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Directed hypergraphs and Horn minimization
Information Processing Letters 128 (2017) 32–37
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Information Processing Letters www.elsevier.com/locate/ipl
Directed hypergraphs and Horn minimization Kristóf Bérczi ∗ , Erika R. Bérczi-Kovács Department of Operations Research, Eötvös University, Budapest, Hungary
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Article history: Received 13 March 2017 Received in revised form 13 July 2017 Accepted 22 July 2017 Available online 7 August 2017 Communicated by A. Tarlecki Keywords: Combinatorial problems Graph algorithms Directed hypergraphs GD basis Horn minimization
a b s t r a c t A Boolean function given in a conjunctive normal form is Horn if every clause contains at most one positive literal, and it is pure Horn if every clause contains exactly one positive literal. Due to their computational tractability, Horn functions are studied extensively in many areas of computer science and mathematics such as combinatorics, artificial intelligence, database theory, algebra and logic. The present paper considers the problem of finding minimal representations of pure Horn functions. We give a new proof for a recent min–max result of Boros et al. regarding body-minimal representations. The proof is algorithmic and finds the so called Guigues–Duquenne basis. We also describe a new construction that combines two existing representations into a third one. © 2017 Elsevier B.V. All rights reserved.
1. Introduction As a subclass of Boolean functions, Horn functions play an important role in different areas of mathematics due to their interesting computational properties. The satisfiability problem for this subclass of Boolean functions can be solved in linear time and the equivalence of Horn formulas can be decided in polynomial time [12]. This concept appears as lattices and closure systems in algebra, as implicational systems in artificial intelligence, as directed hypergraphs in graph theory, and is also used for representing knowledge base in propositional expert systems. Informally, the Horn minimization problem is to find a minimal representation that is equivalent to a given Horn formula. For example, such a representation can be used to reduce the size of the knowledge base in a propositional expert system, thus improving the performance of the system. The size of a formula can be measured in many different ways (see [5]). Unfortunately, it is NPhard to find an optimal representation for almost all of
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Corresponding author. E-mail addresses: [email protected] (K. Bérczi), [email protected] (E.R. Bérczi-Kovács). http://dx.doi.org/10.1016/j.ipl.2017.07.013 0020-0190/© 2017 Elsevier B.V. All rights reserved.
these measures. There is however an interesting exception, called body-minimal representation, for which polynomial time algorithms were independently discovered [5,11,13]. In [7], Boros et al. gave an explanation why this measure is so different from the others in terms of tractability by providing a min–max result on the minimum number of bodies appearing in the representation of a Horn function. Their proof is algorithmic and it actually determines a canonical body-minimal representation called the Guigues–Duquenne basis. A common aspect of previous algorithms for determining a body-minimal representation is that they are using frameworks different from that of directed hypergraphs, for example, functional dependencies or implication systems. For this reason, the steps of these algorithms are difficult to follow and they do not reveal the structure of body-minimal representations. Our aim was to give a better understanding of the min–max result of [7] by using a purely graph theoretical approach. In contrast to body-minimal representations, edgeminimal representations are not only hard to find but even hard to approximate. Bhattacharya et al. [6] showed that ( 1 −ε )
(n) this problem is inapproximable within a factor 2 O (log assuming N P D T I M E (n polylog (n) ), while Boros and Gru-
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ber showed that it is inapproximable within a factor 1−o (1)
n) 2 O (log assuming P N P , where n denotes the number of variables. However, the existence of an O (nc ) approximation for some 0 < c < 1 is a rather interesting open problem; such an approximation algorithm would immediately find a wide list of applications. We present a surprising result, which given two pure Horn formulas 1 and 2 , constructs a new one such that the bodies and heads of form subsets of the bodies of 1 and the heads of 2 , respectively. We hope that this observation may help us in finding a good approximation for the edge-minimal representation. The rest of the paper is organized as follows. A brief introduction into Horn logic is given in Section 2. We give a new algorithmic proof of the min–max result of Boros et al. in Section 3. In Section 4, we show that the bodyminimal representation provided by the algorithm is in fact the GD basis. Finally, we show how a new representation from two given ones can be constructed in Section 5.
