On the probabilistic logical modelling of quantum and geometrically-inspired IR

Information Retrieval approaches can mostly be classed into probabilistic, geometric or logic-based. Recently, a new unifying framework for IR has emerged that integrates a probabilistic description within a geometric framework, namely vectors in Hilbert spaces. The geometric model leads naturally to a predicate logic over linear subspaces, also known as quantum logic. In this paper we show the relation between this model and classic concepts such as the Generalised Vector Space Model, highlighting similarities and differences. We also show how some fundamental components of quantum-based IR can be modelled in a descriptive way using a well-established tool, i.e. Probabilistic Datalog

Foundations research in information retrieval inspired by quantum theory

In the information age information is useless unless it can be found and used, search engines in our time thereby form a crucial component of research. For something so crucial, information retrieval (IR), the formal discipline investigating search, can be a confusing area of study. There is an underlying difficulty, with the very definition of information retrieval, and weaknesses in its operational method, which prevent it being called a 'science'. The work in this thesis aims to create a formal definition for search, scientific methods for evaluation and comparison of different search strategies, and methods for dealing with the uncertainty associated with user interactions; so that one has the necessary formal foundation to be able to perceive IR as "search science". The key problems restricting a science of search pertain to the ambiguity in the current way in which search scenarios and concepts are specified. This especially affects evaluation of search systems since according to the traditional retrieval approach, evaluations are not repeatable, and thus not collectively verifiable. This is mainly due to the dependence on the method of user studies currently dominating evaluation methodology. This evaluation problem is related to the problem of not being able to formally define the users in user studies. The problem of defining users relates in turn to one of the main retrieval-specific motivations of the thesis, which can be understood by noticing that uncertainties associated with the interpretation of user interactions are collectively inscribed in a relevance concept, the representation and use of which defines the overall character of a retrieval model. Current research is limited in its understanding of how to best model relevance, a key factor restricting extensive formalization of the IR discipline as a whole. Thus, the problems of defining search systems and search scenarios are the principle issues preventing formal comparisons of systems and scenarios, in turn limiting the strength of experimental evaluation. Alternative models of search are proposed that remove the need for ambiguous relevance concepts and instead by arguing for use of simulation as a normative evaluation strategy for retrieval, some new concepts are introduced that can be employed in judging effectiveness of search systems. Included are techniques for simulating search, techniques for formal user modelling and techniques for generating measures of effectiveness for search models. The problems of evaluation and of defining users are generalized by proposing that they are related to the need for an unified framework for defining arbitrary search concepts, search systems, user models, and evaluation strategies. It is argued that this framework depends on a re-interpretation of the concept of search accommodating the increasingly embedded and implicit nature of search on modern operating systems, internet and networks. The re-interpretation of the concept of search is approached by considering a generalization of the concept of ostensive retrieval producing definitions of search, information need, user and system that (formally) accommodates the perception of search as an abstract process that can be physical and/or computational. The feasibility of both the mathematical formalism and physical conceptualizations of quantum theory (QT) are investigated for the purpose of modelling the this abstract search process as a physical process. Techniques for representing a search process by the Hilbert space formalism in QT are presented from which techniques are proposed for generating measures for effectiveness that combine static information such as term weights, and dynamically changing information such as probabilities of relevance. These techniques are used for deducing methods for modelling information need change. In mapping the 'macro level search' process to 'micro level physics' some generalizations were made to the use and interpretation of basic QT concepts such the wave function description of state and reversible evolution of states corresponding to the first and second postulates of quantum theory respectively. Several ways of expressing relevance (and other retrieval concepts) within the derived framework are proposed arguing that the increase in modelling power by use of QT provides effective ways to characterize this complex concept. Mapping the mathematical formalism of search to that of quantum theory presented insightful perspectives about the nature of search. However, differences between the operational semantics of quantum theory and search restricted the usefulness of the mapping. In trying to resolve these semantic differences, a semi-formal framework was developed that is mid-way between a programmatic language, a state-based language resembling the way QT models states, and a process description language. By using this framework, this thesis attempts to intimately link the theory and practice of information retrieval and the evaluation of the retrieval process. The result is a novel, and useful way for formally discussing, modelling and evaluating search concepts, search systems and search processes

