Cogn Process DOI 10.1007/s10339-014-0619-z

REVIEW

Probing recursion David J. Lobina

Received: 26 March 2014 / Accepted: 22 April 2014 Ó Marta Olivetti Belardinelli and Springer-Verlag Berlin Heidelberg 2014

Abstract The experimental probing of recursion in human performance is faced with non-trivial problems. Here, I analyse three case studies from the literature and argue that they tell us little about the underlying mental processes at play within each of these domains: (a) the question of whether experimental participants employ recursive rules in parsing artificial strings of nonsense syllables; (b) the role of self-embedded structures in reasoning and general cognition; and (c) the reputed connection between structural features of a given object and the corresponding, recursive rules needed to represent/generate it. I then outline what a recursive process would actually look like and how one could go about probing its presence in human behaviour, concluding, however, that recursive processes in performance are very unlikely, at least as far as fast, mandatory, and automatic modular processes are concerned. Keywords Recursive functions  Self-embedded structures  Recursive processes

Introduction This paper focuses on the rather narrow question of how to discern whether there are recursive mental processes in cognition; specifically, whether any domain of human performance exhibits operations that call themselves during their procedure, thereby producing chains of deferred suboperations that are architecturally equivalent but simpler than the original, ’calling’ operation. Such a topic, of

course, has not featured much in the cognitive science literature, scholars having been preoccupied with the role of self-embedded mental representations instead. Indeed, the literature has mainly focused on whether the mind has and uses such representations, including whether these are particular to one given domain (say, language) or instead generalise across the whole of cognition (say, mental ascription abilities). I would want to suggest, however, that the representational and computational paradigm so central to cognitive science mandates that we pay some attention to the sort of computations cognitive systems implement, and that includes the ’shape’ a computation can take, that is, the manner in which a computation applies as opposed to the operations it carries out. Such a take on cognitive studies, I will argue below, can be rather informative of the type of operations working memory allows, a source of data that has not received as much attention as it perhaps deserves. One of the points to be defended here, in fact, will be that the presence of self-embedded structures1 does not necessarily mean that the underlying operations, be those of a domain’s competence (in the sense of Chomsky 1965) or those involved in real-time processing (Chomsky’s aforementioned performance), are ipso facto recursive in the sense that will engage me here. This equivocation, which could be termed the conflation fallacy, has been discussed in various places but in very different terms, given that oftentimes scholars employ different definitions of recursion (e.g. Lobina and Garcı´a-Albea 2009; Lobina 2011b; Luuk and Luuk 2011; Watumull et al. 2014).2 Here, the 1

D. J. Lobina (&) Faculty of Philosophy, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK e-mail: [email protected]

A wide-ranging phenomenon, I will claim, even if manifested in particular ways across domains. 2 The conflation fallacy is rather widespread in the linguistics literature, in fact; see Lobina (2011b) for discussion of a number of studies that claim that recursive structures necessarily require recursive mechanisms.

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meaning of recursion will be firmly based on the formal sciences (mainly, mathematical logic and computer science), where this concept first originated. According to Brainerd and Landweber (1974), it is useful to define functions ‘using some form of induction scheme..., a general scheme...which we call recursion’ (p. 54). Many functions have been defined in this manner, a technique that has come to be called a definition by induction. Also known as a recursive definition, it consists in ‘defining a function by specifying each of its values in terms of previously defined values’ (Cutland 1980, p. 32), a self-referential characteristic (Tomalin 2006, p. 61). As an example, take the factorial class (factðnÞ ¼ n n  1  n  2  . . .1), which can be recursively defined in the two-equation system so common of such definitions as follows: if n ¼ 1, then factðnÞ ¼ 1 (base case); if n [ 1, then factðnÞ ¼ n  factðn  1Þ (recursive step). Note, then, that the recursive step involves another invocation of the factorial function. Thus, in order to calculate the factorial of, say, 4 (i.e. 4  factð3Þ), the function must return the result of the factorial of 3, and so on until it reaches the factorial of 1, the base case, effectively terminating the recursion. A number of classes of recursive functions—the primitive, the general, the partial—have been outlined over the years, and some of these proved to be rather central in attempts at formalising the class of computable functions from the 1930s onwards.3 In fact, recursion itself proved to be very important in such attempts, as other recursively specified formalisms, such as Post’s production systems (see infra), also proved to be central to characterising computational systems. For these and some other reasons, the mathematical logic field came to draw perhaps too close a connection between the idea of a computation and recursive techniques, to the point that recursion was taken to be synonymous with computation and recursive with computable, as Soare (1996) has amply shown. As I will show below, it is precisely this aspect of mathematical logic that greatly influenced Chomsky’s introduction of recursion into linguistics. From a related but at the same time distinct set of interests, computer scientists also use recursive techniques in their studies, and in much the same terms as it is employed in mathematical logic. In a programming language such as LISP, for instance, the step-by-step list of instructions for completing a task—the procedure—can contain steps that are defined in terms of previous steps, much as recursive definitions of functions are (Abelson and Sussman 1996). Of more interest here, however, 3 Church (1936) identified the general class of recursive functions with the class of computable functions, but mistakenly so (see Soare 1996 for comments). Kleene (1938) modified this class into what he called the partial recursive functions, which correctly formalise the computable class of functions.

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procedures give rise to computational processes, the realtime implementations of the rules specified in a procedure. For the time being, and this will be certainly expanded as I proceed, the recursive definition of the factorial class provides an approximation to what the corresponding LISP procedure would look like, whilst the expansion of the factorial of 4 I outlined earlier gives an idea of the selfcalls and chains of deferred operations involved in recursive processes (ibid., pp. 33–34). Relatedly, and in contrast to mathematical logic, computer scientists talk of recursive data structures in terms of the internal constitution of the objects computational processes manipulate, and not in terms of how they are generated and/or manipulated. Wirth (1986), for instance, shows that there exists a data structure that corresponds to recursive procedures, the ‘[v]alues of such a recursive data type [containing] one or more components belonging to the same type as itself’ (p. 172), whilst Roberts (2006) discusses data types and arrays that ‘can be recursive in that the definitions of [this] class can include references to objects of the same class’ (p. 133). Naturally, one could not be faulted for suspecting that there must be a close correspondence between recursive structures and recursive processes, but as Wirth (1986) stresses, this correspondence is a matter of demonstration, not conflation, and much more will be said about this point in what follows, given that it is to the question of whether such processes are present in human performance that this essay will be partly devoted.4 The main aim of this piece, however, is to provide a critical analysis of how scholars have gone about unearthing actual, real-time recursive processes. To that end, the next section very briefly chronicles the introduction and application of recursion within linguistic studies in order to be rather specific as to, first, what its role in competence is, so that I can then describe how it applies, or ought to apply, in studies of performance. That discussion will be complemented by the analysis of three case studies from the literature in the ‘Recursion in performance’ section, all of them showing, so I claim, very little indeed in the way of elucidating underlying recursive processes. That part of the essay will then be re-evaluated from a different angle in the last section. Therein, I proceed to, first, show what a recursive process would actually look like and then explain how one could probe whether there are any such processes in human performance; finally, I cast doubt on whether the search for recursive mental processes will be a fruitful one.

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As I will show below, linguists and cognitive scientists at large also make use of recursive structures in their studies, but it is important to stress that recursive mechanisms and recursive structures are two different things.