2. Preliminaries 2.1. Horn logic Let V be a set of n variables. Members of V are called positive while their negations are called negative literals. A Boolean function is a mapping f : {0, 1} V → {0, 1}. For a subset Z ⊆ V let χ Z denote the characteristic vector of Z , that is, χ Z ( v ) = 1 if v ∈ Z and 0 otherwise. Then Z is called true for f if f (χ Z ) = 1 and false otherwise. The sets of true and false sets of f are denoted by T f and F f , respectively. It is known that any Boolean function can be represented by a conjunctive normal form (CNF). A CNF is a conjunction of clauses, where a clause is a disjunction of literals. A clause is Horn if at most one of its literals is positive, and is pure Horn (or definite Horn) if it contains exactly one positive literal. We usually denote the set of clauses appearing in a representation by C . A CNF = ( V , C ) is pure Horn if all of its clauses are pure Horn. Finally, a Boolean function h is pure Horn if it can be represented by a pure Horn CNF. For a subset ∅ = B ⊆ V and v ∈ V \ B we write ( B → v ) to denote the pure Horn clause C = v ∨ u ∈ B u. Here B and v are called the body and head of the clause, respectively. The set of bodies and set of heads appearing in a CNF representation are denoted by B () and H(), respectively. It is known that for any pure Horn function h, Th is closed under intersection and contains V (see e.g. [9]). Vice versa, for any set T of subsets of V which is closed under intersection and contains V , there exists a pure Horn function h with Th = T . Indeed, the CNF = ( V , C ) with C = {( B → v ) : T ∈ T s.t. B ⊆ T , v ∈ / T } represents such a pure Horn function h. As any Boolean function is uniquely determined by its set of true sets, the above implies that there is a one-to-one correspondence between pure Horn functions and sets of subsets of V closed under intersection and containing V . Given a pure Horn function h, the forward chaining closure of a set Z ⊆ V is the unique smallest true set containing Z and is denoted by F h ( Z ). If is a pure Horn CNF
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representation of h then the forward chaining closure can 0 be obtained by the following method. Set F ( Z ) := Z . In i i a general step, if F ( Z ) is a true set then F h ( Z ) = F ( Z ). Otherwise take an arbitrary violated implication ( B → v ) i +1 i of and set F := F ( Z ) + v. Note that ( B → v ) is vii i i ( Z ) but v ∈ / F ( Z ). olated by F ( Z ) if and only if B ⊆ F The result of the process depends neither on the particular choice of the representation nor on the order in which violated implications are chosen, but only on the underlying function h. 2.2. Directed hypergraphs Directed hypergraphs are generalizations of directed graphs and can be defined in several ways [10,14]. In our investigations we will use the following notation. A directed hypergraph is a pair H = ( V , E ) where V is a set of nodes and E is a set of hyperedges. A hyperedge is a pair ( B , v ) where ∅ = B ⊆ V is the body and v ∈ V \ B is the head of the hyperedge. The set of bodies and set of heads appearing in H are denoted by B ( H ) and H( H ), respectively. We say that a hyperedge ( B , v ) ∈ E covers a set Z ⊆ V if B ⊆ Z and v ∈ / Z . The hypergraph H covers a family P of subsets of V if for each Z ∈ P there exists an edge in E covering Z . A subset Z ⊆ V is called true if H does not cover Z and false otherwise. The sets of true and false sets are denoted by T H and F H , respectively. Given a node v ∈ V , let H − v denote the hypergraph obtained from H by deleting each hyperedge containing v (either as a body node or a head node). We say that a node v ∈ V is reachable from a set Z ⊆ V in H if either v ∈ Z or there exists a hyperedge ( B , v ) such that each node in B is reachable from Z in H − v. The set of nodes reachable from Z in H is denoted by F H ( Z ). 2.3. Pure Horn functions and directed hypergraphs There is a natural one-to-one correspondence between pure Horn CNFs and directed hypergraphs. Namely, a CNF = ( V , C ) and a hypergraph H = ( V , E ) correspond to each other if ( B → v ) ∈ C if and only if ( B , v ) ∈ E . Let h be a pure Horn function, be a pure Horn CNF representing h and H be the corresponding hypergraph. It is easy to see that Th = T H , Fh = F H , B () = B ( H ), H() = H( H ) and F h ( Z ) = F H ( Z ) for every Z ⊆ V . Hence the problem of finding a body-minimal representation of h is equivalent to finding a hypergraph H = ( V , E ) with T H = Th and |B ( H )| being minimal. For a given pure Horn CNF = ( V , C ), we will denote the corresponding directed hypergraph by H = ( V , E ) . 3. Body-minimal representation Let E ∗ denote the set of all possible hyperedges on V , that is, E ∗ := {( B , v ) : ∅ = B ⊂ V , v ∈ V \ B }. Let h be a pure Horn function. An hyperedge ( X , v ) ∈ E ∗ is called valid if it covers none of the true sets in Th . Observe that a hypergraph H = ( V , E ) represents h if and only if it covers Fh and only has valid hyperedges. A true set Y is said to separate false sets X 1 and X 2 if X 1 ∩ X 2 ⊆ Y and either
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Y ⊂ X 1 or Y ⊂ X 2 . Two sets X 1 and X 2 are called independent if they can not be covered by valid hyperedges having the same body. Claim 1. Two false sets X 1 and X 2 are independent if and only if either they are separated by a true set or X 1 ∩ X 2 = ∅. Proof. Assume first that X 1 and X 2 are independent. If X 1 ∩ X 2 = ∅, then let Y denote the unique minimal true set containing X 1 ∩ X 2 . If none of X 1 and X 2 contain Y properly, then there exist nodes v i ∈ Y − X i for i = 1, 2. But then ( X 1 ∩ X 2 , v 1 ) and ( X 1 ∩ X 2 , v 2 ) are valid hyperedges with the same body covering X 1 and X 2 , contradicting the independence of X 1 and X 2 . Now we prove the other direction. If X 1 ∩ X 2 = ∅, then the two sets are clearly independent. Assume now that X 1 ∩ X 2 = ∅ and let Y be a true set separating X 1 and X 2 , say, X 1 ∩ X 2 ⊆ Y ⊂ X 1 . Let ( X , v 1 ) and ( X , v 2 ) be hyperedges covering X 1 and X 2 , respectively. Then X ⊆ X 1 ∩ X 2 and v i ∈ V − X i . However, this means that ( X , v 1 ) is not a valid hyperedge as it covers Y , thus concluding the proof. 