Advances in quantum machine learning

Here we discuss advances in the field of quantum machine learning. The following document offers a hybrid discussion; both reviewing the field as it is currently, and suggesting directions for further research. We include both algorithms and experimental implementations in the discussion. The field's outlook is generally positive, showing significant promise. However, we believe there are appreciable hurdles to overcome before one can claim that it is a primary application of quantum computation.Comment: 38 pages, 17 Figure

The Measurement Problem Is a Feature, Not a Bug--Schematising the Observer and the Concept of an Open System on an Informational, or (Neo-)Bohrian, Approach

I flesh out the sense in which the informational approach to interpreting quantum mechanics, as defended by Pitowsky and Bub and lately by a number of other authors, is (neo-)Bohrian. I argue that on this approach, quantum mechanics represents what Bohr called a ``natural generalisation of the ordinary causal description'' in the sense that the idea (which philosophers of science like Stein have argued for on the grounds of practical and epistemic necessity) that understanding a theory as a theory of physics requires that one be able to ``schematise the observer'' within it is elevated in quantum mechanics to the level of a postulate in the sense that interpreting the outcome of a measurement interaction, as providing us with information about the world, requires as a matter of principle, the specification of a schematic representation of an observer in the form of a `Boolean frame' -- the Boolean algebra representing the yes-or-no questions associated with a given observable representative of a given experimental context. I argue that the approach's central concern is with the methodological question of how to assign physical properties to what one takes to be a system in a given experimental context, rather than the metaphysical question of what a given state vector represents independently of any context, and I show how the quantum generalisation of the concept of an open system may be used to assuage Einstein's complaint that the orthodox approach to quantum mechanics runs afoul of the supposedly fundamental methodological requirement to the effect that one must always be able, according to Einstein, to treat spatially separated systems as isolated from one another.Comment: This paper has been published in Entropy 2023, 25(10), 1410. Because Entropy has an editorial policy against footnotes, this version, which uses footnotes but is otherwise identical (including page numbers) to the published version, may be more readable. Please use the publication details just mentioned when citing this pape

Lexical measurements for information retrieval: a quantum approach

The problem of determining whether a document is about a loosely defined topic is at the core of text Information Retrieval (IR). An automatic IR system should be able to determine if a document is likely to convey information on a topic. In most cases, it has to do it solely based on measure- ments of the use of terms in the document (lexical measurements). In this work a novel scheme for measuring and representing lexical information from text documents is proposed. This scheme is inspired by the concept of ideal measurement as is described by Quantum Theory (QT). We apply it to Information Retrieval through formal analogies between text processing and physical measurements. The main contribution of this work is the development of a complete mathematical scheme to describe lexical measurements. These measurements encompass current ways of repre- senting text, but also completely new representation schemes for it. For example, this quantum-like representation includes logical features such as non-Boolean behaviour that has been suggested to be a fundamental issue when extracting information from natural language text. This scheme also provides a formal unification of logical, probabilistic and geometric approaches to the IR problem. From the concepts and structures in this scheme of lexical measurement, and using the principle of uncertain conditional, an “Aboutness Witness” is defined as a transformation that can detect docu- ments that are relevant to a query. Mathematical properties of the Aboutness Witness are described in detail and related to other concepts from Information Retrieval. A practical application of this concept is also developed for ad hoc retrieval tasks, and is evaluated with standard collections. Even though the introduction of the model instantiated here does not lead to substantial perfor- mance improvements, it is shown how it can be extended and improved, as well as how it can generate a whole range of radically new models and methodologies. This work opens a number of research possibilities both theoretical and experimental, like new representations for documents in Hilbert spaces or other forms, methodologies for term weighting to be used either within the proposed framework or independently, ways to extend existing methodologies, and a new range of operator-based methods for several tasks in IR

Quantum Mathematics in Artificial Intelligence

In the decade since 2010, successes in artificial intelligence have been at the forefront of computer science and technology, and vector space models have solidified a position at the forefront of artificial intelligence. At the same time, quantum computers have become much more powerful, and announcements of major advances are frequently in the news. The mathematical techniques underlying both these areas have more in common than is sometimes realized. Vector spaces took a position at the axiomatic heart of quantum mechanics in the 1930s, and this adoption was a key motivation for the derivation of logic and probability from the linear geometry of vector spaces. Quantum interactions between particles are modelled using the tensor product, which is also used to express objects and operations in artificial neural networks. This paper describes some of these common mathematical areas, including examples of how they are used in artificial intelligence (AI), particularly in automated reasoning and natural language processing (NLP). Techniques discussed include vector spaces, scalar products, subspaces and implication, orthogonal projection and negation, dual vectors, density matrices, positive operators, and tensor products. Application areas include information retrieval, categorization and implication, modelling word-senses and disambiguation, inference in knowledge bases, and semantic composition. Some of these approaches can potentially be implemented on quantum hardware. Many of the practical steps in this implementation are in early stages, and some are already realized. Explaining some of the common mathematical tools can help researchers in both AI and quantum computing further exploit these overlaps, recognizing and exploring new directions along the way.Comment: Adding journal reference, recommended by JAIR editors upon publicatio