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Recursion in linguistic theory Anyone reviewing the foundational texts of generative grammar would be left with little doubt as to either the centrality or the intended meaning of ’recursion’ within such a theory. Chomsky himself has been rather clear that a grammar is underlain by a computational system that effects a recursive enumeration of linguistic objects, a point that is repeated twice in the first edition of his 2006 book, originally published in 1966: ‘in general, a set of rules that recursively define an infinite set of objects may be said to generate this set’ (p. 112; my emphasis); and a ‘generative grammar recursively enumerates structural description of sentences’ (p. 165). By a recursively enumerable set, we ought to understand, following the terminology of recursion function theory, a field of study that so influenced Chomsky at the time, a set that falls within the range of a recursively defined function (Soare 1996, p. 300); or in other words, a set is recursively enumerable if there is an algorithm that can list the members of the set.5 An early formulation of the recursive enumeration of linguistic structures was provided in Chomsky and Miller (1963); adopting the generative, or production, systems pioneered by Emil Post in the 1940s (for instance, in his 1944 and 1947 papers), Chomsky and Miller declared the ! relationship between two structures to be a conversion of some structure /1 ; . . ./n into some structure /nþ1 that ought to be interpreted as ‘expressing the fact that if our process of recursive specification generates the structures /1 ; . . ./n , then it also generates the structure /nþ1 ’ (p. 284). It should be noted that such a characterisation, which Pullum (2011) sees as no more than a summary of Post’s production systems (p. 288), was meant to apply to all rules of grammar. Crucially, a production system qua formalisation of a computational system ‘naturally lends itself to the generating of sets by the method of definition by induction’ (Post 1943, p. 201). It is this property that justifies every stage of a derivation a production system generates—that is, every set so generated is a recursively defined set. Note, then, that this recursive definition of language was proposed as the general format of the grammar, making a generative grammar a type of recursive 5

Elsewhere, Chomsky has noted that generative grammar developed within ‘a particular mathematical theory, namely recursive function theory’ (p. 101 in Piattelli-Palmarini 1980). As mentioned, this terminology is somewhat confusing, as logicians in the 1930s and 1940s used the terms recursive and recursively to mean computable and computably. Soare (1996) calls this state of affairs the Recursion Convention, which Chomsky has always tacitly followed, as I reported in Lobina (2011b, p. 155, ft. 5). Thus, I am here claiming that Chomsky’s employment of recursion was clear, not that these issues were not confusing within mathematical logic.

function—unambiguously locating, we might add, the place of recursion within linguistic studies. Furthermore, the recursive definition of language does not seem to have translated, or mutated, with subsequent developments of the theory. Chomsky (2000, p. 19) explicitly states that all recursion means is the need to enumerate the potentially infinite number of expressions, a property that in the current shape of generative grammar, the so-called minimalist programme, is ascribed to the sole generative procedure underlying the language faculty: merge. A recent description delineates merge as a settheoretic operation in which repeated applications over one element yield a potentially infinite set of structures, drawing an analogy between the way merge applies and the successor function (Chomsky 2008; cf. Kleene 1952, p. 21). The successor function also underlies what is known as the ’iterative conception of set’ (Boolos 1971), a process in which sets are ‘recursively generated at each stage’ (ibid., p. 223), a model that seems to fit Chomsky’s conception of merge. Thus, production systems and merge are recursive for the very same reason (they are both underlain by a definition by induction), demonstrating a certain consistency, at least in Chomsky’s individual writings, regarding the role and placement of recursion within linguistic theory, an interesting exegetical point that needs expansion but that I will simply note here.6 This general, recursive property of the grammar should not be confused with an internal application of recursion within rewriting rules, a particularity of the systems outlined in the 1950s and early 1960s that has featured extensively in the literature. I am of course referring to those rules in which the same symbol appears on both sides of the arrow—as in a noun phrase (NP) rule in which another NP appears in the rule’s yield: NP ! N ? NP. Given that rules such as these can generate the so-called self-embedded, or recursive, sentences (such as John’s brother’s teacher’s book... for the rule just specified), most of the current literature has focused on such sentences and the corresponding rules when discussing the locus of recursion in language, ignoring the wider historical record chronicled supra. Even though the literature on whether all languages exhibit self-embedded sentences (and, consequently, recursive rules in the sense just described) is massive, it will not concern us much here. The point to keep in mind now is that, as stated in the preceding paragraphs, recursion was introduced as a central property of the mapping function at the heart of a generative grammar; it certainly did not, and still does not, depend on either specific rules or operations of merge or on the presence or

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I have treated this point more extensively in Lobina (2014), where I furthermore expand upon the conflation fallacy mentioned earlier.

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absence of self-embedded sentences in a given language (cf. Tomalin 2011).7 The recursive property of a grammar is central to understanding what the subject matter of linguistics is. As such, generative grammarians have taken the ’function in intension’ that generates sound-meaning pairs as their topic of research, thereby focusing on the properties of the internal generative mechanism that ‘computes a certain recursive function’ (Pylyshyn 1973, p. 44). Under such a framework, consequently, linguistic knowledge is to be accounted for procedurally; as Pylyshyn (1973) puts it whilst discussing computer science and the infinitude of the natural numbers, a computer scientist would not bother to endow a machine with a list specifying that such or such integer belongs to the set of the natural numbers; instead, knowledge of a recursive definition of the successor function would suffice. Similarly, generativists do not wish to endow the linguistic system with a list of the expressions a grammar is capable of generating (if that is even possible), and a recursive mechanism is instead postulated. A grammar so conceived, then, specifies the function that is computed in language use, but not the procedure that is in fact employed to do so in real-life interaction. The whole point of a ’theory of the computation’— Marr’s (1982) computational level, akin to Chomsky’s competence—is that the function that effects these pairs can be provided, and should be provided, prior to figuring out how the function is computed in real-time processing (Marr’s algorithmic level, akin, once again, to Chomsky’s performance). After all, the function in intension and the process that computes it need not bear a transparent relation to each other whatsoever (Matthews 1992; Steedman 2000). This is not to say that the function in intension that underlies the language faculty does not play a role in performance; indeed, that grammatical sentences are at all processed follows from the fact that they are generated by this function in intension. Having said that, however, there is no guarantee that this function will be literally present in real-time language processing. As a result, it will be here taken for granted that there is no such thing as experimentally probing the recursive function that the grammar specifies, at least not directly, even if many other aspects of a grammar (some of its operations, the structures generated, and so on) can be

experimentally investigated; psycholinguists, ultimately, aim to discover how the recursive function is implemented, a slightly different matter. If cognitive psychology is to study the role of recursion in cognition, then it will have to either focus upon the manner in which the right interpretation of self-embedded sentences/structures is computed or embark upon a search for potentially recursive mental processes (the latter to be explicitly delineated below)—not two mutually exclusive undertakings, of course. The labours of the cognitive psychologist, such as they are, are not free of complications; here, three case studies are discussed in order to evaluate how they fare in relation to the two undertakings being delineated: the role of recursion in artificial grammar learning (AGL); the effect of interpreting self-embedded expressions in reasoning; and the reputed nexus between structural features of a given object (or rather, how these features are mentally represented) and the possibly corresponding recursive rules needed to represent/generate such objects. All three, it will here be argued, are afflicted with non-trivial problems.

