Relation Algebras for Reasoning about Time and Space

This paper presents a brief introduction to relation algebras, including some examples motivated by work in computer science, namely, the ‘interval algebras’, relation algebras that arose from James Allen’s work on temporal reasoning, and by ‘compass algebras’, which are designed for similar reasoning about space. One kind of reasoning problem, called a ‘constraint satisfaction problem’, can be defined for arbitrary relation algebras. It will be shown here that the constraint satisfiability problem is NP-complete for almost all compass and interval algebras. 1. Relation Algebras Composition of binary relations was introduced to logic by Augustus De Morgan [28] [29] (see [30, pp. 55–57, 208, 221, etc.]). De Morgan observed that the syllogism “every A is a B, every B is a C, so every A is a C” remains valid if the copula “is” is replaced by any transitive relation L. De Morgan went further, noting that if LM is the composition of the relation L with the relation M , that is, A is an LM of B just in case A is an L of an M of B, then the following syllogism is valid: “if every A is an L of a B, and every B is an M of a C, then every A is an LM of a C.” De Morgan [29] (see [30, p. 222]) denoted the converse of the relation L by L−1 and its contrary by not-L, and observed that these operations commute: the converse of the contrary of L is the contrary of the converse of L. Around the same time, George Boole [7] [6] created algebra from the logic of classes. Starting with [31] in 1870, Charles Sanders Peirce applied Boole’s ideas to create algebra from De Morgan’s logic of relations, “and after many attempts produced a good general algebra of logic, together with another algebra specially adapted to dyadic relations (Studies in Logic, by members of the Johns Hopkins University, 1883, Note B, 187–203). Schröder developed the last in a systematic manner” in [35] (quotation from [26]). Peirce [32] laid out his calculus in 17 pages. F. W. K. Ernst Schröder’s investigation [35] extended this to 649 pages. His book remains today the only exhaustive treatise on the calculus of relations. For additional survey and historical material on relation algebras see [8], [11], [12], [13], [14], [20], [21], [22], [23], [24], [37], and [38]. Consider an arbitrary classs, called the ‘universe of discourse’ or simply the ‘universe’. The universe could, depending on the situation and purposes, contain all possible mathematical objects, or all states of a machine, or all real numbers, or just a finite set of letters. The fundamental operations of the calculus of relations are natural set-theoretical operations on binary relations over the universe. In addition to the Boolean operations of union, intersection, and complementation, there are the ‘relative’ (as Peirce calls them), or ‘Peircean’ (as Tarski calls them) operations, namely the binary operation of ‘relative addition’ (Peirce’s name), the binary operation of ‘relative multiplication’ (Peirce’s name) or ‘composition’ (De Morgan’s name) and the unary operation of conversion. There are also four distinguished relations, namely the universal relation, the empty relation, the identity relation, and the diversity relation. The definitions of these operations and distinguished relations are listed below. In these definitions, x and y are arbitrary binary relations on the universe. By a binary relation we simply mean a class of ordered pairs. The ordered pair whose first element is p and whose second element is q is denoted 〈p, q〉. Thus p is an x of q if and only if 〈p, q〉 ∈ x. x+ y = union of x and y = {〈p, q〉 : 〈p, q〉 ∈ x or 〈p, q〉 ∈ y} x · y = intersection of x and y = {〈p, q〉 : 〈p, q〉 ∈ x and 〈p, q〉 ∈ y} x = complement of x = {〈p, q〉 : p, q are in the universe, but 〈p, q〉 / ∈ x} x†y = relative sum of x and y = {〈p, r〉 : for every q in the universe, 〈p, q〉 ∈ x or 〈q, r〉 ∈ y} x;y = relative product of x and y = {〈p, r〉 : for some q, 〈p, q〉 ∈ x and 〈q, r〉 ∈ y} x̆ = converse of x = {〈q, p〉 : 〈p, q〉 ∈ x} 1 = universal relation = {〈p, q〉 : p, q are in the universe} 0 = empty relation = ∅ 1, = identity relation = {〈p, p〉 : p is in the