2 The next min–max result first appeared in [7] in a slightly different form. We give a new proof here using directed hypergraphs. The advantage of using the hypergraph terminology is that both the statement of the theorem and the main steps of the algorithmic proof are easier to interpret. Theorem 2. Let h be an arbitrary pure Horn function. The minimum number of bodies appearing in a hypergraph representation H = ( V , E ) of h equals the maximum number of pairwise independent false sets. Proof. Take an arbitrary representation H = ( V , E ) of h and a family I of pairwise independent false sets. For each X ∈ I , there must be a valid hyperedge in E that covers X . As no two members of I can be covered by valid hyperedges having the same body, the number of different bodies appearing in the representation is at least |I |, showing |B ( H )| ≥ |I |. By choosing H to be body-minimal and I to be maximal, we get that the minimum is at least the maximum. Hence, in order to prove equality, it suffices to show a representation H = ( V , E ) of h and a family I of pairwise independent false sets such that |B ( H )| = |I |. Procedure MinMax constructs such a representation. For now, we do not care about how the pure Horn function is given or how the steps of the procedure can be executed; our aim is to verify the min–max relation. (However, in Theorem 4 we will show that our proof is in fact algorithmic if h is given by a CNF.) At the beginning, we set H := ( V , ∅). At a general step of the algorithm, take an inclusionwise minimal false set X ∈ Fh not covered by H and let Y ∈ Th be the minimal true set containing X . Note that Y is uniquely determined as Th is closed under intersection and V ∈ Th . Add ( X , v ) to E for each v ∈ Y − X . We repeat these steps as long as possible. Let H = ( V , E ) be the resulting hypergraph, let X 1 , . . . , X t denote
the bodies in H in the order they got into H and let Y i be the unique minimal true set containing X i for i = 1, . . . , k. Clearly, H covers every false set in Fh and contains only valid hyperedges. In addition, X i ∈ Fh for i = 1, . . . , t. We claim that these false sets are pairwise independent. Indeed, take two sets, say X i and X j with i < j. If X i ⊂ X j then Y i separates them, otherwise one of the hyperedges {( X i , v ) : v ∈ Y i − X i } would cover X j , hence X j could not appear as a body in the representation. Assume now that none of X i − X j , X i ∩ X j and X j − X i is empty. We claim that Y = X i ∩ X j is a true set. Assume indirectly that Y is false. Then Y became covered no later than X i and X j . However, a hyperedge covering Y also covers at least one of X i and X j , contradicting that both of them are bodies in the final hypergraph. Hence Y is a true set which separates X i and X j . Thus we conclude that X 1 , . . . , X t are independent false sets, finishing the proof. 2 Procedure MinMax. Input : A pure Horn function h. Output: A body-minimal representation H = ( V , E ) of h. 1 E := ∅ 2 H := ( V , E ) 3 while ∃ false set not covered by H do 4 Among the false sets not covered by H choose an 5 6
inclusionwise minimal one X . Let Y be the unique minimal true set containing X . E := E ∪ {( X , v ) : v ∈ Y − X }
7 end 8 Output H = ( V , E ).
Now we show that Theorem 2 is equivalent to the min– max result of Boros et al. [7]. Let = ( V , C ) be a pure Horn CNF. For a subset S ⊆ V , define E S = {( B → v ) ∈ C : B ⊆ S, v ∈ / S } and call such a set essential if it is nonempty. Two essential sets E S 1 and E S 2 where S 1 = S 2 are body-disjoint if no two nodes v 1 ∈ F ( S 1 ∩ S 2 ) \ S 1 and v 2 ∈ F ( S 1 ∩ S 2 ) \ S 2 exist simultaneously. (In fact both essentiality and body-disjointness are defined slightly differently in [7], but for now we can think of these sets as mentioned above.) ˇ Makino). Let h be an arbitrary pure Theorem 3 (Boros, Cepek, Horn function. Then the minimum number of bodies appearing in a pure Horn CNF representation of h equals the maximum number of pairwise body-disjoint essential sets. It is not difficult to see that E S 1 and E S 2 are bodydisjoint essential sets if and only if they are false and either S 1 ∩ S 2 = ∅ or F ( S 1 ∩ S 2 ) is a true set separating S 1 and S 2 . Hence the equivalence of the two theorems follows. By using the notation of the proof of Theorem 2, we get that the pure Horn CNF
=
t
( Xi → v )
(1)
i =1 v ∈ Y i \ X i
is a body minimal representation of h. Hence the proof immediately suggests a direct algorithm for determining
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a body-minimal representation of a pure Horn function h given by a pure Horn representation . Theorem 4. Let h be a pure Horn function given by a pure Horn CNF representation . Then a body-minimal pure Horn representation defined by (1) can be determined in O (|B ()|2 · ||) time, where || denotes the total number of literal occurrences in . Proof. The set of steps of the algorithm while the hypergraph H is unchanged will be called a phase. Consider a phase of the procedure and let H = ( V , E ) denote the hypergraph constructed so far. Observe that F H ( Z ) ⊆ F H ( Z ) for every Z ⊆ V as we only added hyperedges of form ( X i , v ) where v ∈ F h ( X i ). In Step 4, we have to find a minimal set X which is covered by H but uncovered by H . Such a set X surely contains a body B ∈ B ( H ). As H does not cover X , necessarily we have F H ( B ) ⊆ X . On the other hand, F H ( B ) is covered by H unless F H ( B ) = F H ( B ). Hence X can be chosen to be a minimal set among the sets F H ( B ) for B ∈ B ( H ) with F H ( B ) = F H ( B ). We can find such a set by applying the forward chaining procedure for each body B ∈ B () in both H and H . As forward chaining can be performed in O (||) time [8], the set X in Step 4 can be determined in O (|B ()| · ||) time. The unique minimal true set containing a given false set X is just F h ( X ) which can be determined by using forward chaining (based on ), hence Step 5 can be performed in O (||) time. As the number of bodies B ∈ B ( H ) with F H ( B ) = F H ( B ) decreases in each phase, there are at most |B ()| phases. By the above, the steps within a phase can be performed in O (|B ()| · ||) time. We can conclude that the algorithm terminates after O (|B ()|2 · ||) steps. 2 A short description of the direct algorithm is presented by Procedure BodyMinimal. Procedure BodyMinimal. Input : A pure Horn CNF representation = ( V , C ) of h. Output: A body-minimal representation of h.