Document ranking with quantum probabilities

In this thesis we investigate the use of quantum probability theory for ranking documents. Quantum probability theory is used to estimate the probability of relevance of a document given a user's query. We posit that quantum probability theory can lead to a better estimation of the probability of a document being relevant to a user's query than the common approach, i.e. the Probability Ranking Principle (PRP), which is based upon Kolmogorovian probability theory. Following our hypothesis, we formulate an analogy between the document retrieval scenario and a physical scenario, that of the double slit experiment. Through the analogy, we propose a novel ranking approach, the quantum probability ranking principle (qPRP). Key to our proposal is the presence of quantum interference. Mathematically, this is the statistical deviation between empirical observations and expected values predicted by the Kolmogorovian rule of additivity of probabilities of disjoint events in configurations such that of the double slit experiment. We propose an interpretation of quantum interference in the document ranking scenario, and examine how quantum interference can be effectively estimated for document retrieval. To validate our proposal and to gain more insights about approaches for document ranking, we (1) analyse PRP, qPRP and other ranking approaches, exposing the assumptions underlying their ranking criteria and formulating the conditions for the optimality of the two ranking principles, (2) empirically compare three ranking principles (i.e. PRP, interactive PRP, and qPRP) and two state-of-the-art ranking strategies in two retrieval scenarios, those of ad-hoc retrieval and diversity retrieval, (3) analytically contrast the ranking criteria of the examined approaches, exposing similarities and differences, (4) study the ranking behaviours of approaches alternative to PRP in terms of the kinematics they impose on relevant documents, i.e. by considering the extent and direction of the movements of relevant documents across the ranking recorded when comparing PRP against its alternatives. Our findings show that the effectiveness of the examined ranking approaches strongly depends upon the evaluation context. In the traditional evaluation context of ad-hoc retrieval, PRP is empirically shown to be better or comparable to alternative ranking approaches. However, when we turn to examine evaluation contexts that account for interdependent document relevance (i.e. when the relevance of a document is assessed also with respect to other retrieved documents, as it is the case in the diversity retrieval scenario) then the use of quantum probability theory and thus of qPRP is shown to improve retrieval and ranking effectiveness over the traditional PRP and alternative ranking strategies, such as Maximal Marginal Relevance, Portfolio theory, and Interactive PRP. This work represents a significant step forward regarding the use of quantum theory in information retrieval. It demonstrates in fact that the application of quantum theory to problems within information retrieval can lead to improvements both in modelling power and retrieval effectiveness, allowing the constructions of models that capture the complexity of information retrieval situations. Furthermore, the thesis opens up a number of lines for future research. These include (1) investigating estimations and approximations of quantum interference in qPRP, (2) exploiting complex numbers for the representation of documents and queries, and (3) applying the concepts underlying qPRP to tasks other than document ranking

Cellular Automata

Modelling and simulation are disciplines of major importance for science and engineering. There is no science without models, and simulation has nowadays become a very useful tool, sometimes unavoidable, for development of both science and engineering. The main attractive feature of cellular automata is that, in spite of their conceptual simplicity which allows an easiness of implementation for computer simulation, as a detailed and complete mathematical analysis in principle, they are able to exhibit a wide variety of amazingly complex behaviour. This feature of cellular automata has attracted the researchers' attention from a wide variety of divergent fields of the exact disciplines of science and engineering, but also of the social sciences, and sometimes beyond. The collective complex behaviour of numerous systems, which emerge from the interaction of a multitude of simple individuals, is being conveniently modelled and simulated with cellular automata for very different purposes. In this book, a number of innovative applications of cellular automata models in the fields of Quantum Computing, Materials Science, Cryptography and Coding, and Robotics and Image Processing are presented