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Shutao Zhang

Publications and source records attributed to Shutao Zhang.

22 records · Page 2Linked to original sources
arXiv2021

On the Strong Equivalences for LPMLN Programs

LPMLN is a powerful knowledge representation and reasoning tool that combines the non-monotonic reasoning ability of Answer Set Programming (ASP) and the probabilistic reasoning ability of Markov Logic Networks (MLN). In this paper, we study the strong equivalence for LPMLN programs, which is an important tool for program rewriting and theoretical investigations in the field of logic programming. First of all, we present the notion of p-strong equivalence for LPMLN and present a model-theoretical characterization for the notion. And we investigate the relationships among the p-strong equivalence and other existing notions of strong equivalences for LPMLN. Then, we investigate several properties of the p-strong equivalence from the following four aspects. Firstly, we investigate two relaxed notions of the p-strong equivalence according to practical scenarios of program rewriting, and present corresponding characterizations for the notions. Secondly, we analyze the computational complexities of deciding strong equivalences for LPMLN programs. Thirdly, we investigate the relationships among the strong equivalences of LPMLN and two extensions of ASP: ASP with weak constraints and ordered disjunctions. Finally, we investigate LPMLN program simplification via the p-strong equivalence and present some syntactic conditions that decide the p-strong equivalence between a single LPMLN rule and the empty program. The contributions of the paper are as follows. Firstly, all of the results presented in this paper provide a better understanding of LPMLN programming, which helps us further explore the properties of LPMLN. Secondly, the relationships among the strong equivalences open a way to study the strong equivalences for some logic formalisms by translating into LPMLN. Thirdly, the program simplification can be used to enhance the implementations of the LPMLN solvers ...

cs.LO↗
arXiv2019

On the Strong Equivalences of LPMLN Programs

By incorporating the methods of Answer Set Programming (ASP) and Markov Logic Networks (MLN), LPMLN becomes a powerful tool for non-monotonic, inconsistent and uncertain knowledge representation and reasoning. To facilitate the applications and extend the understandings of LPMLN, we investigate the strong equivalences between LPMLN programs in this paper, which is regarded as an important property in the field of logic programming. In the field of ASP, two programs P and Q are strongly equivalent, iff for any ASP program R, the programs P and Q extended by R have the same stable models. In other words, an ASP program can be replaced by one of its strong equivalent without considering its context, which helps us to simplify logic programs, enhance inference engines, construct human-friendly knowledge bases etc. Since LPMLN is a combination of ASP and MLN, the notions of strong equivalences in LPMLN is quite different from that in ASP. Firstly, we present the notions of p-strong and w-strong equivalences between LPMLN programs. Secondly, we present a characterization of the notions by generalizing the SE-model approach in ASP. Finally, we show the use of strong equivalences in simplifying LPMLN programs, and present a sufficient and necessary syntactic condition that guarantees the strong equivalence between a single LPMLN rule and the empty program.

cs.LO↗
arXiv2019

Sharp Sobolev inequalities on the complex sphere

This paper is devoted to establish a class of sharp Sobolev inequalities on the unit complex sphere as follows: 1) Case $0<d<Q=2n+2$: for any $f\in C^\infty$ and $2\leq q \leq \frac{2Q}{Q-d}$, \begin{equation*} \|f\|_q^2\leq \frac{8(q-2)}{d(Q-d)} \frac{Γ^2((Q-d)/4+1)} {Γ^2((Q+d)/4)}\left( \int_{\mathbb{S}^{2n+1}} f\mathcal{A}_df dξ -\frac{Γ^2((Q+d)/4)} {Γ^2((Q-d)/4)} \int_{\mathbb{S}^{2n+1}} |f|^2 dξ\right) +\int_{\mathbb{S}^{2n+1}} |f|^2 dξ; \end{equation*} 2) Case $d=Q$: for any $f\in C^\infty \cap\mathbb{R}\mathcal{P}$ and $2\leq q< +\infty$, \begin{equation*} \|f\|_q^2\leq \frac{q-2}{(n+1)!} \int_{\mathbb{S}^{2n+1}} f \mathcal{A}'_Q f dξ+\int_{\mathbb{S}^{2n+1}} |f|^2 dξ, \end{equation*} where $\mathcal{A}_d(0<d<Q)$ are the intertwining operator, $\mathcal{A}'_Q$ is the conditional intertwinor introduced in \cite{BFM2013}, and $dξ$ is the normalized surface measure of $\mathbb{S}^{2n+1}$.

math.AP↗