Automated Quantum Reasoning: Non Logic - Semi-Logic - Hyper-Logic

Quantum theory does not necessitate the breakdown of full-blown deduction. On the contrary, it comes with substantially enhanced logical reasoning power as compared to its classical counterpart. It features a codeductive mechanism besides a deductive one, resulting in a sound purely graphical calculus which admits an information-flow interpretation. (Abramsky & Coecke 2004; Coecke 2005a; 2005b; Coecke & Pavlovic 2006). The key physical concept represented by the logic is the interaction of quantum systems i.e. the tensor product structure, contra (Birkhoff & von Neumann 1936)-logic which only addresses individual systems. The trace structure, important in IR applications (van Rijsbergen 2004), is an intrincic part of the logic, together with many other quantitative concepts, again contra BvNlogic where the trace only arises indirectly via Gleason’s theorem. Hence we provide a powerful highlevel formalism for designing, controlling and even automating quantum informatic tasks, which can involve multiple agents. We also mention several existing applications to non-quantum domains such as linguistics, multi-agent systems and concurrency. Introduction This paper is concerned with the logical mechanisms which govern the observable behaviour of physical systems subject to quantum mechanics, be it for the purpose of either: • simulation, analysis and design of computational and communicational devices build out of quantum hardware, • the passage from quantum informatics to quantum AI and more general, any form of automated quantum reasoning, • or, for applying the quantum formalism to other areas of science, exploiting its relaxed and at the same time more versatile features as compared to its classical counterpart. For about 70 years, following the seminal paper (Birkhoff & von Neumann 1936), there was the belief that “quantum logic” implies breakdown of deductive mechanisms. However, recent progress has shown that quantum reasoning is not about decreased but increased deductive power as compared to its classical counterpart. We provide a comprehenCopyright c © 2007, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. sive story on the recent progress which has been made concerning, and discuss some of its applications to other fields. Much of the work towards “quantum hyper-logic” and its applications to quantum informatics was done jointly with, respectively, S. Abramsky (Oxford), D. Pavlovic (Kestrel) and E. O. Paquette (Montréal), the results on the preceding “quantum semi-logic” were obtained mainly with I. Stubbe (Antwerp), and the non-physics applications were developed with A. Baltag (Oxford) and M. Sadrzadeh (Southampton). Quantum logicians have used the physical concepts of observable and measurement as vehicles to extract logical structure (Birkhoff & von Neumann 1936; Piron 1976; Foulis & Randall 1972; Randall & Foulis 1973; Coecke, Moore, & Wilce 2000). On the other hand, we focussed on the concept of quantum interaction, that is, any situation which involves more than one system – and, as we shall show, encompasses measurement situations. Importantly, it was exactly quantum interaction which, besides the decrease of logical mechanisms, has been a stumblingblock for quantum logicians. Indeed, even today the community’s main goal is to find a useful logical counterpart to Hilbert space tensor product. As a result, the pay-off of the progress in quantum logic to the physics community has remained quite obscure, resulting in the physics community and in particular von Neumann himself to denounce the field (Rédei 1997). More specifically, while Logic has become a part of Computer Science rather than of pure mathematics, traditional quantum logic never had any pay-off to Quantum Computer Science (QCS). On the other hand, our approach recently contributed to an 1,600,000 EUR Specific Targeted Research Project to be granted by the EC to a consortium coordinated by the present author and including leading European QCS-pioneers. This is how the “hyper-logical” description and correctness derivation of (tripartite) quantum teleportation looks: