CS 173 Lecture 18: Relations on a Set

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1 CS 173 Lecture 18: Relations on a Set José Meseguer University of Illinois at Urbana-Champaign 1 Relations on a Set 1.1 The Data Type RelpAq Many algorithms, in particular the so-called graph algorithms, manipulate data structures that are binary relations on a set A. Therefore the question: What is the data type of binary relations on a set A? is not an idle question at all, since it is at the heart of many important problems and algorithms in Computer Science. Have we seen this before? Indeed, we have. This data type is just the data type of typed relations RelpA, Aq. Recall that for any sets A and B, the set Rel pa, Bq of typed relations from A to B was defined as the set of triples pa, R, Bq such that R P PpA ˆ Bq. Therefore, the data type RelpA, Aq of binary relations on A is just the set of triples pa, R, Aq such that R P PpA 2 q. Abbreviations. Since A B, the second A in the triple pa, R, Aq is redundant, so we can abbreviate a relation R on A to just a pair pa, Rq without any loss of information. Likewise, we can abbreviate RelpA, Aq to just RelpAq. In this abbreviated form, the data type RelpAq has a very simple description. It is just the product type: RelpAq def tau ˆ PpA 2 q whose elements are the pairs pa, Rq with R P PpA 2 q. This means that, up to a small, bijective change of data representation, the data type RelpAq is equivalent to the powerset data type PpA 2 q. Indeed, the second projection function p 2 : PpA 2 q Ñ PpA 2 q such that p 2 pa, Rq R has an obvious inverse function mapping each R P PpA 2 q to the pair pa, Rq P RelpAq. This also mens that RelpAq is a Bolean Algebra, since we can define union, intersection and complement operations as follows: pa, Rq Y pa, Gq def pa, R Y Gq pa, Rq X pa, Gq def pa, R X Gq pa, Rq def pa, Rq.

2 2 J. Meseguer 1.2 Logical and Graphical Perspectives on Rel paq There are two mutually reinforcing perspectives from which we can look at a binary relation pa, Rq P RelpAq: 1. Logically, we look at pa, Rq as a binary predicate on A. 2. Graphically, we look at pa, Rq as a directed graph on A. By a binary predicate on A we usually mean a function p : A 2 Ñ tt, Fu. However, since the data types ra 2 ÑtT, F us and PpA 2 q are equivalent thanks to the bijective change of representation pred2set : ra 2 ÑtT, Fus Ñ PpA 2 q where pred2setppq tpx, yq P A 2 ppx, yq Tu, in First-Order Logic the set of pairs tpx, yq P A 2 ppx, yq Tu is also called a binary predicate on A. These are just two equivalent representations (what we have called the flashlight and the subset representations) for the same concept. Example. The greater than predicate ą: N 2 Ñ tt, Fu on the natural numbers has the equivalent representation as the binary relation pn, ąq, where ą def tpx, yq P N 2 x ą y Tu The slides for this lecture give several examples of how the logical and graphical perspectives can reinforce each other, including the graphical views for the following logical relations: the greater than relation pn, ąq. the relation pn, sq, where s is the successor function the relation pn, pq, where p is the predecessor function. 1.3 Children and Degree When we view pa, Gq P RelpAq as a graph, we call each a P A a node in the graph, and each pa, bq P G a directed edge of the graph pa, Gq. Also, for each node a P A we call the set Gras the set of children nodes of node a, and call the cardinality Gras the degree of node a, denoted deg G paq. Exercise 1. Prove that pa, fq P RelpAq is a function if and only if for all a P A deg f paq Inverse Relation/Inverse Graph Since for any pa, Rq P RelpAq we also have pa, R 1 q P RelpAq, we call pa, R 1 q the inverse relation (resp. the inverse graph) of pa, Rq. Graphically, pa, R 1 q has a very intuitive description: we just reverse the direction of all the edges in pa, Rq.

3 2 Paths in a Graph and Transitive Relations 2.1 Paths in a Graph Relations on a Set 3 Intutively, a path (also called a walk) in a graph is a sequence of the form a 1 Ñ a 2 Ñ a 3... a n 1 Ñ a n where for each i, 1 ď i ă n we have a i`1 P Gra i s. That is, each node a i`1 is a child of the previous node a i. More precisely given a relation pa, Rq P RelpAq, we can define the set PathpA, Rq as the following set of strings: PathpA, Rq def ta 1... a n P A n ě 2^@i P N pp0 ă i^i ă nq ñ a i`1 P Rra i squ Notation. When we adopt a logical perspective, a path a 1... a n P PathpA, Rq is depicted as follows: a 1 R a 2... a n 1 R a n For example, for the ą relation on natural numbers 10 ą 7 ą 5 ą 3 ą 2 ą 1 ą 0 is a path. Instead, when we adopt a graphical perspective, as already mentioned, a path a 1... a n P PathpA, Rq is depicted as follows: a 1 Ñ a 2 Ñ a 3... a n 1 Ñ a n. Note that, since paths are special strings of lenght ě 2, any path in PathpA, Rq has the form a w b for some (possibly empty) string w P A. Whe then call a the source and b the target of the path a w b P PathpA, Rq. Note also that if a w b and b w 1 c are paths in PathpA, Rq, then a w b w 1 c is also a path in PathpA, Rq, called the composition of a w b and b w 1 c. Notation. Given a binary relation pa, Rq and elements a, b P A, the set of all paths from a to be b is, by definition, the set: PathpA, Rqpa, bq def ta w b P A a w b P PathpA, Rqu The slides for this lecture give a simple example of a set of paths PathpA, Gqpa, bq that happens to be infinite, even though the graph pa, Gq is finite. 2.2 Closed Paths, Cycles, and Loops Given a relation pa, Rq P RelpAq: A closed path is a path of the form a w a P PathpA, Rq for some a P A. A cycle is a closed path of the form a a 1... a n a P PathpA, Rq with n ě 1 such that ta, a 1,..., a n u n ` 1. That is, the only repeated element in the string a a 1... a n a is a itself. A loop is a closed path of the form a a P PathpA, Rq for some a P A. The slides give several examples of closed paths, cycles, and loops in a graph.

