Basics of Relation Algebra. Jules Desharnais Département d informatique et de génie logiciel Université Laval, Québec, Canada

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1 Basics of Relation Algebra Jules Desharnais Département d informatique et de génie logiciel Université Laval, Québec, Canada

2 Plan 1. Relations on a set 2. Relation algebra 3. Heterogeneous relation algebra 4. References

3 Notation from here

4 1 Relations on a set 1. A relation on a set S is a subset of S S 2. The set of relations on S is P(S S) 3. Notation: srt (s, t) R, V = S S 4. Operations on relations Set-theoretical operations:,, Relational operations,, V Composition (relative product): sq ;Ru ( t : sqt tru) Converse: sr t trs Identity relation: sit s = t

5 Representations of relations: sets of ordered pairs, graphs, matrices Let S def = {1, 2, 3}. Q ; R = Q ;R {(1, 2), (2, 2), (2, 3)} ; {(1, 2), (2, 1), (2, 3)} = {(1, 1), (1, 3), (2, 1), (2, 3)} ; = ( ) ; ( ) = ( )

6 Relations on a set satisfy many laws Increasing priority: (, ), ;, Q R = R Q P (Q R) = (P Q) R (Q R) = Q R I ;R = R P ;(Q R) = P ;Q P ;R P ;(Q R) (P ;Q) ;R (Q ;R) = R ;Q R V ;R;V = V P ;Q R P ;R Q R ;Q P P ;Q R (P R ;Q ) ;(Q P ;R)...

7 2 Relation algebra Relation algebra (RA): Aims at characterizing relations on a set by means of simple equational axioms. It is a structure A = A,,, ;,, I such that (1) Q R = R Q (2) P (Q R) = (P Q) R (3) Q R Q R = Q Boolean algebra axioms

8 2 Relation algebra Relation algebra (RA): Aims at characterizing relations on a set by means of simple equational axioms. It is a structure A = A,,, ;,, I such that (1) Q R = R Q (2) P (Q R) = (P Q) R (3) Q R Q R = Q Another axiomatisation of Boolean algebra: Add,, to the signature, replace Huntington s axiom (3) by Q R = R Q P (Q R) = (P Q) R Q (Q R) = Q Q (Q R) = Q P (Q R) = (P Q) (P R) R = R R = R R R = R R =

9 2 Relation algebra Relation algebra (RA): Aims at characterizing relations on a set by means of simple equational axioms. It is a structure A = A,,, ;,, I such that (1) Q R = R Q (2) P (Q R) = (P Q) R (3) Q R Q R = Q (4) P ;(Q ;R) = (P ;Q) ;R (5) (P Q) ;R = P ;R Q ;R (6) R ;I = R (7) R = R (8) (Q R) = Q R (9) (Q ;R) = R ;Q (10) Q ;Q;R R = R Boolean algebra axioms

10 2 Relation algebra Relation algebra (RA): Aims at characterizing relations on a set by means of simple equational axioms. It is a structure A = A,,, ;,, I such that (1) Q R = R Q (2) P (Q R) = (P Q) R (3) Q R Q R = Q (4) P ;(Q ;R) = (P ;Q) ;R (5) (P Q) ;R = P ;R Q ;R (6) R ;I = R (7) R = R (8) (Q R) = Q R (9) (Q ;R) = R ;Q (10) Q ;Q;R R = R Boolean algebra axioms Ordering Define Q R Q R = R. Then (10) can be written Q ;Q;R R.

11 2 Relation algebra Relation algebra (RA): Aims at characterizing relations on a set by means of simple equational axioms. It is a structure A = A,,, ;,, I such that (1) Q R = R Q (2) P (Q R) = (P Q) R (3) Q R Q R = Q (4) P ;(Q ;R) = (P ;Q) ;R (5) (P Q) ;R = P ;R Q ;R (6) R ;I = R (7) R = R (8) (Q R) = Q R (9) (Q ;R) = R ;Q (10) Q ;Q;R R = R Boolean algebra axioms Derived operators,, Q R = Q R = I I = I I

12 Laws that can be proved from the axioms Relation algebra Corresponding laws, relations on sets P (Q R) = (P Q) R P (Q R) = (P Q) R (Q R) = Q R (Q R) = Q R I ;R = R I ;R = R I = I I = I = V = V ; = V ;V = V P ;(Q R) = P ;Q P ;R P ;(Q R) = P ;Q P ;R P ;Q R P ;R Q P ;Q R P ;R Q R ;Q P R ;Q P P ;Q R (P R ;Q ) ;(Q P ;R) P ;Q R (P R ;Q ) ;(Q P ;R)??? R V ;R;V = V

