Recursive Languages Research Articles

Recursion across domains ed. by Luiz Amaral et al.

Reviewed by: Recursion across domains ed. by Luiz Amaral et al. Daniel L. Everett and Edward Gibson Recursion across domains. Ed. by Luiz Amaral, Marcus Maia, Andrew Nevins, and Tom Roeper. Cambridge: Cambridge University Press, 2018. Pp. 404. ISBN 9781108418065. $125 (Hb).1 [ Clarification ] 1. Introduction The purpose of this review article is to evaluate the research reported in Recursion across domains (henceforth RAD) in the context of wider issues that impinge upon it, especially the nature of grammar and its functions in human languages. RAD resulted from a conference organized by the editors and authors on the topic of recursion in cognition and language at the Federal University of Rio de Janeiro in August 2013. RAD assumes one particular approach to recursion in language, in which recursion is purported to be a genetically supplied feature of the syntax of human languages (Hauser, Chomsky, & Fitch 2002; henceforth HC&F). The book consists of a foreword and an introductory chapter, followed by eighteen chapters organized in four sections. Four of the chapters across the sections discuss fieldwork or experiments investigating the Pirahã language, ten of the chapters discuss particular constructions involving embedding of various kinds in South American languages, and the remaining four chapters discuss the acquisition of recursive structures in other languages. In what follows, we first review three perspectives on the nature and locus of recursion: the syntactic one that the authors of RAD assume (due to HC&F), and two precursors, which are not specifically syntactic: Peirce’s (1865) semantic proposal, and a special case of Simon’s (1962) proposal that hierarchy is the most economic way to organize information. Next, we examine the basic findings of the non-Pirahã chapters of RAD taken as a group, looking at the empirical lessons they contribute within the larger theoretical context. Section 4—the bulk of our review—discusses the four chapters about Pirahã. We focus on Pirahã because this is the language of most theoretical interest: it is unique among the languages that are investigated here in that it has been claimed to lack syntactic recursion. 2. Different conceptions of recursion RAD addresses theoretical and empirical aspects of recursion for the understanding of human languages. The book assumes Hauser, Chomsky, and Fitch’s (2002) approach to recursion in language, where recursion is assumed to be a genetically supplied feature of the syntax of human languages. This view contrasts with two prominent precursors in linguistics and psychology, neither of which assumes that recursion is specific to the syntax of human language. [End Page 777] In 1865, borrowing a term from the thirteenth-century Modistae, Charles Peirce reintroduced the phrase ‘universal grammar’ into linguistics and semiotics, arguing that the grammar of meaning required recursion for reasons of logic (Peirce 1982 [1865]). Peirce considered grammar to be a device for supporting the accurate interpretation of signs (icons, indexes, and symbols, as manifested in words, propositions, phrases, phonemes, photographs, semaphores, etc.). In Peirce’s version of universal grammar, recursion was a logical requirement on interpretation, not on grammatical structures. For example, a sign like bachelor is interpreted via other signs (e.g. unmarried and man), and this interpretative procedure produces a chain of interpretation of one sign in terms of another, of arbitrary depth. There can be no language without recursion in Peircean semiotics. But recursion in this Peircean sense is not the syntactic phenomenon intended by HC&F. Peirce does not predict that all languages will manifest evidence of syntactic recursion (e.g. coordination, morphosyntactic embedding, factive predicates, etc.; see Everett 2012 for a more complete list). For Peirce, neither nature nor nurture is the source of interpretative recursion. Peirce’s recursion is the sole consequence of logical constraint on semiotic interpretation (see also Everett 2019). In a second influential work on recursion, Simon (1962) argued that hierarchy emerges from a constraint of efficiency of information processing across all domains, because hierarchical structures are inherently more efficient and stable than other ways of organizing a complex system. Hierarchy is found in atomic structure, in the organization of societies, in the way that we process information, in the organization of galaxies, in management...

