Galois Connections and Fixed Point Calculus
نویسنده
چکیده
Fixed point calculus is about the solution of recursive equations de ̨ned by a monotonic endofunction on a partially ordered set. This tutorial presents the basic theory of ̨xed point calculus together with a number of applications of direct relevance to the construction of computer programs. The tutorial also presents the theory and application of Galois connections between partially ordered sets. In particular, the intimate relation between Galois connections and ̨xed point equations is amply demonstrated. ∗School of Computer Science and Information Technology, University of Nottingham, Nottingham NG8 1BB, England
منابع مشابه
Lattices of Fixed Points of Fuzzy Galois Connections
We give a characterization of the fixed points and of the lattices of fixed points of fuzzy Galois connections. It is shown that fixed points are naturally interpreted as concepts in the sense of traditional logic.
متن کاملCalculating Graph Algorithms by Abstract Interpretation
We present a technique to calculate iterative, polynomial-time graph algorithms directly from the definition of the properties. We do this by fixed-point fusion of (1) a least fixed point expressing all finite paths through a directed graph and (2) Galois connections that capture the properties of interest. We demonstrate the technique by constructing three algorithms from the literature: a tra...
متن کاملCalculating Graph Algorithms for Dominance and Shortest Path
We calculate two iterative, polynomial-time graph algorithms from the literature: a dominance algorithm and an algorithm for the single-source shortest path problem. Both algorithms are calculated directly from the definition of the properties by fixed-point fusion of (1) a least fixed point expressing all finite paths through a directed graph and (2) Galois connections that capture dominance a...
متن کاملFuzzy Galois Connections
The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided. 1...
متن کاملClosure Operators and Polarities
Basic results are obtained concerning Galois connections between collections of closure operators (of various types) and collections consisting of subclasses of (pairs of) morphisms in M for an 〈E,M 〉 -category X . In effect, the “lattice” of closure operators on M is shown to be equivalent to the fixed point lattice of the polarity induced by the orthogonality relation between composable pairs...
متن کامل