E := ∅ H := ( V , E ) := ∅ while ∃ B ∈ B( H ) : F H ( B ) = F H ( B ) do X := argmin{ F H ( B ) : B ∈ B( H ), F H ( B ) = F H ( B )} Y := F H ( B ) E := E ∪ {( X , v ) : v ∈ Y − X } 8 := ∧ v ∈Y \ X ( X → v ) 1 2 3 4 5 6 7
9 end 10 Output .
The proofs of Theorems 2 and 4 imply the following, somewhat surprising result. Theorem 5. Let h be a pure Horn function given by a pure Horn CNF representation . Then there exists a body-minimal pure Horn representation such that B () ⊆ B ().
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Proof. Let X be a set determined in Step 4 of the algorithm and let B ∈ B ( H ) be a body for which X = F H ( B ). Such a body exists according to the proof of Theorem 4. Then in Step 6 of the algorithm hyperedges {( B , v ) : v ∈ Y − B } could be added to H instead of {( X , v ) : v ∈ Y − X } where Y = F H ( X ). Indeed, for every v ∈ Y − X , ( B , v ) is a valid hyperedge and covers every set that is covered by ( X , v ), proving the theorem. 2 4. Guigues–Duquenne basis In [11], a canonical body-minimal representation of pure Horn functions has been introduced called as Guigues–Duquenne basis (GD basis). Algorithms for determining the GD basis of a pure Horn function h given by an arbitrary pure Horn CNF were proposed in [3,7]. In what follows, we show that our algorithm also finds the GD basis. The uniqueness of the GD basis lies in its saturation, a notion that has been introduced already in [2,4]. A pure Horn CNF representation = ( V , C ) is called right-saturated if for every clause ( B → v ) ∈ C we have ( B → v ) ∈ C for every v ∈ F ( B ) \ B, and is called leftsaturated if B 1 ⊂ B 2 for ( B 1 → v 1 ), ( B 2 → v 2 ) ∈ C implies v 1 ∈ B 2 . Finally, is saturated if it is both left- and rightsaturated. These definitions can be naturally extended to directed hypergraphs: H = ( V , E ) is right-saturated if ( B , v ) ∈ E implies ( B , v ) ∈ E for every v ∈ F H ( B ) \ B, and H is left-saturated if ( B 2 ⊂ B 1 ) for ( B 1 , v 1 ), ( B 2 , v 2 ) ∈ E implies v 2 ∈ B 1 . Finally, H is saturated if it is both leftand right-saturated. It is easy to check that is left- or right-saturated if and only if H is left- or right-saturated, respectively. For sake of completeness, we prove that pure Horn functions have a unique saturated representation. Theorem 6. A pure Horn function has a unique saturated representation. Proof. Assume indirectly that the pure Horn function h has two different saturated representations H 1 = ( V , E1 ) and H 2 = ( V , E2 ). Let ( B , v ) be a hyperedge in the symmetric difference of E1 and E2 with | B | being minimal. Without loss of generality, assume that ( B , v ) ∈ E1 . Then B∈ / B ( H 2 ) as otherwise H 2 is not right-saturated. As B ∈ Fh , there exists a hyperedge ( B , w ) ∈ E2 covering B. By the choice of ( B , v ), we have ( B , u ) ∈ E1 , thus H 1 is not left-saturated, a contradiction. 2 Now we show that the algorithm determines the GD basis. Theorem 7. The output of Procedure BodyMinimal is the GD basis of h. Proof. By Theorem 6, it suffices to show that the output of the algorithm is both left- and right-saturated. Let H = ( V , E ) denote the directed hypergraph constructed by the algorithm and let X be a body of . That is,
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X = F H ( B ) for some B ∈ B ( H ) where H denotes the hypergraph constructed by the algorithm before considering X in Step 5. As H represents , F H ( X ) = F H ( X ). Indeed, F H ( X ) is the unique smallest true set in T that contains X while F H ( X ) is the unique smallest true set in T H containing X , hence they must coincide. Similarly, F H ( B ) = F H ( B ). But H is a subhypergraph of H , hence F H ( B ) = X implies F H ( B ) = F H ( X ). Concluding these observations, we get F H ( X ) = F H ( B ). Step 7 of the algorithm ensures that ( X , v ) ∈ E for v ∈ F H ( X ) \ X . Thus H , and in turn are indeed right-saturated. Now consider two clauses ( X 1 → v 1 ) and ( X 2 → v 2 ) of such that X 1 ⊂ X 2 . By Theorem 2, X 1 and X 2 must be independent false sets, hence there exists a true set Y ∈ T H separating X 1 and X 2 , that is, X 1 ⊂ Y ⊂ X 2 . As T H and T H coincide, H may contain only hyperedges not covering any true set in T H , hence v 1 ∈ X 2 as required. Thus H , and in turn are left-saturated. 2 5. Edge-minimal representations While trying to give a good approximation algorithm for finding an edge-minimal representation, we came to the following interesting result which may be useful in further examinations. Theorem 8. Assume that H 1 = ( V , E1 ) and H 2 = ( V , E2 ) are two hypergraph representations of a pure Horn function h. Then there exists a hypergraph representation H = ( V , E ) of h such that |E | ≤ |E1 |, H( H ) ⊆ H( H 1 ) and B ( H ) ⊆ B ( H 2 ). Proof. We may assume that for every hyperedge in E1 there exists a false set covered only by that hyperedge, as otherwise the hyperedge could be simply deleted. Our proof is based on the following lemma. Lemma 9. For every hyperedge ( B , v ) ∈ E1 such that B ∈ / B( H 2 ), there exists a body B ∈ B( H 2 ) such that ( V , E1 − ( B , v ) + ( B , v )) is also a representation of h. Proof. Let ( B , v ) ∈ E1 be a hyperedge of H 1 such that B∈ / B ( H 2 ) and let M ⊆ V be an inclusionwise maximal set such that ( V , E1 − ( B , v ) + ( M , v )) is also a representation of h. We distinguish two cases. Case 1: M ∈ B ( H 2 ). In this case B = M satisfies the requirements of the lemma.