Context-sensitive languages and G-automata

For a given finitely generated group [Formula: see text], the type of languages that are accepted by [Formula: see text]-automata is determined by the word problem of [Formula: see text] for most of the classical types of languages. We observe that the only exceptions are the families of context-sensitive and recursive languages. Thus, in general, to ensure that the language accepted by a [Formula: see text]-automaton is in the same classical family of languages as the word problem of [Formula: see text], some restriction must be imposed on the [Formula: see text]-automaton. We show that restricting to [Formula: see text]-automata without [Formula: see text]-transitions is sufficient for this purpose. We then define the pullback of two [Formula: see text]-automata and use this construction to study the closure properties of the family of languages accepted by [Formula: see text]-automata without [Formula: see text]-transitions. As a further consequence, when [Formula: see text] is the product of two groups, we give a characterization of the family of languages accepted by [Formula: see text]-automata in terms of the families of languages accepted by [Formula: see text]- and [Formula: see text]-automata. We also give a construction of a grammar for the language accepted by an arbitrary [Formula: see text]-automaton and show how to get a context-sensitive grammar when [Formula: see text] is finitely generated with a context-sensitive word problem and the [Formula: see text]-automaton is without [Formula: see text]-transitions.

Closure properties for fuzzy recursively enumerable languages and fuzzy recursive languages

There are several variations of fuzzy Turing machines in the literature, many of them require a t-norm in order to establish their accepted language. This paper generalize the concept of non-deterministic fuzzy Turing machine - NTFM, replacing the t-norm operator for several aggregation functions. We establish the languages accepted by these machines, called fuzzy recursively enumerable languages or simply LFRE and show, among other results, which classes of LFRE are closed under unions and intersections.

On Boolean Closed Full Trios and Rational Kripke Frames

We study what languages can be constructed from a non-regular language L using Boolean operations and synchronous or non-synchronous rational transductions. If all rational transductions are allowed, one can construct the whole arithmetical hierarchy relative to L. In the case of synchronous rational transductions, we present non-regular languages that allow constructing languages arbitrarily high in the arithmetical hierarchy and we present non-regular languages that allow constructing only recursive languages. A consequence of the results is that aside from the regular languages, no full trio generated by a single language is closed under complementation. Another consequence is that there is a fixed rational Kripke frame such that assigning an arbitrary non-regular language to some variable allows the definition of any language from the arithmetical hierarchy in the corresponding Kripke structure using multimodal logic.

Debates with Small Transparent Quantum Verifiers

We study a model where two opposing provers debate over the membership status of a given string in a language, trying to convince a weak verifier whose coins are visible to all. We show that the incorporation of just two qubits to an otherwise classical constant-space verifier raises the class of debatable languages from at most NP to the collection of all Turing-decidable languages (recursive languages). When the verifier is further constrained to make the correct decision with probability 1, the corresponding class goes up from the regular languages up to at least E. We also show that the quantum model outperforms its classical counterpart when restricted to run in polynomial time, and demonstrate some non-context-free languages which have such short debates with quantum verifiers.

Is Your Database System a Semantic Web Reasoner?

Databases and semantic technologies are an excellent match in scenarios requiring the management of heterogeneous or incomplete data. In ontology-based query answering, application knowledge is expressed in ontologies and used for providing better query answers. This enhancement of database technology with logical reasoning remains challenging—performance is critical. Current implementations use time-consuming pre-processing to materialise logical consequences or, alternatively, compute a large number of large queries to be answered by a database management system (DBMS). Recent research has revealed a third option using recursive query languages to “implement” ontological reasoning in DBMS. For lightweight ontology languages, this is possible using the popular Semantic Web query language SPARQL 1.1, other cases require more powerful query languages like Datalog, which is also seeing a renaissance in DBMS today. Herein, we give an overview of these areas with a focus on recent trends and results.