13 Properties of the equational axiomatisation Because RAs are defined by equations, the class of RAs is a variety: it is closed under products, homomorphic images and subalgebras. def def Example. Consider the relations on S 2 = {1, 2} and S 3 = {1, 2, 3} or, equivalently, the subsets of ( ) ( ) def 1 1 def V 2 = 1 1 and V 3 = Products The set of pairs of relations A 2,3 = {(R 2, R 3 ) R 2 V 2 R 3 V 3 } is an RA with identity (I 2, I 3 ). Operations are defined pointwise. E.g., (Q 2, Q 3 ) ;(R 2, R 3 ) = (Q 2 ;R 2, Q 3 ;R 3 ) and (R 2, R 3 ) = (R 2, R 3 ). The top relation is (V 2, V 3 ) or, on an isomorphic matrix form, V 2,3 =

14 2. Homomorphic images Let f : A 2,3 P(S 2 S 2 ) be defined by f((r 2, R 3 )) = R 2. Then f is a homomorphism. An RA homomorphism is defined by the following properties. f : A B f(q A R) = f(q) B f(r) f(r A ) = f(r) B f(q ; A R) = f(q) ; B f(r) f(r A ) = (f(r)) B f(i A ) = I B f : A 2,3 P(S 2 S 2 ) f((q 2, Q 3 ) (R 2, R 3 )) = Q 2 R 2 f((r 2, R 3 )) = R 2 f((q 2, Q 3 ) ;(R 2, R 3 )) = Q 2 ;R 2 f((r 2, R 3 ) ) = (R 2 ) f((i 2, I 3 )) = I 2 The image of an RA homomorphism is an RA.

15 3. Subalgebras An RA is a subalgebra of an RA B = B,,, ;,, I A = A,,, ;,, I if A B (note: the operations are the same). For instance, {, I, I, },,, ;,, I is a subalgebra of every RA with at least 4 elements. Given B A, a subalgebra can be generated by closing B under the operations of A.

16 Models of the axioms 1. Relations on a set S where the universal relation V is an equivalence relation. For instance, all relations included in V 2,3 = form a relation algebra. Now let R 2,3 = and consider the composition V 2,3 ; R 2,3 ; V 2,3.

17 V 2,3 ; R 2,3 ; V 2,3 = = ; ; V 2,3 Thus, RAs do not in general satisfy the Tarski rule R ;R; =. Those that do are simple RAs (with only two homomorphic images: themselves and the trivial RA with one element; they are not closed under products). For concrete relations on a set S, they are those with V = S S.

18 Models of the axioms 2. Let A = A,,, ;,, I be an RA. Let M n be the set of n n matrices with elements of A as entries. Define the following (red) operations on M n. Operation I Definition (M N)[i, j] def = M[i, j] N[i, j] M[i, j] def = M[i, j] ; (M ;N)[i, j] def = ( k : M[i, k] ;N[k, j]) M [i, j] def = (M[j, i]) { I[i, j] def = I if i = j if i j Then is an RA. M n,,, ;,, I

19 This model can be used for the description of programs. R 1 t 2 t Q 3 4 t t Q R The matrix (graph) represents the control structure. The entries of the matrix (labels of the graph) are relations describing how the state changes by a transition.

20 Expressing properties of relations Property Definition Set-theoretical expression R total R ; = ( x : ( y : xry)) I R ;R R R ;I R functional R ;R I ( x, y, z : xry xrz y = z) (or univalent) R ;I R R reflexive I R ( x : xrx) R antisymmetric R R = I ( x, y : xry yrx x = y) R transitive R ;R R ( x, y, z : xry yrz xrz).. Using the relational instead of the set-theoretical definitions leads to equational proofs that are more compact and easier to verify..

