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

Relational and Algebraic Methods in Computer Science: 14th International Conference, RAMiCS 2014, Marienstatt, Germany, April 28-May 1, 2014, Proceedings

This book constitutes the proceedings of the 14th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2014 held in Marienstatt, Germany, in April/May 2014. The 25 revised full papers presented were carefully selected from 37 submissions. The papers are structured in specific fields on concurrent Kleene algebras and related formalisms, reasoning about computations and programs, heterogeneous and categorical approaches, applications of relational and algebraic methods and developments related to modal logics and lattices

Relational lattices: from databases to Universal Algebra

Relational lattices are obtained by interpreting lattice connectives as natural join and inner union between database relations. Our study of their equational theory reveals that the variety generated by relational lattices has not been discussed in the existing literature. Furthermore, we show that addition of just the header constant to the lattice signature leads to undecidability of the quasiequational theory. Nevertheless, we also demonstrate that relational lattices are not as intangible as one may fear: for example, they do form a pseudoelementary class. We also apply the tools of Formal Concept Analysis and investigate the structure of relational lattices via their standard contexts. Furthermore, we show that the addition of typing rules and singleton constants allows a direct comparison with the monotonic relational expressions of Sagiv and Yannakakis