Tractable subsets of first-order logic are a central topic in AI research. Several these formalisms have been used as the basis for probabilistic languages. However, intractable, losing original motivation. Here we propose first non-trivially tractable language. It is subset Markov logic, and uses class part hierarchies to control complexity. We call it TML (Tractable Logic). show that knowledg...

We propose a novel logic, Frame Logic (FL), that extends first-order logic and recursive definitions with construct Sp (·) captures the implicit supports of formulas—the precise subset universe upon which their meaning depends. Using such supports, we formulate proof rules facilitate frame reasoning elegantly when underlying model undergoes change. show is expressive by capturing several data-s...

We introduce PHFL, a probabilistic extension of higher-order fixpoint logic, which can also be regarded as temporal logics such PCTL and the $\mu^p$-calculus. show that PHFL is strictly more expressive than $\mu^p$-calculus, model-checking problem for finite Markov chains undecidable even $\mu$-only, order-1 fragment PHFL. Furthermore full far expressive: we give translation from Lubarsky's $\m...