universe} 0, = diversity relation = {〈p, q〉 : p, q are in the universe, p 6= q} Both De Morgan and Peirce denoted the composition of x and y simply by “xy ”, but Schröder [35] used “x;y ”, as is done here. The notation “x|y ” was used by Whitehead and Russell [45] and adopted by Tarski and his school [10]. Peirce introduced the notation “ x̆ ” for the converse of x. Schröder introduced “ 1, ” and “ 0, ” for the identity and diversity relations. Here are some laws in the calculus of relations. These laws hold for every possible universe, and all possible binary relations x, y, and z. (i) (x+ y) + z = x+ (y + z) (ii) x+ y = y + x (iii) x = x+ y + x+ y (iv) x · y = x+ y (v) 1 = x+ x (vi) 0 = 1 (vii) x;(y ;z) = (x;y);z (viii) x;1, = x (ix) (x+ y);z = x;z + y ;z (x) ̆̆ x = x (xi) (x+ y)̆ = x̆+ y̆ (xii) (x;y)̆ = y̆ ;x̆ (xiii) x̆;x;y + y = y (xiv) 0, = 1, (xv) x†y = x;y A relation algebra is an algebra of the form A = 〈A,+, ·,−, 0, 1, † , ;, ̆, 0,, 1,〉 that satisfies the identities (i)-(xv) above. Identities (i)–(vi) say that 〈A,+, ·,−, 0, 1〉 is a Boolean algebra (called the Boolean part or Boolean reduct of A). One of the most significant laws of the calculus of relations is De Morgan’s “Theorem K” (see [30, pp. 186–7, 224] or [24, p. 434–5]), which asserts that the following statements are equivalent: x;y ≤ z, x̆;z ≤ y, z ; y̆ ≤ x. After minor Boolean transformations Theorem K becomes the cycle law, that the following statements are equivalent: x;y · z = 0, x̆;z · y = 0, z ; y̆ · x = 0. The cycle law and De Morgan’s Theorem K hold in every relation algebra because they can be proved from axioms (i)–(xv). There are many other equivalent axiomatizations for relation algebras. For example, equations (ix)–(xiii) can be replaced with the cycle law or with Theorem K. Also, equations (i)–(vi) can be replaced with some other set of equations that define Boolean algebras. The algebra containing all binary relations on the universe U is denoted Re(U). Identities (i)–(xv) hold in Re(U), so Re(U) is a relation algebra. Relation algebras are defined by equations, so it follows that subalgebras, homomorphic images, and direct products of relation algebras are again relation algebras. The algebras that can be obtained from algebras of the form Re(U) by forming subalgebras, homomorphic images, and direct products are called representable relation algebras. Roger Lyndon [18] showed that not all relation algebras are representable. It follows that the axioms (i)–(xv) are incomplete, in the sense that there are equations which hold in every algebra of the form Re(U) but cannot be derived from (i)–(xv). J. Donald Monk [27] proved that the equations that hold in every algebra of the form Re(U) does not have a finite axiomatization. Let A be a relation algebra. An element x of A is an atom if x is not 0 and there is no other element between x and 0, that is, either x · y = x or x · y = 0 for every y in A. Let AtA be the set of atoms of A. It is easy to prove that the converse of an atom is an atom, i.e., , if x ∈ AtA then x̆ ∈ AtA. The relation algebra A is said to be atomic if there is an atom below every nonzero element, that is, if y 6= 0 then there is some x ∈ AtA such that x ≤ y. A is said to be complete if every subset X of A has a least upper bound ∑ X and greatest lower bound ∏ X. It turns out that if A is both complete and atomic, then the structure of A is entirely determined by its atoms and the action of the relative operations on the atoms. For a precise statement of this fact, let C(A) = {〈a, b, c〉 : a, b, c ∈ AtA and a ;b ≥ c} and I(A) = {a : a ∈ AtA and a ≤ 1,}. C(A) is the set of cycles of A and I(A) is the set of identity atoms of A. Define the atom structure of A to be AtA = 〈AtA,C(A), ̆, I(A)〉. Any two complete atomic relation algebras with the isomorphic atom structures are isomorphic. For any a, b, c ∈ AtA, let [a, b, c] = { 〈a, b, c〉 , 〈ă, c, b〉 , 〈b, c̆, ă〉 , 〈 b̆, ă, c̆〉, 〈 c̆, a, b̆〉, 〈c, b̆, a〉 } . By the cycle law, the set C(A) of cycles of A is a union of sets of the form [a, b, c]. We refer to such sets as cyclesets. The identity element, the converse of x, and the relative product of x and y can be computed from the atom structure as follows: 1, = ∑ I(A), x̆ = ∑ {ă : x ≥ a ∈ AtA}, and x;y = ∑ {c : there are a, b ∈ AtA with a ≤ x, b ≤ y, and 〈a, b, c〉 ∈ C(A)}. Hence to specify a complete atomic relation algebra it suffices to list its atoms, to list its identity atoms, to indicate the converse of each atom, and, finally, to list the cyclesets [a, b, c]. This is especially convenient when A is finite. We present several examples of relation algebras using this method. 