Reanalysis of ERP Studies on EFL Learners' Language Recursion-based Sentence Parsing

This paper will investigate the possibility that EFL can learn the sentence parsing algorithms. This approach stems from Kim et al.s (2013, 2014) ERP studies which show some changes of EFL learners parsing results by educating them English parsing strategies. Referring to these studies, this paper will assume that learning parsing algorithms can lead the results of learning to be changed into getting closer to the level of English L1 speakers. To justify this assumption, this paper will identify the relevant evidence that the essence of the sentence parsing can be learned while we reanalyze EFL leaners syntactic responses from the previous ERP studies. Furthermore, this paper will also present the theoretical foundation to assume the possibility of learning parsing algorithms resulted from the language recursion: a property of human languages (Hauser et al. 2002, Jackendoff and Pinker 2005, Pinker and Jackendoff 2005). Through this investigation, this paper will claim that, though parsing algorithms vary according to the particular grammars, not only might parsing operations be correlated to the recursion of language, but also parsing algorithms can be learned by EFL due to the language property common to human-beings.

On language equations with concatenation and various sets of Boolean operations

Systems of equations of the form X i = ϕ i (X 1 , ... , X n ), for 1 ⩽ i ⩽ n , in which the unknowns X i are formal languages, and the right-hand sides ϕ i may contain concatenation and union, are known for representing context-free grammars. If, instead of union only, another set of Boolean operations is used, the expressive power of such equations may change: for example, using both union and intersection leads to conjunctive grammars [A. Okhotin, J. Automata, Languages and Combinatorics 6 (2001) 519–535], whereas using all Boolean operations allows all recursive sets to be expressed by unique solutions [A. Okhotin, Decision problems for language equations with Boolean operations, Automata, Languages and Programming, ICALP 2003, Eindhoven, The Netherlands, 239–251]. This paper investigates the expressive power of such equations with any possible set of Boolean operations. It is determined that different sets of Boolean operations give rise to exactly seven families of formal languages: the recursive languages, the conjunctive languages, the context-free languages, a certain family incomparable with the context-free languages, as well as three subregular families.

On the Coding of a Turing Machines and Non- Recursive Languages

On the Coding of a Turing Machines and Non- Recursive Languages

How children perceive fractals: Hierarchical self-similarity and cognitive development

The ability to understand and generate hierarchical structures is a crucial component of human cognition, available in language, music, mathematics and problem solving. Recursion is a particularly useful mechanism for generating complex hierarchies by means of self-embedding rules. In the visual domain, fractals are recursive structures in which simple transformation rules generate hierarchies of infinite depth. Research on how children acquire these rules can provide valuable insight into the cognitive requirements and learning constraints of recursion.Here, we used fractals to investigate the acquisition of recursion in the visual domain, and probed for correlations with grammar comprehension and general intelligence. We compared second (n=26) and fourth graders (n=26) in their ability to represent two types of rules for generating hierarchical structures: Recursive rules, on the one hand, which generate new hierarchical levels; and iterative rules, on the other hand, which merely insert items within hierarchies without generating new levels. We found that the majority of fourth graders, but not second graders, were able to represent both recursive and iterative rules. This difference was partially accounted by second graders’ impairment in detecting hierarchical mistakes, and correlated with between-grade differences in grammar comprehension tasks. Empirically, recursion and iteration also differed in at least one crucial aspect: While the ability to learn recursive rules seemed to depend on the previous acquisition of simple iterative representations, the opposite was not true, i.e., children were able to acquire iterative rules before they acquired recursive representations. These results suggest that the acquisition of recursion in vision follows learning constraints similar to the acquisition of recursion in language, and that both domains share cognitive resources involved in hierarchical processing.

On the syntax-centric theories of human creativity

This article criticizes recent attempts to explain creative thought as an off-shoot of syntactic mechanisms, such as recursion in language, and proposes an alternative. It agrees with the syntax-centric hypothesis on two fronts. First, it is claimed that a supramodal notion of creative thought, or what paleoarcheologists call human innovativeness, emerged in the human evolution in a sudden quantum transformation that occurred between 200,000–50,000 years ago. Second, it is argued that human innovativeness is linked with certain formal linguistic computations. Where it perceives weakness is in the claim that formal recursion by itself could transform a finite mind into an infinite one. A new hypothesis is proposed, according to which recursive symbol manipulation provides a controlled, volitional access to a more primitive, pre-existing creative non-discursive thought potential by means of non-veridical “wild” triggering.

