Residual Nominal Automata

Nominal automata are models for accepting languages over infinite alphabets. In this paper we refine the hierarchy of nondeterministic nominal automata, by developing the theory of residual nominal automata. In particular, we show that they admit canonical minimal representatives, and that the universality problem becomes decidable. We also study exact learning of these automata, and settle questions that were left open about their learnability via observations

The price of a perfect system: learnability and the distribution of errors in the speech of children learning English as a first language

This study reports on a strictly-cognitive and symptomatic approach to the treatment of phonological disorders, by an effect which can also be reproduced in most normally- developing children. To explain how this works, it is necessary to address certain asymmetries and singularities in the distribution of children's speech errors over the whole range of development. Particular words occasion particular errors. In early phonology there is 'fronting' with Coronal displacing Dorsal, and harmonies where Coronal is lost. In the middle of phonological acquisition, the harmonic pattern changes with coronal harmony coming to prevail over other forms. As well as these asymmetries, there is also the case of harmonic or migratory errors involving the property of affrication, but not the affricate as a whole, i.e. ignoring the property of voicing. Many of these asymmetries and singularities and the harmony or movement of affrication are described here for the first time. They are all difficult to explain in current theoretical models, especially in 'bottom-up' models. On the basis of the 'top-down' notion of 'parameters' from recent work in phonology, I shall assume that: A) finite learnability has to be ensured; B) there can be no privileged information about the learnability target; and C) phonological theory and the study of speech development (normal and otherwise) have an object in common. I shall propose: A) a Parameter Setting Function, as part of the human genome, possibly a defining part; B) Phonological Parapraxis', as a way of characterising the generalisations here about incompetent phonology by the general mechanisms of floating' and 'non-association'; C) a Stage (_n-1) as a necessary construct in the theory of acquisition, typically not reached before 8;6; D) a' Representability Inspection' relating normal competence to Chomsky's Articulatory/ Perceptual interface', sensitive to a relation between featural properties such as roundness or labiality and prosodic properties such as the foot and syllable; E) a syndrome. Specific Speech and Language Impairment, SSLI, extending the notion of Specific Language Impairment, SLI.I shall hypothesise that: A) segmental and suprasegmental representations interact; B) the phonological learnability space is uniform and consistent; C) it is the very minimality of the learnability system which makes it vulnerable to SSLI. This: A) side-steps the implausible inference that development proceeds by the loss of 'processes'; B) accounts for at least some of the asymmetries noted above; C) lets parameters set' a degree of abstract exponence; D) makes it possible to abolish 'processes' such as fronting, lisping, consonant harmony, in favour of successive degrees of imprecision in the parameterisation; E) provides a conceptual mechanism for the cognitive and symptomatic therapy, mentioned above: the therapy effects an increase in the set of phonological structures which are 'representable' by the child

Substitutional quantification and set theory

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43184/1/10992_2004_Article_BF00258434.pd

Sentence alignment of Hungarian-English parallel corpora using a hybrid algorithm

We present an efficient hybrid method for aligning sentences with their translations in a parallel bilingual corpus. The new algorithm is composed of a length-based and anchor matching method that uses Named Entity recognition. This algorithm combines the speed of length-based models with the accuracy of anchor finding methods. The accuracy of finding cognates for Hungarian-English language pair is extremely low, hence we thought of using a novel approach that includes Named Entity recognition. Due to the well selected anchors it was found to outperform the best two sentence alignment algorithms so far published for the Hungarian-English language pair

Topological Aspects of Epistemology and Metaphysics

The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I’ll deal with “Cassirer’s problem” that may be characterized as an early forrunner of Goodman’s “grue-bleen” problem. On a larger scale, topology turns out to be useful in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish “natural” from “not-so-natural” concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz’s famous principle of the identity of indiscernibles is closely related to some well-known topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz’s principle

Exponential separations between classical and quantum learners

Despite significant effort, the quantum machine learning community has only demonstrated quantum learning advantages for artificial cryptography-inspired datasets when dealing with classical data. In this paper we address the challenge of finding learning problems where quantum learning algorithms can achieve a provable exponential speedup over classical learning algorithms. We reflect on computational learning theory concepts related to this question and discuss how subtle differences in definitions can result in significantly different requirements and tasks for the learner to meet and solve. We examine existing learning problems with provable quantum speedups and find that they largely rely on the classical hardness of evaluating the function that generates the data, rather than identifying it. To address this, we present two new learning separations where the classical difficulty primarily lies in identifying the function generating the data. Furthermore, we explore computational hardness assumptions that can be leveraged to prove quantum speedups in scenarios where data is quantum-generated, which implies likely quantum advantages in a plethora of more natural settings (e.g., in condensed matter and high energy physics). We also discuss the limitations of the classical shadow paradigm in the context of learning separations, and how physically-motivated settings such as characterizing phases of matter and Hamiltonian learning fit in the computational learning framework.Comment: this article supersedes arXiv:2208.0633

Testing product states, quantum Merlin-Arthur games and tensor optimisation

We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses two copies of psi. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarising channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous result of Aaronson et al, this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged sections, added discussion of connections to classical CS. Final version to appear in J of the AC

A Theory of Emergent In-Context Learning as Implicit Structure Induction

Scaling large language models (LLMs) leads to an emergent capacity to learn in-context from example demonstrations. Despite progress, theoretical understanding of this phenomenon remains limited. We argue that in-context learning relies on recombination of compositional operations found in natural language data. We derive an information-theoretic bound showing how in-context learning abilities arise from generic next-token prediction when the pretraining distribution has sufficient amounts of compositional structure, under linguistically motivated assumptions. A second bound provides a theoretical justification for the empirical success of prompting LLMs to output intermediate steps towards an answer. To validate theoretical predictions, we introduce a controlled setup for inducing in-context learning; unlike previous approaches, it accounts for the compositional nature of language. Trained transformers can perform in-context learning for a range of tasks, in a manner consistent with the theoretical results. Mirroring real-world LLMs in a miniature setup, in-context learning emerges when scaling parameters and data, and models perform better when prompted to output intermediate steps. Probing shows that in-context learning is supported by a representation of the input's compositional structure. Taken together, these results provide a step towards theoretical understanding of emergent behavior in large language models