Fixed-point elimination in the intuitionistic propositional calculus

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $\mu$-calculus based on intuitionistic logic is trivial, every $\mu$-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $\mu$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal

Belief functions on lattices

We extend the notion of belief function to the case where the underlying structure is no more the Boolean lattice of subsets of some universal set, but any lattice, which we will endow with a minimal set of properties according to our needs. We show that all classical constructions and definitions (e.g., mass allocation, commonality function, plausibility functions, necessity measures with nested focal elements, possibility distributions, Dempster rule of combination, decomposition w.r.t. simple support functions, etc.) remain valid in this general setting. Moreover, our proof of decomposition of belief functions into simple support functions is much simpler and general than the original one by Shafer

Interpolation in local theory extensions

In this paper we study interpolation in local extensions of a base theory. We identify situations in which it is possible to obtain interpolants in a hierarchical manner, by using a prover and a procedure for generating interpolants in the base theory as black-boxes. We present several examples of theory extensions in which interpolants can be computed this way, and discuss applications in verification, knowledge representation, and modular reasoning in combinations of local theories.Comment: 31 pages, 1 figur

Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted minimum and maximum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions

*-Continuous Kleene ω\omega-Algebras for Energy Problems

Energy problems are important in the formal analysis of embedded or autonomous systems. Using recent results on star-continuous Kleene omega-algebras, we show here that energy problems can be solved by algebraic manipulations on the transition matrix of energy automata. To this end, we prove general results about certain classes of finitely additive functions on complete lattices which should be of a more general interest.Comment: In Proceedings FICS 2015, arXiv:1509.0282

The Power of Linear Programming for Valued CSPs

A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation. Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo