Speech-controlled user interfaces such as Amazon’s Alexa, Apple’s Siri or Cortana by Microsoft substantially facilitate the operation of devices and household functions to laymen. Instead of using keyboard and display as input-output interfaces, the operator pronounces requests or instructions to the device and listens to its responses.
State-of-the-art language technology scans the acoustically analyzed speech signal for relevant keywords that are subsequently inserted into semantic frames  to interpret the user’s intent. This slot filling procedure [2, 3, 4]
is based on large language corpora that are evaluated by standard machine learning methods, such as conditional random fields4], for instance. The necessity to overcome traditional slot filling techniques by proper cognitive information and communication technologies has already been emphasized by Allan . His research group trains semantic parsers from large language data bases such as WordNet or VerbNet that are constrained by hand-crafted expert knowledge and semantic ontologies [2, 6, 7].
One particular demand on cognitive user interfaces are the processing and understanding of numerals, e.g. in instructions like “increase the heating to 22.5 degrees
”, where the device may probably respond with a sensor registration: “the current room temperature is 18.3 degrees” . Numerals are an important research domain in cognitive linguistics and language technology [9, 10, 11, 12, 13, 14]. They exhibit typological differences among languages but share a simple arithmetic semantics. Decent examples are different morphologies in German () or English (), and also different base systems in German () or French () . Linguistically, numerals are regarded as modifiers  with a particular syntactic morphology that should be described by a suitable grammar formalism. This grammar must store numeral morphemes together with their arithmetic semantics in a data base, called the mental lexicon. It should be complex enough to account for the wealth of linguistic typology and constrained enough to exclude ungrammatical compositions such as zweizig in German or twoty in English .
Recent research in computational linguistics has demonstrated that quite different grammar formalisms, such as categorial grammar , tree-adjoining grammar , multiple context free grammar (MCFG) , range concatenation grammar , and minimalist grammar [19, 20] converge toward universal description models [21, 22]. Minimalist grammar has been developed by Stabler  to mathematically codify Chomsky’s Minimalist Program  in the generative grammar framework. A minimalist grammar (MG) consists of a mental lexicon storing linguistic signs as arrays of syntactic, phonetic and semantic features, on the one hand, and of two structure-building functions, called “merge” and “move”, on the other hand. Syntactic features in the lexicon are, e.g., the linguistic base categories noun (n), verb (v), adjective (a), or, in the present context, numeral (num). These are syntactic heads selecting other categories either as complements or as adjuncts. The structure generation is controlled by selector categories that are “merged” together with their selected counterparts. Moreover, one distinguishes between licensors and licensees, triggering the movement of maximal projections. An MG does not comprise any phrase structure rules; all syntactic information is encoded in the feature array of the mental lexicon. Furthermore, syntax and compositional semantics can be combined via the lambda calculus [24, 25], while MG parsing can be implemented by compilation into an equivalent MCFG .
One important property of MG is their effective learnability in the sense of Gold’s formal learning theory . Specifically, MG can be acquired by positive examples [28, 29] from linguistic dependence graphs [30, 31], which is consistent with psycholinguistic findings on early-child language acquisition [32, 33, 34]. However, learning through positive examples only, could easily lead to overgeneralization. According to Pinker  this could effectively be avoided through reinforcement learning [35, 36]. Although there is only little psycholinguistic evidence for reinforcement learning in human language acquisition [37, 38], we outline a machine learning algorithm for the acquisition of an MG mental lexicon of numeral morphology and semantics through reinforcement learning in this contribution.
Ii Numeral Grammar
Our language acquisition approach for numeral grammar combines methods from computational linguistics, formal logic, and abstract algebra. Starting point of our algorithm are utterance meaning pairs (UMP)
where is the spoken or written utterance, given as the exponent of a linguistic sign . Technically, exponents are strings taken from the Kleene hull of some finite alphabet, , i.e. . The sign’s semantics is a logical term, usually expressed by means of lambda calculus.
Ii-a Numeral Semantics
The straightforward meaning of a numeral, say fourtytwo, is a number concept, such as . However, from a computational point of view, the UMP simply relates a symbolic string fourtytwo to another symbolic string , without making the exponent and the semantics of the sign operationally accessible. This is achieved by interpreting digit strings in a -adic number system. In the decimal system with , we have
with coefficients ( the number of digits).
Equation (2) can directly be written as a tree-like arithmetic term structure
Using the binary operators and , and writing them in the unary Schönfinkel representation
where is regarded as a function , and as another function , respectively, we obtain an expression of the arithmetic term algebra  in Polish notation
Ii-B Minimalist Grammar
Following Kracht , we regard a linguistic sign as an ordered triple
with the same exponent and semantics as in the UMP (1). In addition, is a syntactic type that we encode by means of minimalist grammar (MG) in its chain representation . The type controls the generation of syntactic structure and hence the order of lambda application, analogously to the typed lambda calculus in Montague semantics.
An MG consists of a data base, the mental lexicon, containing signs as arrays of syntactic, phonetic and semantic features, and of two structure-generating functions, called “merge” and “move”. Syntactic features are the basic types from a finite set , with , etc, together with a set of their respective selectors that are unified by the “merge” operation. Moreover, one distinguishes between a set of licensers and another set of their corresponding licensees triggering the “move” operation. is another finite set of movement identifiers. is called the feature set. Finally, one has a two-element set of categories, where “::” indicates simple, lexical categories while “:” denotes complex, derived categories. The ordering of syntactic features is prescribed as regular expressions, i.e. is the set of syntactic types [19, 20]. The set of linguistic signs is then given as .