Recursion in performance Case study I: artificial grammar learning In very general terms, an AGL task presents subjects with regular patterns of nonsense syllables (strings) in order to figure out if they can extrapolate the underlying grammar employed to generate these strings.8 Fitch and Hauser (2004) conducted a study that aimed to probe the expressive power of the grammar that subjects had internalised during the experiment, quite explicitly stating that they did not study the different strategies that could have been employed to parse the strings, what they called the ’performance variables’ (ibid., p. 378). According to their data, human subjects could correctly parse An Bn strings, demonstrating mastery of a context-free grammar (context-free because of rewriting rules of the following type, where S stands for a sentence and capital letters for strings: S ! AðSÞB), whereas tamarin monkeys could not.9 Perhaps unsurprisingly, the subsequent literature has analysed both the data and the method employed by Fitch and Hauser (2004) in the very terms they intended to ignore—that is, by focusing on performance variables.

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In fact, one of the main problems with the conflation fallacy is that some scholars do not seem to recognise that recursive mechanisms and recursive structures are two different things—or indeed that recursion is a property that can apply to different constructs. As a case in point, Pinker and Jackendoff (2005) state that recursion ‘refers to a procedure that calls itself, or to a constituent that contains a constituent of the same kind’ (p. 203), the accompanying footnote suggesting that the ’or’ in that sentence is not to be understood as exclusive disjunction, and that is certainly a mistake.

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The material of this section is based on Lobina (2011a), where these issues are given a lengthier discussion. 9 I will completely ignore the issue of whether other species ’do’ recursion here. Moreover, it ought to be added that even though Fitch and Hauser (2004) do not use the word ’recursion’ at all in their paper, the crucial property of context-free grammars lies in recursive rules such as the one shown in the text and that justifies the attention some scholars have devoted to this aspect of their study.

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Indeed, subsequent AGL studies have attempted to probe if subjects literally and directly employ the corresponding grammars in the processing of the different strings, with ‘true recursion’ being demonstrated, Corballis (2007, p. 702) tells us, if subjects were to realise that some A syllables are paired with some B syllables within the An Bn strings; that is, ‘recursive syntactic parsing’ would be operative if subjects bound AB pairs from the outside inwards (ibid.). In any case, the actual results the AGL literature reports regarding whether subjects meet Corballis’s condition are equivocal. On the one hand, some studies conclude that subjects are not capable of processing long-distance dependencies, focusing on partitions and chunks instead (Poletiek 2002). Other studies report that subjects are indeed able to process long-distance dependencies (viz., Friederici et al. 2006; Bahlmann et al. 2008), but these claims are controversial. Regarding Friederici et al. (2006), their conclusion seems to be based on brain imaging data, which purports to show that the frontal operculum is activated during the processing of both finite-state and context-free strings, whilst Brodmann’s Area 44/45 (i.e. Broca’s area) is additionally only activated during the processing of context-free strings, an area they take to be operative in hierarchical processing. In de Vries et al. (2008), these experiments were replicated and no evidence was found for the conclusion that subjects were in fact processing the hierarchical structure of the strings; instead, they could have merely counted the As and matched them with the Bs, failing to meet Corballis’s condition for true recursion. It is only in Bahlmann et al. (2008) that we find a more conscious attempt to match the corresponding pairs by employing the phonetic features [voice] and [place of articulation]; that is, by making sure that each AB pair shares the same features. As a consequence, they claimed, subjects were prevented from counting and matching, which seems to have been borne out in the results. The neuroimaging data of Friederici et al. (2006) were replicated, and this suggests, to them at least, that ‘the activity in [the latter] regions [is] correlated with hierarchical structure building’ (Bahlmann et al. 2008, p. 533). Naturally, hierarchical structure building does not mean recursive structure building, for one can think of hierarchical processes in which some operations precede others without the presence of any operations calling themselves; hierarchical structure building does not mean the correct processing of ’recursive structures’ either, for a structure may be hierarchical without being recursive. Since, in the last experiment, the AB pairs were linked up by phonetic features, it is these very features that subjects had to keep in memory in order to link up the different pairs, but this does not mean that the processing is recursive in any sense. As computer scientists are well aware, the memory load exerted by real-time recursive processes

results from self-calls (that is, an operation calling itself) and the chains of deferred operations thereby created (see Abelson and Sussman 1996, pp. 33–34 for an exposition of how this pans out in the LISP programming language), but this does not seem to be related to the general strategy of keeping the right phonetic feature in memory and linking its bearing element with the next element that carries this same feature. After all, an operation that searches for a specific phonetic feature does not call itself at all, it simply searches a feature until it finds it. More importantly, matching features among long-distance elements bears no relation to the recursive rewriting rules that are supposed to be literally employed in the processing of paired elements; that is, by linking certain elements by phonetic feature, and then eliciting subjects to construct the right pairs, one is in fact changing the operation that is supposed to be under analysis. As mentioned, a recursive process results when a given procedure calls itself, but this self-call is simpliciter; in the case of the factorials, the factorial of 4 becomes [4  (factorial 3)] and then the factorial of 3 turns into [3  (factorial 2)], and so on until it reaches the simplest case, the factorial of 1, for which the base case immediately returns a value. As a consequence of this, an internal hierarchy among the operations develops so that the factorial of 4 cannot be calculated until the factorial of 3 is, and the latter will not be completed until the factorial of 2 is, and so on; it is the operations, in other words, that are hierarchical. This is not the case for the feature-linking operation in either respect. Firstly, a simpler self-call does not take place; instead, the same operation—featurematching—applies to different variables. Secondly, no hierarchy among the operations develops as, quite clearly, a string such as A1 A2 A3 does not necessitate that the B elements appear in any particular order for the correct linking of features to take place; this is the case in some experiments merely as an artefact of the way the experimenter creates and presents the materials. The resultant memory load therefore follows from this linking of features and not from the parser rewriting an S into AðSÞB, and then the resultant S into another AðSÞB, and so on and on. The problem, to regress a bit, is that Corballis (2007) is extrapolating the recursive character of the actual parsing operation from the correct processing of hierarchical structures, which conflates structures and mechanisms into one phenomenon. This very equivocation, this conflation, is the main problem with the wider literature, and it is important to make clear that we are talking of two very different things. In order to make this point clearer, consider that non-recursive mechanisms are perfectly capable of processing recursive structures; even though An Bn strings can be parsimoniously generated by such recursive rules as S ! AðSÞB, they can equally be produced by non-

