into a discussion of creativity, linearity, piecewise, and recursive definitions of functions. This pattern to ponder provided rich mathematical opportunities for all students in my middle school classroom.

Solving constrained Horn clauses modulo algebraic data types and recursive functions

This work addresses the problem of verifying imperative programs that manipulate data structures, e.g., Rust programs. Data structures are usually modeled by Algebraic Data Types (ADTs) in verification conditions. Inductive invariants of such programs often require recursively defined functions (RDFs) to represent abstractions of data structures. From the logic perspective, this reduces to solving Constrained Horn Clauses (CHCs) modulo both ADT and RDF. The underlying logic with RDFs is undecidable. Thus, even verifying a candidate inductive invariant is undecidable. Similarly, IC3-based algorithms for solving CHCs lose their progress guarantee: they may not find counterexamples when the program is unsafe. We propose a novel IC3-inspired algorithm Racer for solving CHCs modulo ADT and RDF (i.e., automatically synthesizing inductive invariants, as opposed to only verifying them as is done in deductive verification). Racer ensures progress despite the undecidability of the underlying theory, and is guaranteed to terminate with a counterexample for unsafe programs. It works with a general class of RDFs over ADTs called catamorphisms. The key idea is to represent catamorphisms as both CHCs, via relationification , and RDFs, using novel abstractions . Encoding catamorphisms as CHCs allows learning inductive properties of catamorphisms, as well as preserving unsatisfiabilty of the original CHCs despite the use of RDF abstractions, whereas encoding catamorphisms as RDFs allows unfolding the recursive definition, and relying on it in solutions. Abstractions ensure that the underlying theory remains decidable. We implement our approach in Z3 and show that it works well in practice.

Iterated multiplication in $$ VTC ^0$$

We show that $VTC^0$, the basic theory of bounded arithmetic corresponding to the complexity class $\mathrm{TC}^0$, proves the $IMUL$ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the $\mathrm{TC}^0$ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, $VTC^0$ can also prove the integer division axiom, and (by our previous results) the RSUV-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories $\Delta^b_1$-$CR$ and $C^0_2$. As a side result, we also prove that there is a well-behaved $\Delta_0$ definition of modular powering in $I\Delta_0+WPHP(\Delta_0)$.

Design, Semiotics, Anticipation

From the Why semiotics? question to the specific aspects of the semiotic underpinning of design the journey is one of discovery. Indeed, design is discovery, i.e., it is anticipation-driven. Therefore, nothing qualifies as its foundation. Design as a process does not require secure preliminaries (theories) from which to set out. It does not require a “place to stand,” a necessary reference, from which to start the adventure. Design assumptions are by their nature questions guiding anticipatory action. They are circumstances of conf lict (the old, the current, the new), which semiotics registers very much like a seismograph. The “earthquake,” i.e., the creative design, is not the graph of the Earth shaking, but a new landscape. The interruptive character of design inquiry, i.e., its disruptive nature, is especially significant when subjected to the après-design analytic moment. This is when recursive definitions (of aesthetic nature, semiotic, economic, cultural, political, etc. significance) are used as a metric of design, creating the illusion that they can become some norm. Actually, design creates a context for meaningful interactions. Design’s self-motivating nature of inquiry escapes such exercises. The activity called design is constitutive of the new, not the celebration of the past. Therefore, we present not only what has so far been learned from a design informed by semiotics, but also what might better serve designers in the context of the rapid change we experience in our days.

Catalan recursion on externally ordered bases of unit interval positroids

The Catalan numbers form a sequence that counts over 200 combinatorial objects. A remarkable property of the Catalan numbers, which extends to these objects, is its recursive definition; that is, we can determine the $n^{th}$ object from previous ones. Matroids are combinatorial objects that generalize the notion of linear independence and have connections with other fields of mathematics. A family of matroids, called unit interval positroids (UIP), are Catalan objects induced by the antiadjacency matrices of unit interval orders. Associated to each UIP is the set of externally ordered bases, which due to Las Vergnas, produces a lattice after adjoining a bottom element. We study the poset of externally ordered UIP bases and the implied Catalan-induced recursion. Explicitly, we describe an algorithm for constructing the lattice of a rank $n$ UIP from the lattice of lower ranks. Using their inherent combinatorial structure, we define a simple formula to enumerate the bases for a given UIP.