Let be exponents, semantic terms in the lambda calculus, one feature identifier, feature strings compatible with the regular types in , and sequences of signs, then and form signs in the sense of (4). A sequence of signs is called a minimalist expression, and the first sign of an expression is called its head, controlling the structure building through “merge” and “move” as follows.
The MG function “merge” is defined through inference schemata
Correspondingly, “move” is given through
Iii Reinforcement Learning
The language learner is a cognitive agent in a state , to be identified with ’s mental lexicon at training time . At time , is initialized as a tabula rasa with empty lexicon
and exposed to UMPs produced by a continuously counting teacher . The first UMPs given by are , , , and so forth. Note that we assume presenting already complete UMPs and not singular utterances to . Thus we avoid the symbol grounding problem of firstly assigning meanings to uttered exponents , which will be addressed in future research. Moreover, we assume that is instructed to reproduce ’s counting based on its own numeric understanding. This provides a feedback loop and therefore applicability of reinforcement learning [35, 36].
As long as is not able to detect patterns or common similarities in ’s UMPs, it simply adds new entries directly to its mental lexicon, assuming that all numerals have base type num. Hence, ’s state evolves according to the update rule
when is the UMP presented at time by .
In this way, the mental lexicon of simplex numerals in Tab. I has been acquired at time .
The learner is so able to perfectly reproduce the learned entries directly via data base query. As a consequence, the teacher rewards thus signalling that it has correctly learned the lexicon .
When the teacher continues counting: , ,
and so on, the learner’s pattern matching faculty detects a common affixteen in the exponents, and a common function in the semantics of UMPs .
Thus, in a first step UMP is still added to the lexicon according to update rule (11),
such that in (13) the previously learned lexicon is revised by removing the entry for the composite thirteen, followed by adding the complex morpheme in (14), and completed in (15). For the morpheme is already contained in the lexicon, further updating is not required at this time.
Next, has to correctly reproduce the UMPs and by invoking its utterance-meaning transducer (UMT) . Consider , which is now ambiguous with respect to the lexicon entries for . First, may access data base entries and and derive the following UMP according to the MG rules (5 – 9)
This yields the correct semantics with the lambda calculus
and the uttered exponent thirteen, generated by the UMT , is well-formed and will be rewarded by the teacher.
However, may alternatively select data base entries and as well. Then
will be derived instead. Although it has the correct semantics , uttering the exponent threeteen will be rejected by . Upon the resulting punishment, has to reconfigure its mental lexicon by introducing additional licenser/licensee pairs, here denoted as [28, 29]. Table II displays the result of this reorganization process at some time later than when all possible ungrammaticalities have been abandoned.
Now only the data base selection and leads to a grammatical derivation of the UMT ,
while its ambiguous counterpart
cannot be further processed due to a lacking licensee -k.
The same argument applies to the ambiguous entries and where only the latter successfully derives . Note that the currently learned grammar also derives the exponent eightteen instead of eighteen; this could be corrected by either learning an additional entry and revising , or, perhaps more appropriately, by introduction of additional phonotactical rules operating on abstract graphon representations . Moreover, since simplex numerals such as four, six, seven, and nine must not possess any other features than num, they would be doubled in a more rigorous treatment, resulting in four additional lexicon entries.
From a semantic point of view, the lexicon state in Tab. II is not yet satisfactory, because another step of lambda abstraction can be applied to entry , entailing the semantics of plain addition
Incorporating this into the training process gives another updating dynamics
such that (21) removes the original teen from the lexicon which is subsequently replaced by the phonetically void addition operator and a new representative .
Table III shows the updated lexicon at some even later time .
Now, the correct derivation of thirteen reads
By virtue of lexicon the learner is able to correctly reproduce numerals , employing its UMT . This will be rewarded by the teacher. Later, the teacher utters the UMPs , , etc. Again, the learner will first incorporate according to rule (11) into the lexicon. But upon perceiving its pattern matching device produces a common morpheme
through lambda abstraction. Then the essentially same processes of reinforcement learning are repeated as above until the complete numeral system of the language taught by the teacher has been acquired by the learner.
In this contribution we have outlined an algorithm for effectively learning the syntactic morphology and semantics of English numerals . Number words are presented to a cognitive agent by a teacher in form of utterance meaning pairs (UMP) where the meanings are encoded as arithmetic terms from a suitable term algebra. This representation allows for the application of compositional semantics via lambda calculus. For the description of syntactic categories we use Stabler’s minimalist grammar (MG) [19, 20], a powerful computational implementation of Chomsky’s recent Minimalist Program for generative linguistics . Despite the controversy between Chomsky and Skinner , we exploit reinforcement learning [35, 36] as training paradigm. Since MG encodes universal linguistic competence through the five inference rules (5 – 9), thereby separating innate linguistic knowledge from the contingently acquired lexicon, our approach could potentially unify generative grammar and reinforcement learning, hence resolving the abovementioned dispute.
Minimalist grammar can be learned from linguistic dependency structures [28, 29, 30, 31] by positive examples, which is supported by psycholinguistic findings on early human language acquisition [32, 33, 34]. However, as Pinker  has emphasized, learning through positive examples alone, could lead to undesired overgeneralization. Therefore, reinforcement learning that might play a role in children language acquisition as well [37, 38], could effectively avoid such problems. The required dependency structures are directly provided by the semantics in the training UMPs. Thus, our approach is explicitly semantic-driven, in contrast to the algorithm in  that regards dependencies as latent variables for EM training.
As a proof-of-concept we suggested an algorithm for English numerals. However, we also have evidence that it works for German and French number systems as well and hopefully for other languages also. Using attribute-value logics  and its associated term algebra, it should be possible to encode the semantics of arbitrary utterances in a compositional fashion. This will open up an entirely new avenue for the further development of speech-controlled cognitive user interfaces .
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