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recursive chains of rules: A ! aB; B ! aC; C ! aD; D ! bE; E ! bF and F ! b. In other words, when Corballis speaks of ’recursive parsing’, what he is in fact referring to is the correct processing of recursive structures, not the parsing operations themselves, i.e. how these structures are actually processed. Perhaps more tellingly, there is no reason to believe that any of the AGL strings require a recursive process at all, or that these strings are cognised as recursive objects. Granted, A3 A2 A1 B1 B2 B3 strings are presented in a certain order, with certain cues, so as to force a hierarchical feature-linking operation, but this is a hierarchy among the different applications of the same operation. Present the string in another order, and it will result in a different hierarchy of these applications. There is absolutely nothing to suggest that any of these strings are hierarchical, let alone self-embedded, as they exhibit no obvious internal structure—certainly not one that the tested subjects would perceive. Indeed, the different pairs enter in no configurational relation to each other, in stark contrast to the self-embedded linguistic expressions they are meant to model, a curious property of the AGL paradigm, considering that AGL scholars seem to believe that their subjects exercise their language faculties in order to parse the artificial strings of nonsense syllables they are exposed to.10 A fortiori, if subjects are really employing the language faculty to master these formal languages, this is just a blind alley, as there are not in actual fact any natural languages that exhibit long-distance dependencies in terms of elements that share the same phonetic feature.11 In short, the self-embedding property of artificial strings can only be an unwarranted projection onto the data by the experimenter, perhaps the result of creating the experimental materials by employing recursive rewriting rules, but there is no obvious hierarchy in the strings so generated. Case study II: interpreting self-embedded expressions In a paper that aims to outline a way to investigate the role of recursion in cognition, Fitch (2010) puts forward an approach that focuses on how self-embedded expressions or structures are correctly interpreted, what this author terms the ’empirical indicator’ for recursion (by recursion, Fitch actually means a self-embedding operation). However, Chomsky did not introduce recursion into linguistics 10 In a recursive sentence such as The cat the dog the mouse bit chased ran away, dog is the subject of chased but the object of bit, whilst cat is the object of chased and the subject of ran away. No such interrelations exist within AGL strings. 11 In the case of the sentence of the previous footnote, note that the subject–verb configurations are based on rather abstract morphosyntactic features, such as person and number.

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as a particular mechanism for the correct interpretation of recursive sentences, but, rather, as an account for the fact that a grammar specifies an infinite set of elements. Moreover, the recursive property was never defined as a self-embedding operation, even if particular instantiations of recursive generation can give the impression of carrying out self-embedding manoeuvres.12 More to the point, in order to figure out what mechanisms and principles bring about the proper interpretation of any sort of structure, and not only the self-embedded ones, linguists have directed their efforts to studying how featured lexical items are combined into legal and legible phrases and sentences. As such, the explanation for why some sentences are grammatical whilst others are not lies in the operations outlined in a syntactic derivation rather than on the structural features of the completed, derived object; that is, a bracketed representation of a recursive sentence, e.g. the cat [the dog [the mouse bit] chased] ran away, whilst a fitting way to indicate where the boundaries within such a sentence are, and a representation that furthermore seems to suggest the applications of self-embedding operations, is not in reality a transparent manifestation of how such a sentence was actually constructed—of what makes it grammatical. As a matter of fact, I know of no linguistic theory in which recursive structures are generated in the manner some scholars seem to suggest, that is, no theory postulates that the embedded phrase of a recursive sentence is built individually and separately and then embedded into a matrix phrase (a self-embedding operation); that would not characterise the structure appropriately, considering the interconnected nature of the internal phrases.13 Rather, the syntactic derivations for such sentences, like the derivations for any syntactic object in fact, will be a rather intricate phenomenon, with no obvious isomorphism with the final, derived structure.14 A fortiori, there is no requirement whatsoever for individual applications of merge to be recursive 12

As mentioned earlier, recursive rules such as NP ! N ? NP were once-upon-a-time employed to generate such self-embedded sentences as John’s brother’s teacher’s book...; the recursive property of such rewriting rules, however, simply refers to the self-call effected (an NP appears in the yield of an NP rule), but there no selfembedding is at all involved. 13 The main reason for that is that verb–object constituents must enter into a local configuration before they can be displaced. Thus, bit and the dog in the example used in the text cannot enter the derivation as part of different phrases, as a self-embedding operation would have it. 14 Stabler (2010) draws a distinction between derived and derivation tree structures and claims that linguists (and psycholinguists) should focus on the latter, as it is therein that one finds an account of the underlying properties and principles that license some structures and not others.

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in the manner that some rewriting rules were, which is what Fitch actually has in mind.15 In other words, that self-embedded sentences are appropriately interpreted follows from the very fact that they can be generated by the language faculty, that is, that there are wellformed derivations for them. However, this is entirely independent of whether there are any underlying self-embedded operations in these derivations. Of course, there is still the question of what strategies are actually used in comprehending self-embedded structures, but the evidence for self-embedding operations in parsing appears to be scant. I will come back to that issue in the next section and at the end of the paper; for now, it is important to point out that none of this should be interpreted as saying that researching the manner in which we understand self-embedded structures is not a worthwhile endeavour, but we must be clear as to what the actual issues involve. As a case in point, consider the following data on the role of self-embedded expressions in rationality tasks. In a study of what they call ’iterative reasoning’, Cherubini and Johnson-Laird (2004) tested the ability of subjects to answer questions (3) and (4) on the basis of premises (1) and (2): (1) (2) (3) (4)

Everybody loves anyone who loves someone. Anne loves Beth. Does it follow that everyone loves Anne? Does it follow that Carol loves Diane?

According to the mental models approach, these scholars adopt, reasoners ‘make inferences by imagining the possibilities compatible with the premises’ (p. 32), and so a positive answer to question (3) requires little effort, being an immediate conclusion—indeed, it follows after little reflection that things must be such. Question (4) also merits a positive answer, but working this out requires, first, deriving the intermediate conclusion that Diane loves Anne, in order to then reuse the general premise Everybody loves anyone who loves someone to derive the further intermediate conclusion that everybody loves Diane. From the latter, one can safely reach the iterative (their term) conclusion that Carol does indeed love Diane. Note, then, that these chains of inferences implicate embeddings of different kinds and depths, and it is perhaps not surprising that iterative inferences—which could be termed recursive inferences, I suppose—are harder than immediate ones, as Cherubini and Johnson-Laird (2004) show. These data, then, tell us something about the effect these structures have in decision-making, but not on whether there are any underlying recursive operations, for mental models may be compiled in any number of ways and there is no reason to 15

Moreover, Fitch is also wrong in supposing that a recursive rewriting rule implies a self-embedding operation; they are two very different things.

believe, or at least these scholars have not provided any, that recursive mental models are generated recursively. Note that I am not denying that mental models may selfembed; what I am denying, keeping with the general point of this essay, is that self-embedded mental models must be the result of recursive processes. If anything, all we can surmise from these data is that subjects are capable of entertaining recursive mental representations, and that the depth of such representations results in a rather significant strain in conceptualisation, that is, in thought. The same point applies to the phenomena that engage Corballis (2011): theory of mind abilities, tool-making conceptualisation or episodic memory. All these examples exhibit, or may exhibit, some kind of self-embedding—a fact of cognition that may perhaps be subsumed into a general class: thoughts inside other thoughts—but Corballis offers no account of how these mental representations are formed; nor does he tell us how they enter into mental processes (or what effect they have in such mental cogitations). Absent that, there is really not much else to say apart from what we have already said: human beings entertain and use such mental representations. The general point of this section deserves to be repeated: the existence of recursive mental representations does not mean that they are generated or processed recursively, as they can be generated/processed in a number of ways. The issue is to find out how human cognition produces and processes such structures, and that is going to require much more than the simple demonstration that these representations are part of our cognitive repertoire. Case study III: recursive signatures in the output An approach that purports to unearth the application of the recursive rules apparently needed to represent/generate self-similar structures is to be found in Martins (2012). Therein, this author aims to discern the behavioural correlates of specific processes, claiming that particular ’signatures’ in an output can be suggestive of the computations that generate it (p. 2056). In the case of recursion, Martins tells us in the very next sentence that the signatures ‘are usually the presence of structural self-similarity or the embedding of constituents within constituents of the same kind’ (ibid.)—a not too dissimilar proposal to that of Fitch (2010).16 16

In private correspondence with Martins, I am told that his paper focuses exclusively on representational abilities and not on how these representations are generated or processed. I note his point here, but go on to show that his paper says much more than that. In particular, the experimental task employed involves the selection of the correct rule in order to successfully select the next step of a fractal generation task, and that surely points to a claim regarding what operations are at play in such tasks. In any case, I will talk of the ’representation/ generation’ of fractals in this section instead of using the phrase ’the processing of fractals’ in order to be as accommodating as possible.