Recursion in programs, thought, and language

This article presents a theory of recursion in thinking and language. In the logic of computability, a function maps one or more sets to another, and it can have a recursive definition that is semi-circular, i.e., referring in part to the function itself. Any function that is computable - and many are not - can be computed in an infinite number of distinct programs. Some of these programs are semi-circular too, but they needn't be, because repeated loops of instructions can compute any recursive function. Our theory aims to explain how naive individuals devise informal programs in natural language, and is itself implemented in a computer program that creates programs. Participants in our experiments spontaneously simulate loops of instructions in kinematic mental models. They rely on such loops to compute recursive functions for rearranging the order of cars in trains on a track with a siding. Kolmogorov complexity predicts the relative difficulty of abducing such programs - for easy rearrangements, such as reversing the order of the cars, to difficult ones, such as splitting a train in two and interleaving the two resulting halves (equivalent to a faro shuffle). This rearrangement uses both the siding and part of the track as working memories, shuffling cars between them, and so it relies on the power of a linear-bounded computer. Linguistic evidence implies that this power is more than necessary to compose the meanings of sentences in natural language from those of their grammatical constituents.

The concept of “demand” and its derivatives in the modern scientific insurance discourse

In the scientific literature concepts such concepts as “demand for insurance”, “demand for insurance services”, “insurance demand”, “demand for insurance products”, etc. are often used. While being simple in their intuitive reading, they are in fact not as unambiguous as they may seem at first glance. Since there is a semantic conditionality of the meaning of a derived word by the values of its components and they are also not always well-established, then terminological uncertainty arises. The negative effect is enhanced by the permissible mixing of concepts with each other or the creation of scholastic multivariability. Quite often, the definition of demand as a category is given a characteristic of the level of demand, and without taking into account its dimension and type. Moreover, there are defects in the choice of definitions and the construction of definitions, the dominance of recursive definitions that have no value for the full disclosure of the meaning and content of the defined. The author concludes about the inadmissibility of taking liberties in the interpretation of significant concepts and terms, careful construction of new turns with their participation. The problem of using concepts without giving meaning to their scientific content is posed.

Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, and grand canonical and canonical ensembles.

A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.

E-Cyclist: Implementation of an Efficient Validation of FOLID Cyclic Induction Reasoning

Checking the soundness of cyclic induction reasoning for first-order logic with inductive definitions (FOLID) is decidable but the standard checking method is based on an exponential complement operation for B\"uchi automata. Recently, we introduced a polynomial checking method whose most expensive steps recall the comparisons done with multiset path orderings. We describe the implementation of our method in the Cyclist prover. Referred to as E-Cyclist, it successfully checked all the proofs included in the original distribution of Cyclist. Heuristics have been devised to automatically define, from the analysis of the proof derivations, the trace-based ordering measures that guarantee the soundness property.

Querying RDF Databases with Sub-CONSTRUCTs

Graph query languages feature mainly two kinds of queries when applied to a graph database: those inspired by relational databases which return tables such as SELECT queries and those which return graphs such as CONSTRUCT queries in SPARQL. The latter are object of study in the present paper. For this purpose, a core graph query language GrAL is defined with focus on CONSTRUCT queries. Queries in GrAL form the final step of a recursive process involving so-called GrAL patterns. By evaluating a query over a graph one gets a graph, while by evaluating a pattern over a graph one gets a set of matches which involves both a graph and a table. CONSTRUCT queries are based on CONSTRUCT patterns, and sub-CONSTRUCT patterns come for free from the recursive definition of patterns. The semantics of GrAL is based on RDF graphs with a slight modification which consists in accepting isolated nodes. Such an extension of RDF graphs eases the definition of the evaluation semantics, which is mainly captured by a unique operation called Merge. Besides, we define aggregations as part of GrAL expressions, which leads to an original local processing of aggregations.

Higher order Toda brackets

We describe two ways to define higher order Toda brackets in a pointed simplicial model category $${\mathcal {D}}$$ : one is a recursive definition using model categorical constructions, and the second uses the associated simplicial enrichment. We show that these two definitions agree, by providing a third, diagrammatic, description of the Toda bracket, and explain how it serves as the obstruction to rectifying a certain homotopy-commutative diagram in $${\mathcal {D}}$$ .

Qualitative perspectives of the North Carolina healthy food small retailer program among customers in participating stores located in food deserts