21 Other properties Property Definition Expression R surjective R total R ; = (or ;R = ) I R ;R R R ;I (or R I ;R) R injective R functional R ;R I R ;I R (or I ;R R) R mapping R total R ;I = R (or total function) and functional R bijective R injective I ;R = R and surjective R bijective mapping R ;R = I and R ;R = I R partial order R reflexive, antisymmetric and transitive

22 Representing subsets There are three equivalent ways to represent subsets by relations. 1. Vectors: relations of the form R ;. 2. Covectors: relations of the form ;R. 3. Subidentities (tests, types): relations t such that t I. Example: representation of the subset {1, 2} of {1, 2, 3} 1. Vector {(1, 1), (1, 2), (1, 3), (2, 1), (2, 1), (2, 3)} Covector {(1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (3, 2)} Subidentity {(1, 1), (2, 2)}

23 Vectors, covectors, tests form a Boolean algebra Let A = A,,,, ;,,,, I be an RA (,, added to the signature for simplicity). 1. Vectors: {R ; R A},,,,, is a BA. 2. Covectors: { ;R R A},,,,, is a BA. 3. Tests: For t I, define t def = t I. {t t I},,,,, I is a BA. For this BA, s t = s ;t. Tests occur in structures without and, like Kleene algebra with tests.

24 Correspondence between vectors, covectors and tests 1. Vector to covector: R ; ;R 2. Covector to vector: ;R R ; 3. Test to vector: t t ; 4. Vector to test: R ; I R ; Prerestriction and postrestriction prerestriction postrestriction vector R ; Q covector ;R Q test t ;Q Q ;t

25 Representability 1. An RA is representable if it is isomorphic to a subalgebra of a concrete relation algebra (one that consists of the subsets of an equivalence relation). 2. There exist nonrepresentable RAs (even finite ones). 3. The class of representable RAs can only be axiomatised with an infinite number of axioms.

26 3 Heterogeneous relation algebra Relations between different sets Let S = {1, 2, 3}, T = {a, b} and U = {,, }. Q ; R = Q ;R {(1, b), (2, b)} ; {(a, ), (b, ), (b, )} = {(1, ), (1, ), (2, ), (2, )} 1 a 2 3 b ; a b = ( ) ; ( Composition of matrices is possible if sizes match. ) = ( )

27 Axiomatising heterogeneous relation algebra Same axioms, but add typing and typing rules (the definition can also be based on category theory). 1. R S,T : S T (relation R S,T has type S T). 2. Q S,T R S,T : S T. 3. R S,T : S T. 4. Q S,T ;R T,U : S U. 5. R S,T : T S. 6. Q S,T R U,V is defined only if S = U and T = V. 7. Q S,T ;R U,V is defined only if T = U. 8. There are constants I S,S, S,T, S,T for each S and T. Types are usually omitted from expressions. Thus, different instances of I may have different types (same remark for and ).

28 Laws that can be derived Most laws derivable in the homogeneous setting are also derivable in the heterogeneous setting, but there are exceptions. Example Homogeneous RA law: ; =. Proof = I ; ; Heterogeneous RA: S,T ; T,U = S,U cannot be derived. The above proof cannot be used: S,U = I S,S ; S,U S,T ; T,U S,U. And there is a counterexample. The only relation between sets S and is S, = S, Thus S, ;,S = S, ;,S = S,S S,S unless S =.

29 Direct products (internal) A pair of (projection) relations (π 1, π 2 ) is called a direct product iff (a) π 1 ; π 1 = I π 1 functional and surjective (b) π 2 ; π 2 = I π 2 functional and surjective (c) π 1 ; π 2 = any two elements can be paired (d) π 1 ;π 1 π 2 ;π 2 = I π 1, π 2 total and construct all pairs in a unique way Set model π 1 π 2 def = {((s 1, s 2 ), s 1 ) s 1 S 1 s 2 S 2 } def = {((s 1, s 2 ), s 2 ) s 1 S 1 s 2 S 2 } 1. π 1 ; π 1 = {(s, s) s S 1 } = I S1,S 1 2. π 2 ; π 2 = {(s, s) s S 2 } = I S2,S 2 3. π 1 ; π 2 = S 1 S 2 = V S1,S 2 4. π 1 ;π 1 π 2 ;π 2 = {((s 1, s 2 ), (s 1, s 2)) s 1 S 1 s 2, s 2 S 2 } {((s 1, s 2 ), (s 1, s 2 )) s 1, s 1 S 1 s 2 S 2 } = {((s 1, s 2 ), (s 1, s 2)) s 1, s 2 S 1 s 2 S 2 } = I S1 S 2,S 1 S 2

30 Remark: Direct products for empty types Consider the set model on the previous page. If S 1 = and S 2, then both π 1 and π 2 are empty and π 2 cannot be surjective. If empty types have to be dealt with, only functionality of π 1 and π 2 should be required 1, i.e., π 1 ; π 1 I and π 2 ; π 2 I. 1 Thanks to Michael Winter for pointing that out to me.