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To this end, Martins and his colleagues have developed a new experimental paradigm, the Visual Recursion Task. In this technique, subjects are shown the first few steps of a fractal generation process and then are asked to identify the very next step out of two candidates. According to Martins, the representation or generation of geometrical self-similar fractals requires recursive embedding rules, defined early on as the embedding of ‘a member of the ALPHA set in another member of the ALPHA set’ (p. 2058), this apparently resulting in the ‘ability to represent new hierarchical levels...beyond the given’ (p. 2056). In turn, this sort of fractals can be compared to non-recursive ones, the latter being the result of an iterative process that ‘embed[s] constituents within fixed hierarchical levels, without generating new levels’ (p. 2060). More to the point, the Visual Recursion Task, we are told, is based on ‘the properties of geometrical self-similar fractals, which can be generated by applying recursive embedding rules a given number of iterations’ (p. 2060). Thus, in order to correctly choose the right next step in the generation of self-similar fractals, that is, ‘in order to correctly generalise a particular recursive rule’ (ibid.), subjects have to demonstrate a number of hierarchical abilities so that they can ‘apply the abstracted rule one level beyond the given’ (ibid.). Note, then, that the Visual Recursion Task involves extrapolating the recursive rule and applying it a number of times in order to generate/ represent self-similar fractals. According to the data reported in Martins (2012), subjects are in fact less successful, and slower, in correctly identifying the next step of self-similar patterns than they are in the case of nonrecursive fractals. There are reasons to doubt the way in which this approach is being conceptualised, however. First of all, by a recursive rule, Martins simply means a rule that embeds an element into an element of the same kind, which may or may not result in a recursive structure. Such self-embedding, however, makes it not a recursive rule, but simply a self-embedding rule, a completely different matter altogether—that is, the self-embedding property is an aspect of what the rule does, but not of how it proceeds, the latter being the identity condition par excellence of recurrent operations such as recursion and iteration. Indeed, an operation is recursive on account of calling itself and not because it self-embeds elements into other elements.17 17 Tellingly, both Fitch (2010) and Martins (2012), who are part of the same research group, prove incapable of offering any references, to the formal sciences or otherwise, to ground a definition of recursion in such terms, even though both of them correctly identify and chronicle the self-reference so central to recursive functions, references a-plenty, in the first half of their papers. I already commented upon Fitch’s characterisation earlier on, and I really do not understand where this sort of definition comes from, nor do I think that different interpretations of theoretical terms should be encouraged. This is not to say that these studies are not interesting, or indeed informative; I

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Definitions to one side, and more importantly, the examples Martins provides for recursively and iteratively generated fractals are not persuasive. For the self-similar fractals, the first panel of his Figure 4 (p. 2061), reproduced below as Fig. 1, displays a single equilateral triangle, with the second panel showing that other equal but smaller equilateral triangles have been added, at their respective vertices, to the vertices of the previous triangle, the process continuing in this manner in the next few steps (this includes the inside of triangles too). Given that all the equilateral triangles touch at their vertices at precisely their angle bisectors, further additions of smaller triangles eventually result, after a few iterations, in a figure that gives the impression of not only a regular pattern, but of some triangles being embedded into other triangles—under a particular interpretation of the (visual) facts, of course. Regarding the non-recursive fractals of Martins’s Figure 5 (ibid.), reproduced below as Fig. 2, small equilateral triangles are placed on the vertices of other triangles at the mid-point of every side of each small triangle, thereby creating an illusion of irregularity, no doubt because of the resulting overlapping lines. This is not the manner in which Martins describes the state of affairs. According to him, in order to generalise the embedding rule for the generation or representation of selfsimilar fractals (recall, not a recursive rule in my terms), subjects need to do a number of things: develop categorial knowledge of the hierarchically structured constituents, identify the positional similarities at different levels, and finally, extrapolate the rule ‘one level beyond the given’ (p. 2060). This, however, is rather dubious indeed, given that the simple rule I informally described above (viz., that new triangles are to be added, at their vertices, to the vertices of previous triangles) would be capable of solving the task without computing the hierarchical dependencies Martins claims subsume the self-similar fractals. In fact, it is not clear to me that self-similar fractals really exhibit a hierarchical structure, and this is perhaps clearest in the progression from panel 1 to panel 2 of Martins’s Figure 4. Why should we suppose, after all, that the smaller triangles really stand in a subordinate relation towards the dominant, bigger triangle, apart from Martins telling us that this is so? I do not of course doubt that we could assign such structure to those visual figures, but there is nothing intrinsically hierarchical in those objects, so who is to say that subjects Footnote 17 continued just contest their claim that what they call a recursive embedding rule is really related to recursion. I do not think this is a trivial point, in fact; in Martins et al. (2014), the Visual Recursion Task is related to the recursive solution of a Tower of Hanoi puzzle, but the latter is recursive in a way that the Visual Recursion Task actually is not (in terms, that is, of self-calls and deferred operations) and that ought to be taken into consideration.

Cogn Process Fig. 1 Recursively generated fractals? Subjects would be exposed to the first three panels, in succession, and then asked to select one of the two panels in the second row as the correct application of the underlying generative rule

Fig. 2 Iteratively generated fractals?

would in fact interpret them in such terms? In this sense, the contrast with the intricate structure of the self-embedded sentences I discussed earlier on could not be clearer.18 18

Moreover, one could not possibly be faulted for believing that Martins’s recursive embedding rule does not in fact apply from panel 1 to panel 2, as the embedding of an alpha set into another alpha set does not seem to take place in that stage. Martins tells me that the embedding rule does occur because ‘the spatial rule connecting dominant and subordinate constituents is constant across several hierarchical levels’, but that statement, beside being not a little cryptic, seems to be a different issue altogether to the embedding of an alpha set into another alpha set. In any case, the paper under discussion here is not at all clear regarding this point. Martins also tells me that these issues are better described in Martins et al. (2014); in this publication, in addition, the experiments there reported

What I am trying to convey here is that neither hierarchical knowledge nor going ‘one level beyond the given’ are in fact needed in order to choose the right next step of the self-similar fractals; all that is needed is the ability to add smaller triangles to specific positions of the bigger triangles. Finally, I would like to note that Martins’s framework would appear to be predicated on the mistaken view that certain tasks can only be solved either recursively or Footnote 18 continued controlled for the possibility that subjects could have used a heuristic (similar to the simple rule I described in the main text) to solve the tasks, but a discussion of this work would have to await another venue.

