Encoding Logical Words as Quantum Gates : The Higher Dimensional Case

The open problem and topic to be covered in this master thesis is the development of logical words with focus on “NOT” as quantum gates in the three and higher dimensional cases. This is undertaken in the associated framework of vector spaces, pregroups and the diagrammatic calculus of compact closed categories as in [6]. It involves the development of canonical matrices or other maps that work uniformly for higher dimensions. The maps are then validated by means of implementation in well-chosen grammatical sentences in several languages as well as their mathematical meaning calculations. Various approaches will be inspected using linguistic and quantum mechanical intuition. One idea that is investigated was first expressed by supervisor Mehrnoosh Sadrzadeh by having a multi-dimensional gate in the form of a linear map; e.g. to be able to distinguish mathematically the semantic difference among the various uses of these logical words, such as predicative or non-predicative use of the associated verb. This leads to both syntactic, but also semantic differences, such as having or not having a noun phrase as the header of their linguistic tree diagram (cf. [9]). For negation the distinction does not accurately apply as worked out in the thesis, hence the main philosophical and logical interpretations of negation will be investigated. Higher-dimensional maps are set up for each one of them. In some cases it will be proven that no such linear maps exist and non-linear maps are set up instead. A second investigation consists in another possible way for an expansion to higher dimension as aforementioned by means of projections to orthogonal subspaces. This is then also implemented in the diagrammatic calculus. Future work can then contain expansion to other important logical words and sentences in the same manner and application in the categorical approach as undertaken by supervisor Bob Coecke and several of his DPhil students. Eventually, this can contribute significantly to quantum computer science, computational linguistics, information retrieval and A.I. by implementation in quantum protocols and machine translation. Chapter 1 Research Procedure I attended the excellent quantum computer science course by Dr. Andreas Doering and Dr. Bob Coecke during Hilary term with substantial reading in the topic and a significant amount of time invested into the final mini-project and current research. Over the spring break the potential supervisors were contacted and current research papers on the topic (viz. [4], [6], [15] and [17]) identified, read and discussed with the two supervisors to determine the open problem that was to be pursued. The questions took on their current shape in further discussion and analysis during the first two weeks of Trinity term together with the two supervisors as well as advice for the computational linguistics aspect of the thesis from Professor Pulman and DPhil student Edward Grefenstette at the OUCL. Trinity term was invested to pursue the research via the approaches explained in regular contact with the supervisors, besides English linguistics preparation for the GRE and one further successful course at the Mathematics Department. This led to less time available than for co-students in the course, but to successfully getting the last required exam score for a possible distinction depending on a good thesis of 80%. Writing up was partly undertaken on the way for the drafts and results. The final evaluation and major writing up was then done from mid-July throughout August with the viva voce outstanding on 24th September, 2010. Table 1.1 shows the plan for completion of the research. Timeline Work Progress Contact supervisors and define the topic in proposal completed May. Reading and research completed July/August Thesis writing completed September Viva voce outstanding Table 1.1: Plan for completion of my research