31 Tupling and parallel product Set model R 1, R 2 ] def = R 1 ;π 1 R 2 ;π 2 [R 1, R 2 ] def = π 1 ;R 1 ;π 1 π 2 ;R 2 ;π 2 R 1, R 2 ] = {(s, (s 1, s 2 )) sr 1 s 1 sr 2 s 2 } [R 1, R 2 ] = {((s 1, s 2 ), (s 1, s 2)) s 1 R 1 s 1 s 2 R 2 s 2} Compare with the cartesian product: R 1 R 2 = {((s 1, s 1), (s 2, s 2)) s 1 R 1 s 1 s 2 R 2 s 2} Same cardinality, but different structure. More general product π 1 ;R 1 ;ρ 1 π 2 ;R 2 ;ρ 2 where (π 1, π 2 ) and (ρ 1, ρ 2 ) are direct products.

32 Matrix model (an example of parallel product) The axioms: π 1 ; π 1 = I π 2 ; π 2 = I π 1 ; π 2 = π 1 ;π 1 π 2 ;π 2 = I π 1 = π 2 = R 1 = ( ) a b c d R 2 = a,..., m {0, 1} e f g h i j k l m [R 1, R 2 ] = a e a f a g b e b f b g a h a i a j b h b i b j a k a l a m b k b l b m c e c f c g d e d f d g c h c i c j d h d i d j c k c l c m d k d l d m

33 Matrix model (an example of tupling) The axioms: π 1 ; π 1 = I π 2 ; π 2 = I π 1 ; π 2 = π 1 ;π 1 π 2 ;π 2 = I π 1 = π 2 = R 1 = ( ) a b c d R 2 = a,..., j {0, 1} ( e f g h i j ) R 1, R 2 ] = ( a e a f a g b e b f ) b g c h c i c j d h d i d j

34 Unsharpness Let (π 1, π 2 ) be the direct product used in the tuplings and parallel products below. Let [R 1, R 2 def = π 1 ;R 1 π 2 ;R 2. The problem: [Cardoso 1982] Does hold for all relations Q 1, Q 2, R 1, R 2? Q 1, Q 2 ] ;[R 1, R 2 = Q 1 ;R 1 Q 2 ;R 2 1. It holds for concrete algebras of relations and all representable RAs. 2. It holds in RA for many special cases [Zierer 88]. 3. It does not hold in RA [Maddux 1993]. 4. It holds in RA for the special cases Q 1, Q 2 ] ;[R 1, R 2 ] = Q 1 ;R 1, Q 2 ;R 2 ] [Q 1, Q 2 ] ;[R 1, R 2 ] = [Q 1 ;R 1, Q 2 ;R 2 ] [Desharnais 1999]. The last equality was in fact the original problem of Cardoso and it was generalized to the one stated above.

35 Unsharpness Let (π 1, π 2 ) be the direct product used in the tuplings and parallel products below. Let [R 1, R 2 def = π 1 ;R 1 π 2 ;R 2. The problem: [Cardoso 1982] Does hold for all relations Q 1, Q 2, R 1, R 2? Q 1, Q 2 ] ;[R 1, R 2 = Q 1 ;R 1 Q 2 ;R 2 1. It holds for concrete algebras of relations and all representable RAs. 2. It holds in RA for many special cases [Zierer 88]. 3. It does not hold in RA [Maddux 1993]. 4. It holds in RA for the special cases Q 1, Q 2 ] ;[R 1, R 2 ] = Q 1 ;R 1, Q 2 ;R 2 ] [Q 1, Q 2 ] ;[R 1, R 2 ] = [Q 1 ;R 1, Q 2 ;R 2 ] [Desharnais 1999]. The last equality was in fact the original problem of Cardoso and it was generalized to the one stated above.