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iteratively—and hence that one can serve as the control for the other in an experimental setting. This is certainly true in the terms that computer scientists have usually treated the distinction between recursive and iterative processes (my terms too)—viz., these recurrent processes differ in the manner in which they apply, not on what sort of operation they implement—but this is also true, I believe, of the way in which Martins treats this distinction.19 In fact, subjects may well have employed any number of strategies, recursive or iterative, for each step of each task; after all, is the suggestion seriously to be that self-similar fractals cannot be generated iteratively by, as it might be, a Turing Machine-like process? Or that the ’iterative’ fractals could not be generated recursively?20 All we have been told is that subjects perform worse with the regular patterns than with the irregular ones, which suggests just that: that subjects are indeed worse at manipulating regular patterns. To be sure, we should strive to explain why subjects perform in the manner that they do, but Martins’s data do not warrant any conclusions regarding what operations or rules are being employed.21

Probing recursion A computational process is recursive if (and only if) one of its operations calls itself during its procedure, resulting in an internal application of the same operation whilst keeping whatever it was doing before the self-call in memory, thereby generating chains of deferred operations. As such, 19

Martins is clearly aware of this; in page 2056 of his paper, he shows how Fibonacci sequences can be computed both iteratively and recursively, the recursive computation there outlined, by the way, being a case of a function calling itself, and no embedding of any kind takes place. This much is also accepted in Martins et al. (2014), wherein they stress that the rules involved in their two main tasks— Visual Recursion and Embedded Iteration, in their terms—apply iteratively (or in succession). As mentioned above, Martins and his colleagues employ the term recursive rule to refer to what I would call a self-embedding rule. 20 A Turing Machine, whilst extensionally equivalent to, say, the partial recursive functions, in the sense that both formalisms can converge on the same output given the same input, really does proceed iteratively, as there are no recursive rules in its operations (see Moschovakis and Paschalis 2008 for details). The difference between a Turing Machine and the partial recursive functions is intensional, i.e. it lies in the manner in which the output is computed from the input. 21 I should add that Martins reports that verbal working memory seems to be correlated with subjects’ performance on the self-similar version of the experiment, a feature he connects to Carruthers’s (2002) contention that language is the inter-modular language of the mind—or in Martins’s terms, that ‘verbal processing resources...enhance reasoning in non-linguistic domains’ (p. 2061). That, however, stands on very shaky grounds, as Carruthers’s claim—based on data from spatial re-orientation tasks—is both unpersuasive and unlikely (see Samuels 2002, 2005 and Lobina 2012 for discussion).

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a process is recursive not on account of what computation precisely an operation effects, but in terms of how this operation applies, a very different matter. Regarding the particular question that has engaged me here—whether there are mental operations that operate recursively in performance—this will only be the case if the right orbiting circumstances demand it. However, it is not the nature of the acted-upon objects that brings about a recursive process, but the character of the task to carry out (the nature of the problem to solve, that is). Indeed, a given problem must meet three interrelated criteria for a recursive solution to be applicable, namely (1) it should be possible to reduce a complex problem into simpler but architecturally equivalent subproblems, each of which (2) should be solvable in the same manner and whose combination, that is, the combination of all the solutions, (3) should provide a solution for the entire task (Roberts 2006). Such a take on things, in fact, precisely identifies what is wrong with many of the frameworks to be found in the literature; it is not in the architectural features of the final object that we should look for recursive ’signatures’, but in the overall task a postulated mental process has to complete. Naturally, the actual structure of a given task will be partly determined by the objects and representations to be manipulated, but this will be only one factor among many (e.g. memory limitations, attentional resources, and parsing strategies). Moreover, if a given task suggests a recursive process, it stands to reason, as a corollary of the Church Thesis, that such a task can also be solved iteratively—or in other words, that the solution can be computed by both the partial recursive functions and by a Turing Machine (see ft. 20 supra). Clearly, spelling this out in terms of actual, mental processes is not a trivial issue, and much thought should be devoted to the proper conceptualisation of its investigation. What we need to isolate, in any case, is a specific cognitive task for which both a recursive and an iterative solution could be postulated, so that an independent variable—working memory being the crucial variable—could be manipulated in order to work out which process is in fact being executed. Working memory clearly is key here; after all, a recursive process will exert a significant amount of memory, the result of the processor going down a level within the hierarchy of tasks to be completed (these are known as the ’push-downs’, which mark the beginning of a self-call) whilst simultaneously keeping unfinished tasks in storage until all sub-tasks have been completed (after each sub-task is completed, the processor moves up a level, these are the ’pop-ups’), a chain of operations that will eventually allow the processor to round up the overall solution. It is interesting to note, in this sense, that when it comes to musical perception, Fitch (2010) considers, albeit rather briefly, an approach that is precisely along the lines I am

Cogn Process

outlining here. Consider the suggestion, advanced in Hofstadter (1979, p. 129), that we may perceive Bach’s Baroque-style modulations recursively, given that the beginning of a C note may be followed by a D note that commences before the C modulation finishes, therefore giving rise to a hierarchy of note modulations, that is, something like this sort of structure: [C...[D...D]...C]. This is just supposition, to be sure, and there is certainly a hint of the usual conflation of structures and mechanisms, in this case equivocating specific structural properties of embeddings of note modulations with the processing of such objects. Nevertheless, Fitch (2010, p. 82) entertains the possibility of experimentally probing if our cognitive system carries out push-down and pop-up operations during the processing of these modulations, a process that would be recursive in the technical sense here employed, unlike the self-embedding operations apparently needed to interpret self-embedded sentences appropriately, Fitch’s interest in the rest of his piece. How could we find out, though? By ‘probing registers and logic gates with measurement devices’, Fitch (2010, p. 78) advances in the context of figuring out which computations a computer is carrying out at a given moment. True enough, but can such methods be applied to the study of human performance? Consider syntactic parsing as a case in point. It is a perhaps surprising characteristic of the psycholinguistics literature that there is rather little discussion of the actual parsing operations at play in language comprehension. Gibson (1998), for instance, supposes that structure building involves looking up the lexical content of the word being processed, drawing predictions based on this information, and then matching the predictions to the following words. If MacDonald et al. (1994) are right, in turn, the parser has access to diverse bodies of lexically stored information (syntactic, lexical, semantic, contextual, etc.) at any stage of the comprehension process, a possibility that has engaged the literature very much indeed. However, very little it is in fact said about how phrases are put together at all. The study of Grodzinsky and Friederici (2006), an expository article on the neuro-imaging data regarding syntactic parsing, is perhaps illustrative of the overall field; therein, they divide the different stages of language comprehension in very broad terms indeed; to wit: local phrase structures are initially built based on lexical information, dependencies are then formed, and integration finally takes place (p. 243).22 One could not be 22

Perhaps one can think of more quintessential tasks the parser must conduct during speech processing: segmenting the string into units (words, clauses, etc.), assigning syntactic roles to those units (verbal phrase, etc.; and also, subject, object, etc.), establishing relations and dependencies between elements; setting up a correspondence between syntactic and thematic roles (agent, patient, etc.), interacting with the semantics/pragmatic component; et alia.

