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Register a new userNote on paraconsistency and reasoning about fractions
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Added by
Inge Bethke
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We apply a paraconsistent logic to reason about fractions.
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In computability theory and computable analysis, finite programs can compute infinite objects. Presenting a computable object via any program for it, provides at least as much information as presenting the object itself, written on an infinite tape. What additional information do programs provide? We characterize this additional information to be any upper bound on the Kolmogorov complexity of the object. Hence we identify the exact relationship between Markov-computability and Type-2-computability. We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets.
Most modern (classical) programming languages support recursion. Recursion has also been successfully applied to the design of several quantum algorithms and introduced in a couple of quantum programming languages. So, it can be expected that recursion will become one of the fundamental paradigms of quantum programming. Several program logics have been developed for verification of quantum while-programs. However, there are as yet no general methods for reasoning about (mutual) recursive procedures and ancilla quantum data structure in quantum computing (with measurement). We fill the gap in this paper by proposing a parameterized quantum assertion logic and, based on which, designing a quantum Hoare logic for verifying parameterized recursive quantum programs with ancilla data and probabilistic control. The quantum Hoare logic can be used to prove partial, total, and even probabilistic correctness (by reducing to total correctness) of those quantum programs. In particular, two counterexamples for illustrating incompleteness of non-parameterized assertions in verifying recursive procedures, and, one counterexample for showing the failure of reasoning with exact probabilities based on partial correctness, are constructed. The effectiveness of our logic is shown by three main examples -- recursive quantum Markov chain (with probabilistic control), fixed-point Grovers search, and recursive quantum Fourier sampling.
Temporal logics are extensively used for the specification of on-going behaviours of reactive systems. Two significant developments in this area are the extension of traditional temporal logics with modalities that enable the specification of on-going strategic behaviours in multi-agent systems, and the transition of temporal logics to a quantitative setting, where different satisfaction values enable the specifier to formalise concepts such as certainty or quality. We introduce and study FSL---a quantitative extension of SL (Strategy Logic), one of the most natural and expressive logics describing strategic behaviours. The satisfaction value of an FSL formula is a real value in [0,1], reflecting `how much or `how well the strategic on-going objectives of the underlying agents are satisfied. We demonstrate the applications of FSL in quantitative reasoning about multi-agent systems, by showing how it can express concepts of stability in multi-agent systems, and how it generalises some fuzzy temporal logics. We also provide a model-checking algorithm for our logic, based on a quantitative extension of Quantified CTL*.
We introduce a logical framework for the specification and verification of component-based systems, in which finitely many component instances are active, but the bound on their number is not known. Besides specifying and verifying parametric systems, we consider the aspect of dynamic reconfiguration, in which components can migrate at runtime on a physical map, whose shape and size may change. We describe such parametric and reconfigurable architectures using resource logics, close in spirit to Separation Logic, used to reason about dynamic pointer structures. These logics support the principle of local reasoning, which is the key for writing modular specifications and building scalable verification algorithms, that deal with large industrial-size systems.
Representation of defeasible information is of interest in description logics, as it is related to the need of accommodating exceptional instances in knowledge bases. In this direction, in our previous works we presented a datalog translation for reasoning on (contextualized) OWL RL knowledge bases with a notion of justified exceptions on defeasible axioms. While it covers a relevant fragment of OWL, the resulting reasoning process needs a complex encoding in order to capture reasoning on negative information. In this paper, we consider the case of knowledge bases in $textit{DL-Lite}_{cal R}$, i.e. the language underlying OWL QL. We provide a definition for $textit{DL-Lite}_{cal R}$ knowledge bases with defeasible axioms and study their properties. The limited form of $textit{DL-Lite}_{cal R}$ axioms allows us to formulate a simpler encoding into datalog (under answer set semantics) with direct rules for reasoning on negative information. The resulting materialization method gives rise to a complete reasoning procedure for instance checking in $textit{DL-Lite}_{cal R}$ with defeasible axioms.
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