Recursive Compounds and Linking Morpheme

This paper shows that the existence of a linking morpheme is not related to recursion of compounds in the given language, but a linking morpheme does play a role in some ways or the other of recursion. Recursion of compounding is defined as embedding at the edge or in the center of an action or object of an instance of the same type. On the other hand, iteration, is simply unembedded repetition of an action or object (Bisetto 2010). Based on these definitions, it is argued that there are languages with a linking morpheme overtly realized in recursive languages. Second, there are also languages which have genitive compounds with a linking morpheme, although recursive compounds do not have a linking morpheme. On the other hand, there are languages with a genitive compounds and recursive compounds of coordinate VNN or nominal coordinate compounds. In these languages, recursive compounds are not so productive. Finally, Turkish and Greek show that the existence of a linking morpheme is not related to recursion of compounds. They have a linking morpheme in iterated compounds, but not in recursive compounds.

Commentary on Studying Circular Questioning “In Situ”

In their discursive study of circular questioning in a systemic family therapy session, the authors raise some interesting perspectives on the original Milan therapy team's guidelines for the therapist as the conductor of therapy. By emphasizing circularity from a discourse point of view, they suggest these guidelines can be used to help family members hypothesize about their own perspectives on themselves and the other family members in circular terms, and, drawing upon Cecchin's notion of neutrality, by creating a state of curiosity in their talk and maybe in their minds. Their emphasis on circularity also helps us to become more sensitive to ways in which the natural recursion in language can help us to appreciate the new guiding lines circular questions can suggest with previously spoken elements. If we follow this line of argument, then circular questions can be seen as a critical part of a therapy we can call recursive family therapy in which we use the recursive element of all natural living languages to help our clients to recursively change their language and lives naturally.

Mind change speed-up for learning languages from positive data

Within the frameworks of learning in the limit of indexed classes of recursive languages from positive data and automatic learning in the limit of indexed classes of regular languages (with automatically computable sets of indices), we study the problem of minimizing the maximum number of mind changes FM(n) by a learner M on all languages with indices not exceeding n. For inductive inference of recursive languages, we establish two conditions under which FM(n) can be made smaller than any recursive unbounded non-decreasing function. We also establish how FM(n) is affected if at least one of these two conditions does not hold. In the case of automatic learning, some partial results addressing speeding up the function FM(n) are obtained.

Computation with perturbed dynamical systems

This paper analyzes the computational power of dynamical systems robust to infinitesimal perturbations. Previous work on the subject has delved on very specific types of systems. Here we obtain results for broader classes of dynamical systems (including those systems defined by Lipschitz/analytic functions). In particular we show that systems robust to infinitesimal perturbations only recognize recursive languages. We also show the converse direction: every recursive language can be robustly recognized by a computable system. By other words we show that robustness is equivalent to decidability.

Evolutionary scenarios for the emergence of recursion

As we have seen, the evolution of language cannot be addressed from the perspective of a single discipline; others, such as genetics or archaeology, also have a say on this issue. There is still a lot of research needed regarding Neanderthal cognition, an issue that deserves more attention also by biolinguistics. Although there are reasons to think so, the available data do not allow us to exclude recursion either from the cognition of H. neanderthalensis or starlings. Thus, what can be called special in humans or in language? The last part of the next quotation may summarize almost perfectly our own hypothesis about the role of recursion in language and in the whole cerebral architecture related to linguistic and non-linguistic activity.

Inferring descriptive generalisations of formal languages

In the present paper, we introduce a variant of Gold-style learners that is not required to infer precise descriptions of the languages in a class, but that must find descriptive patterns, i.e., optimal generalisations within a class of pattern languages. Our first main result characterises those indexed families of recursive languages that can be inferred by such learners, and we demonstrate that this characterisation shows enlightening connections to Angluinʼs corresponding result for exact inference. Furthermore, this result reveals that our model can be interpreted as an instance of a natural extension of Goldʼs model of language identification in the limit. Using a notion of descriptiveness that is restricted to the natural subclass of terminal-free E-pattern languages, we introduce a generic inference strategy, and our second main result characterises those classes of languages that can be generalised by this strategy. This characterisation demonstrates that there are major classes of languages that can be generalised in our model, but not be inferred by a normal Gold-style learner. Our corresponding technical considerations lead to insights of intrinsic interest into combinatorial and algorithmic properties of pattern languages.