faulted for regarding the first stage as primordial, but the aforementioned sources have rather little to say about what operations—merging/compilation operations, we might assume—carry out the construction of local structures, let alone what properties they exhibit or how exactly they apply. This is also true, to some extent, of theories that have discussed computational mechanisms more extensively. The garden-path model, for instance, an initial version of which was put forward by Frazier and Fodor (1978), divides the parser into two stages: the preliminary phrase packager (PPP) builds phrase structures of roughly six words (7  2, to be more exact, in reference to Miller 1956), whilst the sentence structure supervisor (SSS), the second stage, adds higher nodes into a complete phrase structure (Frazier and Fodor 1978, pp. 291–293).23 Before there was a garden-path theory of parsing, though, there was an analysis-by-synthesis (AxS) approach, a model for speech perception initially proposed by Halle and Stevens (1959, 1962). The AxS is based on the observation that in linguistic communication, the receiver must recover the intended message from a signal for which he knows the ’coding function’—that is, the receiver is perfectly capable of generating the signal himself (Halle and Stevens 1959, p. 2). A good strategy from the receiver would be to ‘guess at the argument [i.e. the structure of the signal, DJL]...and then compare [this guess] with the signal under analysis’ (ibid.). In general terms, the model internally generates patterns, and these are matched to the input signal by employing a number of rules until the final analysis ‘is achieved through active internal synthesis of comparison signals’ (Halle and Stevens 1962, p. 155). One way to output patterns would be to provide the system with a repository of structures (templates), but it is not clear that this would not accommodate the open-endedness of language. Rather, an AxS system must be endowed with a set of generative rules—the coding function aforementioned—plausibly provided by the language faculty. Naturally, if all the available generative rules were to be applied ab initio, the computations would take a very long time to converge to the right interpretation of the signal, but this is clearly not what happens in language comprehension. Instead, the perceptual systems plausibly carry out a preliminary analysis in order to eliminate a large number of comparisons, a step that would make available but a small subset of possible representations. The subsequent comparisons among the internally generated representations would have to be ordered somehow (this would be the role of a control component, as used in 23 See Lewis (1996) for an update on the number of units the parser can keep in memory, which is apparently \7  2; see Lewis (2000), in addition, for a discussion of the processor’s architecture.

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computer science; see Pylyshyn 1989 for its application in cognitive science) and the whole analysis–comparison– control sequence (what Halle and Stevens 1962 call a strategy) may be viewed as the nexus between the parser and the grammar. The AxS model, then, can be viewed as encompassing two main stages, the first of which involves the generation of a candidate representation of the input (the preliminary analysis), followed by a comparison of this candidate representation with the actual input as it is being synthesised. The second stage, in particular, carries out a number of calculations of the perceptual candidates until they are synthesised into the correct representation. Halle and Stevens (1959) proposed the AxS approach as a model of perception, but as they anticipated, it can be applied to many other domains, including syntactic parsing. In fact, an AxS model of parsing was already defended in Miller and Chomsky (1963), a suggestion that was perhaps based on the realisation that intensive pattern recognition can extract different types of skeletons, be they words, phrases, intonational units, et alia (Bever and Poeppel 2010, p. 177). The model was then adopted and developed in Bever (1970), expanded by Fodor et al. (1974), and further polished by Townsend and Bever (2001). Bever (1970), in particular, proposed that the preliminary analysis involves the application of a noun–verb– noun (NVN) template onto the input, a reflection of the statistical distribution of sentences (at least for English). The representation the preliminary stage generates is then further expanded, or indeed revised if it turns out to be mistaken, when the proposed candidates are synthesised. The latter stage involves, according to Townsend and Bever (2001), an application of the derivational rules of the grammar upon the sketch created by the first stage. The overall model, then, starts with the extraction of a skeleton (a template) and is then followed by a syntactic derivation that fills the missing parts of the scheme (i.e. the first step is matched to the second; Bever and Poeppel 2010); a syntaxlast parser, to borrow Townsend and Bever’s phrase. From the perspective of the garden-path model, the application of the PPP can be regarded as being similar to that of an AxS preliminary analysis, as both operations can be insensitive to some aspects of well-formedness (Frazier and Fodor 1978, p. 292). The PPP, composed of two building operations (’minimal attachment’, MA, which incorporates a word into a structure using the fewest syntactic nodes, and ’late closure’, LC, which attaches new material to the node currently being processed) closes and shunts phrases as soon as these are formed (ibid., p. 298), with the result that its ’viewing window’ shifts throughout the sentence (p. 305). The SSS, the second component of the garden-path model, carries out a ’reanalysis’ of the interpretation the PPP returns if the latter is incorrect, eventually closing the whole structure under the Sentence

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node (giving a bottom-up flavour to the overall process; pp. 314–317). In a nutshell, the PPP creates local parses and interpretations that are then put together by the SSS.24 Despite all this detail, we still do not have what we are searching for: a description of the shape of parsing operations, that is, an outline of the manner in which these operations apply. The shape of the parsing process, it should be made clear, is a different matter from the shape or structure of the object that is being processed. These two aspects are not always distinguished in the literature; in fact, it is not clear that such a distinction is widely recognised, or even understood. In Frank et al. (2012), for instance, a paper that aims to downplay the role of hierarchy in language use, these issues get very muddled indeed (this is also true of some responses to it; viz., Rizzi 2013a, b). I will not discuss the finer details of their discussion, even though I find much at fault in Frank et al. (2012) indeed; the issue, rather, is this: even though there are very good reasons to believe that the parser must construct a hierarchical representation of the object it is processing,25 whether the parser goes about its business in a hierarchical manner is another matter completely, and one in need of elucidation. In order to clarify the distinction I am highlighting, and as way to be as explicit as possible regarding what recursive mental processes would look like, let us once more contemplate the computation of the factorials.26 As mentioned earlier, a factorial can be computed both recursively and iteratively, as Table 1 exemplifies. I already described the manner in which a factorial can be computed recursively, shown on the lefthand side below; an iterative computation can be obtained if we first multiply 1 by 2, then the result by 3, then by 4, until we reach n, that is, we keep a running product, together with a counter that counts from 1 up to n. Further, we add the stipulation that n! is the value of the product when the counter exceeds n (the righthand side of Table 1, describing the iterative process, shows the factorial to be computed, the counter, and the product, in that order).

24

In further developments of this theory (e.g. Frazier and Clifton 1996), the parser is augmented with an ’active gap filler’ in order to reconstruct displacement chains (the combination of a moved element and its trace). The resultant language comprehension system, then, starts with syntactic processing and this stage is then followed by the operations of a thematic processor. Regarding the operations of the PPP (MA and LC), these now apply to primary syntactic relations only—that is, an argument over adjuncts take on things—which is meant to capture some cross-linguistic differences in the application of MA and LC (see Frazier and Clifton 1996 for details). 25 Among many reasons, the right interpretation of a syntactic object demands it. 26 I discussed this very point in Lobina (2011a) in the context of AGL tasks.