/ B ( H 2 ). Note that M is a false set, hence there Case 2: M ∈ exists a hyperedge h = ( B , v ) ∈ E2 covering M. Let Y be the forward chaining closure of B , and let B = M ∪ Y . If v ∈ Y , hyperedge ( B , v ) is valid and covers all sets covered by ( M , v ). This means that H = ( V , E1 − ( B , v ) + ( B , v )) is also a representation of h. Assume now that v ∈ / Y . We claim that H = ( V , E1 − ( B , v ) + ( B , v )) is a representation of h, contradicting the maximality of M. To prove this, it suffices to show that hyperedge ( B , v ) covers all false sets that are covered only by ( M , v ) in E1 − ( B , v ) + ( M , v ). Let F be such a false set and assume that it is not covered by ( B , v ), that is,
Y F . Define F := F ∩ Y . By B ⊆ M ⊆ F and B ⊆ Y , we have B ⊆ F . As Y is the forward chaining closure of B , F is a false set. Hence H 1 has a hyperedge covering F . As Y is a true set, this hyperedge has its body in F and head in Y − F , contradicting to our original assumption that F is covered only by ( M , v ), thus concluding the proof of the lemma. 2 We can apply the lemma for each body in B ( H 1 ) −
B( H 2 ), thus the theorem follows. 2
A surprising corollary of the theorem, which also appeared in [1] in a completely different context, is as follows. Corollary 10. Every pure Horn function h has a representation which is both edge-minimal and body-minimal. Proof. Let H 1 and H 2 be edge minimal and body minimal representations of h, respectively. By applying Theorem 8 to H 1 and H 2 , the resulting representation H is both edge and body minimal. 2 Finally, the next corollary may serve as a starting point for approximating edge-minimal representations. Corollary 11. Every pure Horn function h has an edge-minimal representation which is the subset of the GD-basis. Proof. Let H 1 be an edge-minimal representations of h and let H 2 denote the GD basis. As H 2 is right-saturated, the hypergraph provided by Theorem 8 is a subhypergraph of H 2 . 2 Acknowledgements The authors were supported by the MTA-ELTE Egerváry Research Group and by the Hungarian Scientific Research Fund – OTKA, No. K109240. References [1] K. Adaricheva, J.B. Nation, On implicational bases of closure systems with unique critical sets, Discrete Appl. Math. 162 (2014) 51–69. [2] M. Arias, J.L. Balcázar, Query learning and certificates in lattices, in: Algorithmic Learning Theory, Springer, 2008, pp. 303–315. [3] M. Arias, J.L. Balcázar, Canonical Horn representations and query learning, in: Algorithmic Learning Theory, Springer, 2009, pp. 156–170. [4] M. Arias, A. Feigelson, R. Khardon, R.A. Servedio, Polynomial certificates for propositional classes, Inf. Comput. 204 (5) (2006) 816–834. [5] G. Ausiello, A. D’Atri, D. Sacca, Minimal representation of directed hypergraphs, SIAM J. Comput. 15 (2) (1986) 418–431. [6] A. Bhattacharya, B. DasGupta, D. Mubayi, Gy. Turán, On approximate Horn formula minimization, in: ICALP, Part 1, in: Lect. Notes Comput. Sci., vol. 6198, 2010, pp. 438–450. ˇ K. Makino, A combinatorial min–max theorem [7] E. Boros, O. Cepek, and minimization of pure-Horn functions, in: International Symposium on Artificial Intelligence and Mathematics, 2016. [8] W.F. Downling, J.H. Gallier, Linear-time algorithms for testing the satisfiability of propositional Horn formulae, J. Log. Program. (1984) 267–284. [9] T. Eiter, T. Ibaraki, K. Makino, Bidual Horn functions and extensions, Discrete Appl. Math. 96 (1999) 55–88.
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[10] G. Gallo, G. Longo, S. Pallottino, S. Nguyen, Directed hypergraphs and applications, Discrete Appl. Math. 42 (2) (1993) 177–201. [11] J.-L. Guigues, V. Duquenne, Familles minimales d’implications informatives résultant d’un tableau de données binaires, Math. Sci. Hum. 95 (1986) 5–18. [12] H.K. Büning, T. Lettmann, Propositional Logic: Deduction and Algorithms, Cambridge University Press, 1999.