BackgroundThe North Carolina Healthy Food Small Retailer Program (NC HFSRP) was established through a policy passed by the state legislature to provide funding for small food retailers located in food deserts with the goal of increasing access to and sales of healthy foods and beverages among local residents. The purpose of this study was to qualitatively examine perceptions of the NC HFSRP among store customers.MethodsQualitative interviews were conducted with 29 customers from five NC HFSRP stores in food deserts across eastern NC. Interview questions were related to shoppers’ food and beverage purchases at NC HFSRP stores, whether they had noticed any in-store efforts to promote healthier foods and beverages, their suggestions for promoting healthier foods and beverages, their familiarity with and support of the NC HFSRP, and how their shopping and consumption habits had changed since implementation of the NC HFSRP. A codebook was developed based on deductive (from the interview guide questions) and inductive (emerged from the data) codes and operational definitions. Verbatim transcripts were double-coded and a thematic analysis was conducted based on code frequency, and depth of participant responses for each code.ResultsAlthough very few participants were aware of the NC HFSRP legislation, they recognized changes within the store. Customers noted that the provision of healthier foods and beverages in the store had encouraged them to make healthier purchase and consumption choices. When a description of the NC HFSRP was provided to them, all participants were supportive of the state-funded program. Participants discussed program benefits including improving food access in low-income and/or rural areas and making healthy choices easier for youth and for those most at risk of diet-related chronic diseases.ConclusionsFindings can inform future healthy corner store initiatives in terms of framing a rationale for funding or policies by focusing on increased food access among vulnerable populations.

Entailment is Undecidable for Symbolic Heap Separation Logic Formulæ with Non-Established Inductive Rules

Entailment is undecidable in general for Separation (SL) Logic formulæ with inductive definitions, but it has been shown to be decidable [1] if the inductive rules satisfy three conditions, namely progress, connectivity and establishment. We show that entailment is undecidable if the latter condition is dropped, thus drawing a much clearer frontier for (un)decidability.

Countability of Inductive Types Formalized in the Object-Logic Level

The set of integer number lists with finite length, and the set of binary trees with integer labels are both countably infinite. Many inductively defined types also have countably many elements. In this paper, we formalize the syntax of first order inductive definitions in Coq and prove them countable, under some side conditions. Instead of writing a proof generator in a meta language, we develop an axiom-free proof in the Coq object logic. In other words, our proof is a dependently typed Coq function from the syntax of the inductive definition to the countability of the type. Based on this proof, we provide a Coq tactic to automatically prove the countability of concrete inductive types. We also developed Coq libraries for countability and for the syntax of inductive definitions, which have value on their own.

Automating Induction by Reflection

Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires an infinite number of axioms, which is not a feasible input to a computer-aided theorem prover requiring a finite input. Mathematical practice is to specify these infinite sets of axioms as axiom schemes. Unfortunately these schematic definitions cannot be formalized in first-order logic, and therefore not supported as inputs for first-order theorem provers. In this work we introduce a new method, inspired by the field of axiomatic theories of truth, that allows to express schematic inductive definitions, in the standard syntax of multi-sorted first-order logic. Further we test the practical feasibility of the method with state-of-the-art theorem provers, comparing it to solvers' native techniques for handling induction.

Multiparty protocol that usually shuffles

AbstractMultiparty computation is raising importance because its primary objective is to replace any trusted third party in the distributed computation. This work presents two multiparty shuffling protocols where each party, possesses a private input, agrees on a random permutation while keeping the permutation secret. The proposed shuffling protocols are based on permutation network, thereby data‐oblivious. The first proposal is ‐permute that permutes inputs in all possible ways. ‐permute network consists of layers, and in each layer there are gates. Our second protocol is ‐permute shuffling that defines a permutation set where , and the resultant shuffling is a random permutation . The ‐permute network contains less number of layers compare with ‐permute network. Let , the ‐permute network would define layers. The proposed shuffling protocols are unconditionally secure against malicious adversary who can corrupt at most parties. The probability that adversary can learn the outcome of ‐permute is upper bound by . Whereas, the probability that adversary can learn the outcome of ‐permute is upper bounded by , for some positive integer , and a recursive definition of . The protocols allow the parties to build quorums, and distribute the load among the quorums.

Hindsight and Sequential Rationality of Correlated Play

Driven by recent successes in two-player, zero-sum game solving and playing, artificial intelligence work on games has increasingly focused on algorithms that produce equilibrium-based strategies. However, this approach has been less effective at producing competent players in general-sum games or those with more than two players than in two-player, zero-sum games. An appealing alternative is to consider adaptive algorithms that ensure strong performance in hindsight relative to what could have been achieved with modified behavior. This approach also leads to a game-theoretic analysis, but in the correlated play that arises from joint learning dynamics rather than factored agent behavior at equilibrium. We develop and advocate for this hindsight rationality framing of learning in general sequential decision-making settings. To this end, we re-examine mediated equilibrium and deviation types in extensive-form games, thereby gaining a more complete understanding and resolving past misconceptions. We present a set of examples illustrating the distinct strengths and weaknesses of each type of equilibrium in the literature, and prove that no tractable concept subsumes all others. This line of inquiry culminates in the definition of the deviation and equilibrium classes that correspond to algorithms in the counterfactual regret minimization (CFR) family, relating them to all others in the literature. Examining CFR in greater detail further leads to a new recursive definition of rationality in correlated play that extends sequential rationality in a way that naturally applies to hindsight evaluation.