Cogn Process Table 1 Recursive and iterative implementations 4 (factorial 3) 4 (3 (factorial 2))

Factiter

4

1

1

4 (3 (2 (factorial 1)))

Factiter

4

2

1

4 (3 (2  1))

Factiter

4

3

2

4  ð3  2Þ

Factiter

4

4

6

46

Factiter

4

5

24

fact4 fact3

4

fact2

3 2

fact1

Fig. 3 Recursive implementation of the factorials

Table 1 provides a rather schematic view of the shape of both implementations, but the brackets on the left-hand side do point to the hierarchy among the operations of a recursive process, outlining the shape of the derivation. This is not the case for the iterative computation, the process manifests a flat structure, as every operation is resolved at each stage of the process. Figure 3 is a more transparent way of showing the hierarchy underlying the recursive processing of the factorials. Note that such hierarchy is a property that arises among the operations of the task under completion (viz., the calculation of a factorial), it is not a feature of the object the process is calculating (namely, the factorial of 4). In this particular case, there is of course nothing particularly hierarchical, or indeed recursive, about the object being manipulated; after all, in what sense could the factorial of 4, or indeed the set 4, be regarded as recursive?27 The crucial point here is that in those cases in which a task involves a recurrent set of operations, any object, regardless of its internal structure, can in fact be manipulated/ computed both recursively and iteratively.28 Plausibly, recursive processes could be unearthed by attempting to capitalise on the observation (or perhaps simply the belief) that working memory does not only have 27

As noted earlier, mathematical logic reserves the recursive property to rules or functions, it does not apply it to objects, whilst computer science treats objects as recursive by analogy, but independently, to recursive processes. 28 It would also be a mistake to read the diagram as suggesting that the factorial of 3 is embedded into the factorial of 4; the tree structure is supposed to show different stages of the computation (the factorial of 4 is rewritten as 4 factorial of 3, and nothing more), not containment relations.

the capacity to store computing units, it can also impose a limit on the number of operations that can take place at each stage. Indeed, one could attempt to figure out how mental processes are being executed via manipulating (that is, straining) memory load. To be sure, such a take on things faces multiple complications, such as the issue of whether we can really know how mental processes are being applied by straining memory resources, or indeed whether, in the first place, dual-task experiments of this kind would allow us to compartmentalise those very memory resources so as to be informative of the underlying mechanisms. Moreover, working memory is a rather intricate system and this is a very relevant issue to keep in mind regarding the sort of approach I am delineating here. Indeed, a sub-part of working memory is devoted to processing visual information, whilst another involves only verbal information, the operations of these two subsystems being guided and coordinated by the control structure of working memory. Suffice here to say, however, that I am pointing to a distinction between storing values (an iterative process) and storing deferred operations and values (a recursive process), and the latter ought to involve more cognitive load, hopefully irrespective of the actual organisation of working memory. The question here, however, is whether syntactic processes such as the PPP or the computation of local parses are likely to proceed recursively in the first place. In order to approach this issue, consider the field of rationality studies, wherein a certain consensus seems to have obtained regarding what is being called the dual-process theory. According to this consensus, human rationality is subsumed by two different systems. One of these is fast, mandatory, intuitive in nature, and apparently has no direct access to working memory, thereby being, by definition, non-recursive. The second system is much slower and more deliberative, but with access to working memory, permitting the use of recursion (see Johnson-Laird 2010 for brief remarks regarding this state of affairs). According to this division of labour, the speed in which a mental process proceeds is legislating on whether an operation can make sophisticated use of working memory—whether, that is, it can use working memory in order to organise a hierarchy of operations—and that seems a reasonable point to make. Of course, the recursive processes ascribed to the deliberative system have to do with the conceptualisation of recursive solutions and the like, which are clearly higher-order examples of cognition (think of the data discussed in case study II supra) and therefore inapplicable in the study of parsing. Moreover, as indeed it was defended in the analysis of case study II, even if subjects are in fact capable of entertaining and combining self-embedded mental representations of various kinds, there is no reason to believe that the underlying mental processes that give

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rise to these representations, and that furthermore manipulate them, are recursive in the sense that has engaged me here. However, if these scholars are right about the intuitive system, a fast and compulsory process as much as parsing is, then an analogy between the two has much greater import. Syntactic parsing is a very fast phenomenon indeed; more importantly, it appears to be incremental, in the sense that it computes a representation (a meaning) of the incoming input on a word-by-word basis (see Harley 2001 for a textbook discussion of this property). From the point of view of daily conversational interaction, this may not be a surprising feature; after all, the comprehension system receives a great number of ’chunks’, it is not the case that every input is a full sentence, but in every case a structure is nonetheless built by the cognitive system. Crucially for our interests here, this means that in some sense the operations of the parser are resolved at every point during the processing of a sentence, at least to a certain extent, that is, the construction of a representation does not seem to be dependent upon any chains of deferred operations that are kept somewhere in working memory.29 It is very likely, therefore, that the automatic computations carried out by the PPP or whatever mechanism is carrying out the early construction of syntactic phrases really do not have access to working memory—nothing in the literature suggests otherwise—and that means that recursive processes are unlikely to be found at that level of description. This point, I believe, generalises to many other domains of the mind, especially those that exhibit the type of fast and automatic computations I have focused on here, that is, it applies to modular systems, in the sense of Fodor (1983). Fodor (1983), as is well known, postulated sui generis computations and representations for each of his (peripheral) processing modules; that being the case, it would appear to be rather sensible to conclude that all these modules (say, language comprehension, early vision, etc.), given that they carry out such fast computations and that furthermore operate over such restricted information, proceed in an iterative rather than in a recursive manner, even if these

processes are capable of (re)constructing rather intricate, hierarchical objects—indeed, even recursive objects. In any case, most studies from the literature have tended to concentrate on very different issues: either on the nature of the recursive representations present in cognition or on those mechanisms mistakenly supposed to be recursive. Those scholars who have focused on the former construct have been guilty of drawing entirely unwarranted conclusions regarding the underlying mechanisms that conceptualise recursive representations, with a rather astonishing disregard for actually investigating the imagined mechanisms and operations. As for those who have studied the latter construct, these have supposed to be studying recursive mechanisms because they have declared them to be so, but for poorly thought-out reasons. Some of the studies the literature can offer have been analysed here and found wanting. Perhaps unsurprisingly, this literature is somewhat obsessed with all things recursive, including, as it could not otherwise be, a quest to find recursive representations and processes. That the mind has and uses recursive representations of various kinds should not be controversial at all; whether there are any recursive processes is another issue altogether. If mental processes are mostly found to proceed iteratively rather than recursively, and that is in fact very likely for automatic processes, that would be no less worthy a finding than the discovery of any recursive operations in cognition—discovering the inner features of the mind, that is the task of the scholar. None of this has any effect on the role of recursion in competence, of course, for its place therein is due to very different reasons, and I shall leave it at that. Acknowledgments The research presented here was funded, at least in part, by a Beatriu de Pino´s fellowship awarded by the Catalan Research Council (AGAUR; Ref: 2011-BP-A-00127), and by an AGAUR research grant awarded to the Psycholinguistics Research Group at the Universitat Rovira i Virgili in Tarragona, Spain (2009SGR-401). I am thankful to Mr Mark Brenchley for comments on a previous version of this paper and to the two reviewers of this journal (especially, to Mauricio Martins) for making this paper a much better one than it would otherwise have been.

29

This seems to be true even in the case of self-embedded sentences, where the computation of subject–verb pairs would certainly merit the presence of chains of deferred operations. Hudson (1996) provides a good review of the literature and a reasonable account for why some self-embedded sentences are hard, or indeed impossible, to process. According to Hudson (1996, p. 22), hearers cannot handle [N1 ½N2 ½N3  V3 V2 V1 ] structures in which any of the following applies: (a) a finite clause [   N2    V2   ] modifies N1 , (b) N2 is modified by a finite clause [   N3    V3   ], (c) N3 is a common noun, or (d) upon processing N1  N2  N3 , it is hard to establish the meaning of V2 and V3 . This seems to explain why some ungrammatical self-embedded sentences (as in sentences that exhibit three subject Ns followed by only two Vs) are accepted by speakers (the missing verb effect; see Frazier 1985); this could not be the case if there really were deferred N–V pairing operations.

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