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[13] D. Maier, Minimum covers in relational database model, J. ACM 27 (4) (1980) 664–674. [14] M. Thakur, R. Tripathi, Linear connectivity problems in directed hypergraphs, Theor. Comput. Sci. 410 (27) (2009) 2592–2618.
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Information Processing Letters www.elsevier.com/locate/ipl
Directed hypergraphs and Horn minimization Kristóf Bérczi ∗ , Erika R. Bérczi-Kovács Department of Operations Research, Eötvös University, Budapest, Hungary
a r t i c l e
i n f o
Article history: Received 13 March 2017 Received in revised form 13 July 2017 Accepted 22 July 2017 Available online 7 August 2017 Communicated by A. Tarlecki Keywords: Combinatorial problems Graph algorithms Directed hypergraphs GD basis Horn minimization
a b s t r a c t A Boolean function given in a conjunctive normal form is Horn if every clause contains at most one positive literal, and it is pure Horn if every clause contains exactly one positive literal. Due to their computational tractability, Horn functions are studied extensively in many areas of computer science and mathematics such as combinatorics, artificial intelligence, database theory, algebra and logic. The present paper considers the problem of finding minimal representations of pure Horn functions. We give a new proof for a recent min–max result of Boros et al. regarding body-minimal representations. The proof is algorithmic and finds the so called Guigues–Duquenne basis. We also describe a new construction that combines two existing representations into a third one. © 2017 Elsevier B.V. All rights reserved.
1. Introduction As a subclass of Boolean functions, Horn functions play an important role in different areas of mathematics due to their interesting computational properties. The satisfiability problem for this subclass of Boolean functions can be solved in linear time and the equivalence of Horn formulas can be decided in polynomial time [12]. This concept appears as lattices and closure systems in algebra, as implicational systems in artificial intelligence, as directed hypergraphs in graph theory, and is also used for representing knowledge base in propositional expert systems. Informally, the Horn minimization problem is to find a minimal representation that is equivalent to a given Horn formula. For example, such a representation can be used to reduce the size of the knowledge base in a propositional expert system, thus improving the performance of the system. The size of a formula can be measured in many different ways (see [5]). Unfortunately, it is NPhard to find an optimal representation for almost all of
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Corresponding author. E-mail addresses: [email protected] (K. Bérczi), [email protected] (E.R. Bérczi-Kovács). http://dx.doi.org/10.1016/j.ipl.2017.07.013 0020-0190/© 2017 Elsevier B.V. All rights reserved.
these measures. There is however an interesting exception, called body-minimal representation, for which polynomial time algorithms were independently discovered [5,11,13]. In [7], Boros et al. gave an explanation why this measure is so different from the others in terms of tractability by providing a min–max result on the minimum number of bodies appearing in the representation of a Horn function. Their proof is algorithmic and it actually determines a canonical body-minimal representation called the Guigues–Duquenne basis. A common aspect of previous algorithms for determining a body-minimal representation is that they are using frameworks different from that of directed hypergraphs, for example, functional dependencies or implication systems. For this reason, the steps of these algorithms are difficult to follow and they do not reveal the structure of body-minimal representations. Our aim was to give a better understanding of the min–max result of [7] by using a purely graph theoretical approach. In contrast to body-minimal representations, edgeminimal representations are not only hard to find but even hard to approximate. Bhattacharya et al. [6] showed that ( 1 −ε )
(n) this problem is inapproximable within a factor 2 O (log assuming N P D T I M E (n polylog (n) ), while Boros and Gru-
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ber showed that it is inapproximable within a factor 1−o (1)
n) 2 O (log assuming P N P , where n denotes the number of variables. However, the existence of an O (nc ) approximation for some 0 < c < 1 is a rather interesting open problem; such an approximation algorithm would immediately find a wide list of applications. We present a surprising result, which given two pure Horn formulas 1 and 2 , constructs a new one such that the bodies and heads of form subsets of the bodies of 1 and the heads of 2 , respectively. We hope that this observation may help us in finding a good approximation for the edge-minimal representation. The rest of the paper is organized as follows. A brief introduction into Horn logic is given in Section 2. We give a new algorithmic proof of the min–max result of Boros et al. in Section 3. In Section 4, we show that the bodyminimal representation provided by the algorithm is in fact the GD basis. Finally, we show how a new representation from two given ones can be constructed in Section 5.