36 Direct sums (internal) A pair of (injection) relations (σ 1, σ 2 ) is called a direct sum iff (a) σ 1 ;σ 1 = I σ 1 total and injective (b) σ 2 ;σ 2 = I σ 2 total and injective (c) σ 1 ;σ 2 = σ 1 and σ 2 inject elements in disjoint subsets (d) σ 1 ; σ 1 σ 2 ; σ 2 = I σ 1, σ 2 functional and construct all injected elements Set model σ 1 σ 2 def = {(s, (s, 1)) s S 1 } def = {(s, (s, 2)) s S 2 } 1. σ 1 ;σ 1 = I S1,S 1 2. σ 2 ;σ 2 = I S2,S 2 3. σ 1 ;σ 2 = S1,S 2 4. σ 1 ; σ 1 σ 2 ; σ 2 = {((s, 1), (s, 1)) s S 1 } {((s, 2), (s, 2)) s S 2 } = I S1 S 2,S 1 S 2

37 Matrix model (an example of direct sum) The axioms: σ 1 ;σ 1 = I σ 2 ;σ 2 = I σ 1 ;σ 2 = σ 1 ; σ 1 σ 2 ; σ 2 = I R 1 = σ 1 = ( ) ( ) a b c d R 2 = σ 2 = e f g h i j k l m a,..., m {0, 1} σ 1 ; R 1 ;σ 1 σ 2 ; R 2 ;σ 2 = a b c d e f g 0 0 h i j 0 0 k l m

38 Expressivity 1. Maddux, page 33 in [Brink Kahl Schmidt 1997]: An equation is true in every relation algebra iff its translation into a 3-variable sentence can be proved using at most 4 variables. 2. Maddux, page 36 in [Brink Kahl Schmidt 1997]: Already in 1915 Löwenheim presented a proof (taken from a letter by Korselt) that the sentence saying there are at least four elements, namely v 0 v 1 v 2 v 3 ( v 0 Iv 1 v 0 Iv 2 v 1 Iv 2 v 0 Iv 3 v 1 Iv 3 v 2 Iv 3 ) is not equivalent to any relation-algebraic equation. 3. How to increase expressivity? Add projections. Use fork algebra: RA with an additional operator for pairing. See Haeberer et al., Chapter 4 in [Brink Kahl Schmidt 1997].

39 Additional operators 1. Transitive closure : add the axioms [Ng 1984] R R ;R = R ;R (i.e., R R ;R ) (R ;R;R ) = R ;R;R R (R Q) = (R Q) (i.e., monotonicity R (R Q) ) 2. Left residual: largest solution X of X ;Q R Definition by a Galois connection: X ;Q R X R/Q Explicit definition: R/Q = R ;Q 3. Right residual: largest solution X of Q ;X R Definition by a Galois connection: Q ;X R X Q\R Explicit definition: Q\R = Q ;R

40 Complete relation algebras 1. A complete RA is an RA A,,, ;,, I for which (hence T exists too). T exists for all T A 2. In a complete RA, monotonic functions have a least and greatest fixed point. For instance, R can be defined by 3. Useful, e.g., for program semantics. R = (µx : I R ;X). 4. Calculational rules for the manipulation of fixed points can be found in [Backhouse 2000].

41 References [Backhouse 2000] Roland Backhouse. Fixed point calculus. Presented at the Summer School and Workshop on Algebraic and Coalgebraic Methods in the Mathematics of Program Construction, Oxford, April 11 14, Available at [Brink Kahl Schmidt 1997] C. Brink, W. Kahl and G. Schmidt (eds.). Relational Methods in Computer Science, Springer, [Cardoso 1982] R. Cardoso. Untersuchung paralleler Programme mit relationenalgebraischen Methoden. Diplomarbeit, Institut für Informatik, Technische Universität München, [Desharnais 1999] J. Desharnais. Monomorphic characterization of n-ary direct products. Information Sciences An International Journal, 119 (3 4): , October [Hirsch Hodkinson 2002] R. Hirsch and I. Hodkinson. Relation Algebras by Games. Volume 147 of Studies in Logic and the Foundations of Mathematics, North Holland, 2002.

42 [Maddux 1993] R. D. Maddux. On the derivation of identities involving projection functions. Department of Mathematics, Iowa State University, July [Maddux 2006] R. D. Maddux. Relation Algebras, Studies in Logic and the Foundations of Mathematics. Elsevier, [Ng 1984] K. C. Ng. Relation algebras with transitive closure. Ph.D. thesis, University of California, Berkeley, [Schmidt Strohlein 1993] G. Schmidt and T. Ströhlein. Relations and Graphs, EATCS Monographs in Computer Science (Springer-Verlag, Berlin, 1993). [Tarski 1941] A. Tarski. On the calculus of relations. Journal of Symbolic Logic, 6(3):73 89, [Zierer 88] H. Zierer. Programmierung mit Funktionsobjekten: Konstruktive Erzeugung semantischer Bereiche und Anwendung auf die partielle Auswertung. Report TUM-I8803, Institut für Informatik, Technische Universität München, February 1988.