4 4 J. Meseguer 2.3 Transitive Relations A relation pa, Rq P RelpAq is called transitive if and only if for each a, b, c P A arb ^ brc ñ arc where, using a logical notation, we write xry if and only if px, yq P R. A key question about transitive relations is the following: How are the transitivity of a relation pa, Rq P RelpAq and the set of paths PathpA, Rq related to each other? Transitive Closure. Given a relation pa, Rq P Rel paq, its transitive closure, denoted pa, R`q P RelpAq, is the relation defined by the following equivalence for all a, b P A: pa, bq P R` ô def Dw P A pa w b P PathpA, Rqq That is, pa, bq P R` if and only if there is a path in PathpA, Rq of the form: a R a 1 R a 2... a n 1 R b where the string a 1... a n 1 could be the empty string ɛ. Theorem 1. Given any relation pa, Rq P RelpAq the following two properties hold:. 1. pa, R`q, is a transitive relation. 2. pa, Rq, is transitive if and only if pa, Rq pa, R`q. Proof: Let us first prove (1). Let a, b, c P A be such that ar`b and br`c. By definition this means that we have paths a w b and b w 1 c in PathpA, Rq. But then a w b w 1 c is also a path in PathpA, Rq, and this means that ar`c, making R` transitive, as desired. To prove (2), first note that, by definition of pa, R`q, we always have pa, Rq Ď pa, R`q. We just need to show that if pa, Rq, is transitive, then we also have pa, R`q Ď pa, Rq, that is, that for each x, y P A, if x w y P PathpA, Rq then px, yq P R. This can be proved by induction of the lenght w of the string w in x w y P PathpA, Rq. Base Case w 0. Then w ɛ and by the definition of PathpA, Rq this means that if x y P PathpA, Rq then we must have px, yq P R, as desided. Induction Step. Suppose that for all x, y P A if x w y P PathpA, Rq and w ď n then px, yq P R. Now consider for any x, y P A any path x w 1 y P PathpA, Rq with w 1 n ` 1. Then w 1 must be of the form w z for some w with w n. But then x w z P PathpA, Rq and pz, yq P R. But by the induction hypothesis we have xrz, and since we also have zry and R is transitive, we also have xry, as desired. This finishes the proof of the Theorem.

5 Relations on a Set 5 3 Reflexive and Irreflexive Relations, DAGs, Trees, and Forests Reflexive and Irreflexive Relations. A relation pa, Rq P Rel paq is called: 1. Reflexive P A ppa, aq P Rq 2. Not Reflexive iff Da P A ppa, aq R Rq 3. Irreflexive P A ppa, aq R Rq. Exercise 2. Prove that pa, Rq P RelpAq is reflexive (resp. irreflexive) if and only if R R Y id A (resp. R R id A ). Reflexive Closure. Given a relation pa, Rq P RelpAq, its reflexive closure is, by definition, the relation pa, R Y id A q. Note that, by Exercise 2, for any pa, Rq P RelpAq: (i) pa, R Y id A q is reflexive; and (ii) if pa, Rq is reflexive, then pa, Rq pa, R Y id A q. Directed Acyclic Graphs (DAGs). A relation pa, Rq P Rel paq is called a directed acyclic graph (DAG) if an only if its transitive closure pa, R`q is irreflexive. Exercise 3. Prove that a relation pa, Rq P RelpAq is a DAG if and only if there are no closed paths in PathpA, Rq. This explains the mention of acyclic in directed acyclic graph. The slides give examples of acyclic and non-acyclic graphs. Trees. A relation pa, T q P RelpAq is called a tree if and only if: 1. pa, T q is a DAG, and 2. There exists an element a P A called the root of pa, T q such that for any b P A tau there is a unique path a w b P PathpA, Rqpa, bq, that is, PathpA, Rqpa, bq 1. The slides give examples of DAGs that are trees and DAGS that fail to be trees. Forests. If pa 1, T 1 q,..., pa n, T n q are trees, and for 1 ď i ă j ď n we have A i X A j H, then the DAG pa 1 Y... Y A n, T 1 Y... Y T n q is called a forest. The slides give an example of a forest made up of several trees. 4 Strict Partial Orders A relation pa, ąq P RelpAq is called a strict partial order if and only if: 1. ą is irreflexive, and 2. ą is transitive.

6 6 J. Meseguer Note that it follows from the definition of a DAG pa, Rq P RelpAq that its transitive closure pa, R`q is always a strict partial order. The slides give several examples of strict partial orders. Strict Total Orders. A strict partial order pa, ąq is called total if and only if it satisfies the following property: Trichotomy y P A px ą y _ x y _ y ą xq The slides give several examples of strict total orders and of a strict partial order that is not total.

Slides for Faculty Oxford University Press All rights reserved.

Slides for Faculty Oxford University Press All rights reserved. Oxford University Press 2013 Slides for Faculty Assistance Preliminaries Author: Vivek Kulkarni vivek_kulkarni@yahoo.com Outline Following topics are covered in the slides: Basic concepts, namely, symbols,