2. Preliminaries 2.1. Horn logic Let V be a set of n variables. Members of V are called positive while their negations are called negative literals. A Boolean function is a mapping f : {0, 1} V → {0, 1}. For a subset Z ⊆ V let χ Z denote the characteristic vector of Z , that is, χ Z ( v ) = 1 if v ∈ Z and 0 otherwise. Then Z is called true for f if f (χ Z ) = 1 and false otherwise. The sets of true and false sets of f are denoted by T f and F f , respectively. It is known that any Boolean function can be represented by a conjunctive normal form (CNF). A CNF is a conjunction of clauses, where a clause is a disjunction of literals. A clause is Horn if at most one of its literals is positive, and is pure Horn (or definite Horn) if it contains exactly one positive literal. We usually denote the set of clauses appearing in a representation by C . A CNF = ( V , C ) is pure Horn if all of its clauses are pure Horn. Finally, a Boolean function h is pure Horn if it can be represented by a pure Horn CNF. For a subset ∅ = B ⊆ V and v ∈ V \ B we write ( B → v ) to denote the pure Horn clause C = v ∨ u ∈ B u. Here B and v are called the body and head of the clause, respectively. The set of bodies and set of heads appearing in a CNF representation are denoted by B () and H(), respectively. It is known that for any pure Horn function h, Th is closed under intersection and contains V (see e.g. [9]). Vice versa, for any set T of subsets of V which is closed under intersection and contains V , there exists a pure Horn function h with Th = T . Indeed, the CNF = ( V , C ) with C = {( B → v ) : T ∈ T s.t. B ⊆ T , v ∈ / T } represents such a pure Horn function h. As any Boolean function is uniquely determined by its set of true sets, the above implies that there is a one-to-one correspondence between pure Horn functions and sets of subsets of V closed under intersection and containing V . Given a pure Horn function h, the forward chaining closure of a set Z ⊆ V is the unique smallest true set containing Z and is denoted by F h ( Z ). If is a pure Horn CNF
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representation of h then the forward chaining closure can 0 be obtained by the following method. Set F ( Z ) := Z . In i i a general step, if F ( Z ) is a true set then F h ( Z ) = F ( Z ). Otherwise take an arbitrary violated implication ( B → v ) i +1 i of and set F := F ( Z ) + v. Note that ( B → v ) is vii i i ( Z ) but v ∈ / F ( Z ). olated by F ( Z ) if and only if B ⊆ F The result of the process depends neither on the particular choice of the representation nor on the order in which violated implications are chosen, but only on the underlying function h. 2.2. Directed hypergraphs Directed hypergraphs are generalizations of directed graphs and can be defined in several ways [10,14]. In our investigations we will use the following notation. A directed hypergraph is a pair H = ( V , E ) where V is a set of nodes and E is a set of hyperedges. A hyperedge is a pair ( B , v ) where ∅ = B ⊆ V is the body and v ∈ V \ B is the head of the hyperedge. The set of bodies and set of heads appearing in H are denoted by B ( H ) and H( H ), respectively. We say that a hyperedge ( B , v ) ∈ E covers a set Z ⊆ V if B ⊆ Z and v ∈ / Z . The hypergraph H covers a family P of subsets of V if for each Z ∈ P there exists an edge in E covering Z . A subset Z ⊆ V is called true if H does not cover Z and false otherwise. The sets of true and false sets are denoted by T H and F H , respectively. Given a node v ∈ V , let H − v denote the hypergraph obtained from H by deleting each hyperedge containing v (either as a body node or a head node). We say that a node v ∈ V is reachable from a set Z ⊆ V in H if either v ∈ Z or there exists a hyperedge ( B , v ) such that each node in B is reachable from Z in H − v. The set of nodes reachable from Z in H is denoted by F H ( Z ). 2.3. Pure Horn functions and directed hypergraphs There is a natural one-to-one correspondence between pure Horn CNFs and directed hypergraphs. Namely, a CNF = ( V , C ) and a hypergraph H = ( V , E ) correspond to each other if ( B → v ) ∈ C if and only if ( B , v ) ∈ E . Let h be a pure Horn function, be a pure Horn CNF representing h and H be the corresponding hypergraph. It is easy to see that Th = T H , Fh = F H , B () = B ( H ), H() = H( H ) and F h ( Z ) = F H ( Z ) for every Z ⊆ V . Hence the problem of finding a body-minimal representation of h is equivalent to finding a hypergraph H = ( V , E ) with T H = Th and |B ( H )| being minimal. For a given pure Horn CNF = ( V , C ), we will denote the corresponding directed hypergraph by H = ( V , E ) . 3. Body-minimal representation Let E ∗ denote the set of all possible hyperedges on V , that is, E ∗ := {( B , v ) : ∅ = B ⊂ V , v ∈ V \ B }. Let h be a pure Horn function. An hyperedge ( X , v ) ∈ E ∗ is called valid if it covers none of the true sets in Th . Observe that a hypergraph H = ( V , E ) represents h if and only if it covers Fh and only has valid hyperedges. A true set Y is said to separate false sets X 1 and X 2 if X 1 ∩ X 2 ⊆ Y and either