2. Interval Algebras To define the interval algebra IA [1] [2] take the universe U to be the set of all ‘events’, where an event is simply a pair of real numbers, the second of which is larger than the first. The first number in an event is its ‘starting time’, the second its ‘ending time’. (Our model for time here is just the real numbers.) Seven binary relations on events are defined in the list below, where x, x′, y, y′ are real numbers and 〈x, x′〉, 〈y, y′〉 are events. identity: 1, = {〈〈x, x′〉 , 〈y, y′〉〉 : x = y < x′ = y′} precedes: p = {〈〈x, x′〉 , 〈y, y′〉〉 : x < x′ < y < y′} during: d = {〈〈x, x′〉 , 〈y, y′〉〉 : y < x < x′ < y′} overlaps: o = {〈〈x, x′〉 , 〈y, y′〉〉 : x < y < x′ < y′} meets: m = {〈〈x, x′〉 , 〈y, y′〉〉 : x < x′ = y < y′} starts: s = {〈〈x, x′〉 , 〈y, y′〉〉 : x = y < x′ < y′} finishes: f = {〈〈x, x′〉 , 〈y, y′〉〉 : y < x < x′ = y′} These seven relations are studied in [42] and are used in some computer programs [5] [25] [36]. They generate a finite subalgebra of Re(U), called the interval algebra, or simply the IA. The IA has 13 atoms, namely 1,, p, p̆, d, d̆, o, ŏ, m, m̆, s, s̆, f , and f̆ . (It turns out that p alone will generate the IA, and so will each of the elements p̆, m, m̆, o, and ŏ [17] [16, Theorem 4.4].) If we start with the rational numbers instead of the reals, or, in fact, any dense linear ordering without endpoints, then the resulting algebra is isomorphic to the IA. But if we use some other infinite linear ordering, then the relation algebra generated by 1,, p, d, o, m, s, and f may be infinite. This happens, for example, when we use the integers. If we start with a finite linear ordering on U , then the subalgebra generated by 1,, p, d, o, m, s, and f will be Re(U). Any relation algebra obtained in this way will be called an interval algebra (while the IA is the one obtained from the reals or rationals). The IA has 75 cyclesets: [1,, 1,, 1,], [1,, s, s], [1,,m,m], [1,, p, p], [1,, o, o], [1,, f, f ], [1,, d, d], [s, 1,, s], [s, s, s], [s,m, p], [s, p, p], [s, o,m], [s, o, p], [s, o, o], [s, f, d], [s, d, d], [m, 1,,m], [m, s,m], [m,m, p], [m, p, p], [m, o, p], [m, f, s], [m, f, o], [m, f, d], [m, d, s], [m, d, o], [m, d, d], [p, 1,, p], [p, s, p], [p,m, p], [p, p, p], [p, o, p], [p, f, s], [p, f,m], [p, f, p], [p, f, o], [p, f, d], [p, d, s], [p, d,m], [p, d, p], [p, d, o], [p, d, d], [o, 1,, o], [o, s, o], [o,m, p], [o, p, p], [o, o,m], [o, o, p], [o, o, o], [o, f, s], [o, f, o], [o, f, d], [o, d, s], [o, d, o], [o, d, d], [f, 1,, f ], [f, s, d], [f,m,m], [f, p, p], [f, o, s], [f, o, o], [f, o, d], [f, f, f ], [f, d, d], [d, 1,, d], [d, s, d], [d,m, p], [d, p, p], [d, o, s], [d, o,m], [d, o, p], [d, o, o], [d, o, d], [d, f, d], [d, d, d]. Although all relative products in the IA can be computed from the cycles, it is convenient to also have the products in a table. The table of relative products of atoms of the IA is given in Figs. 1 and 2. To save space the + signs are omitted, so, for example, pdoms = p + d + o + m + s. The table appeared first in [2]. It not only shows relative products of atoms in the IA, but also shows containments for the Allen-Hayes algebra [3] [4]. By the Allen-Hayes algebra we mean the direct product of ‘all’ interval algebras, i.e., the direct product of an indexed system of algebras containing one algebra from each isomorphism type of interval algebras. The Allen-Hayes algebra contains the elements 1,, p, p̆, d, d̆, o, ŏ, m, m̆, s, s̆, f , and f̆ . They form a partition, i.e., they are pairwise disjoint and 1 = 1, + p + p̆ + d + d̆ + o + ŏ + m + m̆ + s + s̆ + f + f̆ . Finally, the relative product of any two of them is contained in (and not necessarily equal to) the corresponding entry in the table.