The stochastic interpretation of Parikh＇s game logic should not follow the usual pattern of Kripke models, which in turn are based on the Kleisli morphisms for the Giry monad, rather, a specific and more general appr...The stochastic interpretation of Parikh＇s game logic should not follow the usual pattern of Kripke models, which in turn are based on the Kleisli morphisms for the Giry monad, rather, a specific and more general approach to proba- bilistic nondeterminism is required. We outline this approach together with its probabilistic and measure theoretic basis, in- troducing in a leisurely pace the Giry monad and its Kleisli morphisms together with important techniques for manipu- lating them. Proof establishing specific techniques are given, and pointers to the extant literature are provided. After working through this tutorial, the reader should find it easier to follow the original literature in this and related areas, and it should be possible for her or him to appreciate measure theoretic arguments for original work in the areas of Markov transition systems, and stochastic effectivity func- tions.展开更多
We extend the constraint data model to allow complex objects and study the expressive power of various query languages over this sort of constraint databases.The tools we use come in the form of collapse results which...We extend the constraint data model to allow complex objects and study the expressive power of various query languages over this sort of constraint databases.The tools we use come in the form of collapse results which are well established in the context of first-order logic.We show that the natural-active collapse with a condition and the activegeneric collapse carry over to the second-order logic for structures with o-minimality property and any signature in the complex value relations.The expressiveness results for more powerful logics including monadic second-order logic,monadic second-order logic with fix-point operators,and fragments of second-order logic are investigated in the paper.We discuss the data complexity for second-order logics over constraint databases.The main results are that the complexity upper bounds for three theories,MSO+(LIN),MSO+(POLY),and Inflationary DATALOGact^cv(SC,M)without powerset operator are∪iΣi^NC1,NCH=∪iΣi^NC,and AC^0/poly,respectively.We also consider the problem of query closure property in the context of embedded finite models and constraint databases with complex objects and the issue of how to determine safe constraint queries.展开更多
文摘The stochastic interpretation of Parikh＇s game logic should not follow the usual pattern of Kripke models, which in turn are based on the Kleisli morphisms for the Giry monad, rather, a specific and more general approach to proba- bilistic nondeterminism is required. We outline this approach together with its probabilistic and measure theoretic basis, in- troducing in a leisurely pace the Giry monad and its Kleisli morphisms together with important techniques for manipu- lating them. Proof establishing specific techniques are given, and pointers to the extant literature are provided. After working through this tutorial, the reader should find it easier to follow the original literature in this and related areas, and it should be possible for her or him to appreciate measure theoretic arguments for original work in the areas of Markov transition systems, and stochastic effectivity func- tions.
文摘We extend the constraint data model to allow complex objects and study the expressive power of various query languages over this sort of constraint databases.The tools we use come in the form of collapse results which are well established in the context of first-order logic.We show that the natural-active collapse with a condition and the activegeneric collapse carry over to the second-order logic for structures with o-minimality property and any signature in the complex value relations.The expressiveness results for more powerful logics including monadic second-order logic,monadic second-order logic with fix-point operators,and fragments of second-order logic are investigated in the paper.We discuss the data complexity for second-order logics over constraint databases.The main results are that the complexity upper bounds for three theories,MSO+(LIN),MSO+(POLY),and Inflationary DATALOGact^cv(SC,M)without powerset operator are∪iΣi^NC1,NCH=∪iΣi^NC,and AC^0/poly,respectively.We also consider the problem of query closure property in the context of embedded finite models and constraint databases with complex objects and the issue of how to determine safe constraint queries.