Eye gaze reveals a fast, parallel extraction of the syntax of arithmetic formulas

Mathematics shares with language an essential reliance on the human capacity for recursion, permitting the generation of an infinite range of embedded expressions from a finite set of symbols. We studied the role of syntax in arithmetic thinking, a neglected component of numerical cognition, by examining eye movement sequences during the calculation of arithmetic expressions. Specifically, we investigated whether, similar to language, an expression has to be scanned sequentially while the nested syntactic structure is being computed or, alternatively, whether this structure can be extracted quickly and in parallel. Our data provide evidence for the latter: fixations sequences were stereotypically organized in clusters that reflected a fast identification of syntactic embeddings. A syntactically relevant pattern of eye movement was observed even when syntax was defined by implicit procedural rules (precedence of multiplication over addition) rather than explicit parentheses. While the total number of fixations was determined by syntax, the duration of each fixation varied with the complexity of the arithmetic operation at each step. These findings provide strong evidence for a syntactic organization for arithmetic thinking, paving the way for further comparative analysis of differences and coincidences in the instantiation of recursion in language and mathematics.

Automata theory based on complete residuated lattice-valued logic: Turing machines

Automata theory based on complete residuated lattice-valued logic, called L-valued finite automata (L-VFAs), has been established by the second author in 2001. In view of the importance of Turing machines, in this paper, we establish a theory of Turing machines based on complete residuated lattice-valued logic, which is a continuation of L-VFAs. First, we give the definition of L-valued nondeterministic Turing machines (L-NTMs), and observe that the multitape L-NTMs have the same language-recognizing power as the single-tape L-NTMs. We give some related properties of L-valued Turing machines, and discuss computing with fuzzy letters via L-valued Turing machines. Second, we introduce the concepts of L-valued recursively enumerable languages and L-valued recursive languages, and obtain some equivalent relations. Some results concerning the characterization of n-recursively enumerable sets are given, and the super-computing power of L-valued Turing machines is investigated. We also prove that L-valued deterministic Turing machines and L-NTMs are not equivalent in the sense of recognizing or deciding languages. Finally, we show that there is no universal L-valued Turing machine. However, a universal L-valued Turing machine exists if the membership degrees of L-valued sets are restricted to a finite complete residuated lattice with universal bounds 0 and 1.

Set systems: Order types, continuous nondeterministic deformations, and quasi-orders

By reformulating a learning process of a set system L as a game between Teacher and Learner, we define the order type of L to be the order type of the game tree, if the tree is well-founded. The features of the order type of L (dimL in symbol) are (1) we can represent any well-quasi-order (wqo for short) by the set system L of the upper-closed sets of the wqo such that the maximal order type of the wqo is equal to dimL; (2) dimL is an upper bound of the mind-change complexity of L. dimL is defined iff L has a finite elasticity (fe for short), where, according to computational learning theory, if an indexed family of recursive languages has fe then it is learnable by an algorithm from positive data. Regarding set systems as subspaces of Cantor spaces, we prove that fe of set systems is preserved by any continuous function which is monotone with respect to the set-inclusion. By it, we prove that finite elasticity is preserved by various (nondeterministic) language operators (Kleene-closure, shuffle-closure, union, product, intersection, …). The monotone continuous functions represent nondeterministic computations. If a monotone continuous function has a computation tree with each node followed by at most n immediate successors and the order type of a set system L is α, then the direct image of L is a set system of order type at most n-adic diagonal Ramsey number of α. Furthermore, we provide an order-type-preserving contravariant embedding from the category of quasi-orders and finitely branching simulations between them, into the complete category of subspaces of Cantor spaces and monotone continuous functions having Girard’s linearity between them.