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Y ⊂ X 1 or Y ⊂ X 2 . Two sets X 1 and X 2 are called independent if they can not be covered by valid hyperedges having the same body. Claim 1. Two false sets X 1 and X 2 are independent if and only if either they are separated by a true set or X 1 ∩ X 2 = ∅. Proof. Assume first that X 1 and X 2 are independent. If X 1 ∩ X 2 = ∅, then let Y denote the unique minimal true set containing X 1 ∩ X 2 . If none of X 1 and X 2 contain Y properly, then there exist nodes v i ∈ Y − X i for i = 1, 2. But then ( X 1 ∩ X 2 , v 1 ) and ( X 1 ∩ X 2 , v 2 ) are valid hyperedges with the same body covering X 1 and X 2 , contradicting the independence of X 1 and X 2 . Now we prove the other direction. If X 1 ∩ X 2 = ∅, then the two sets are clearly independent. Assume now that X 1 ∩ X 2 = ∅ and let Y be a true set separating X 1 and X 2 , say, X 1 ∩ X 2 ⊆ Y ⊂ X 1 . Let ( X , v 1 ) and ( X , v 2 ) be hyperedges covering X 1 and X 2 , respectively. Then X ⊆ X 1 ∩ X 2 and v i ∈ V − X i . However, this means that ( X , v 1 ) is not a valid hyperedge as it covers Y , thus concluding the proof. 2 The next min–max result first appeared in [7] in a slightly different form. We give a new proof here using directed hypergraphs. The advantage of using the hypergraph terminology is that both the statement of the theorem and the main steps of the algorithmic proof are easier to interpret. Theorem 2. Let h be an arbitrary pure Horn function. The minimum number of bodies appearing in a hypergraph representation H = ( V , E ) of h equals the maximum number of pairwise independent false sets. Proof. Take an arbitrary representation H = ( V , E ) of h and a family I of pairwise independent false sets. For each X ∈ I , there must be a valid hyperedge in E that covers X . As no two members of I can be covered by valid hyperedges having the same body, the number of different bodies appearing in the representation is at least |I |, showing |B ( H )| ≥ |I |. By choosing H to be body-minimal and I to be maximal, we get that the minimum is at least the maximum. Hence, in order to prove equality, it suffices to show a representation H = ( V , E ) of h and a family I of pairwise independent false sets such that |B ( H )| = |I |. Procedure MinMax constructs such a representation. For now, we do not care about how the pure Horn function is given or how the steps of the procedure can be executed; our aim is to verify the min–max relation. (However, in Theorem 4 we will show that our proof is in fact algorithmic if h is given by a CNF.) At the beginning, we set H := ( V , ∅). At a general step of the algorithm, take an inclusionwise minimal false set X ∈ Fh not covered by H and let Y ∈ Th be the minimal true set containing X . Note that Y is uniquely determined as Th is closed under intersection and V ∈ Th . Add ( X , v ) to E for each v ∈ Y − X . We repeat these steps as long as possible. Let H = ( V , E ) be the resulting hypergraph, let X 1 , . . . , X t denote
the bodies in H in the order they got into H and let Y i be the unique minimal true set containing X i for i = 1, . . . , k. Clearly, H covers every false set in Fh and contains only valid hyperedges. In addition, X i ∈ Fh for i = 1, . . . , t. We claim that these false sets are pairwise independent. Indeed, take two sets, say X i and X j with i < j. If X i ⊂ X j then Y i separates them, otherwise one of the hyperedges {( X i , v ) : v ∈ Y i − X i } would cover X j , hence X j could not appear as a body in the representation. Assume now that none of X i − X j , X i ∩ X j and X j − X i is empty. We claim that Y = X i ∩ X j is a true set. Assume indirectly that Y is false. Then Y became covered no later than X i and X j . However, a hyperedge covering Y also covers at least one of X i and X j , contradicting that both of them are bodies in the final hypergraph. Hence Y is a true set which separates X i and X j . Thus we conclude that X 1 , . . . , X t are independent false sets, finishing the proof. 2 Procedure MinMax. Input : A pure Horn function h. Output: A body-minimal representation H = ( V , E ) of h. 1 E := ∅ 2 H := ( V , E ) 3 while ∃ false set not covered by H do 4 Among the false sets not covered by H choose an 5 6
inclusionwise minimal one X . Let Y be the unique minimal true set containing X . E := E ∪ {( X , v ) : v ∈ Y − X }
7 end 8 Output H = ( V , E ).
Now we show that Theorem 2 is equivalent to the min– max result of Boros et al. [7]. Let = ( V , C ) be a pure Horn CNF. For a subset S ⊆ V , define E S = {( B → v ) ∈ C : B ⊆ S, v ∈ / S } and call such a set essential if it is nonempty. Two essential sets E S 1 and E S 2 where S 1 = S 2 are body-disjoint if no two nodes v 1 ∈ F ( S 1 ∩ S 2 ) \ S 1 and v 2 ∈ F ( S 1 ∩ S 2 ) \ S 2 exist simultaneously. (In fact both essentiality and body-disjointness are defined slightly differently in [7], but for now we can think of these sets as mentioned above.) ˇ Makino). Let h be an arbitrary pure Theorem 3 (Boros, Cepek, Horn function. Then the minimum number of bodies appearing in a pure Horn CNF representation of h equals the maximum number of pairwise body-disjoint essential sets. It is not difficult to see that E S 1 and E S 2 are bodydisjoint essential sets if and only if they are false and either S 1 ∩ S 2 = ∅ or F ( S 1 ∩ S 2 ) is a true set separating S 1 and S 2 . Hence the equivalence of the two theorems follows. By using the notation of the proof of Theorem 2, we get that the pure Horn CNF
=
t
( Xi → v )
(1)
i =1 v ∈ Y i \ X i
is a body minimal representation of h. Hence the proof immediately suggests a direct algorithm for determining
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a body-minimal representation of a pure Horn function h given by a pure Horn representation . Theorem 4. Let h be a pure Horn function given by a pure Horn CNF representation . Then a body-minimal pure Horn representation defined by (1) can be determined in O (|B ()|2 · ||) time, where || denotes the total number of literal occurrences in . Proof. The set of steps of the algorithm while the hypergraph H is unchanged will be called a phase. Consider a phase of the procedure and let H = ( V , E ) denote the hypergraph constructed so far. Observe that F H ( Z ) ⊆ F H ( Z ) for every Z ⊆ V as we only added hyperedges of form ( X i , v ) where v ∈ F h ( X i ). In Step 4, we have to find a minimal set X which is covered by H but uncovered by H . Such a set X surely contains a body B ∈ B ( H ). As H does not cover X , necessarily we have F H ( B ) ⊆ X . On the other hand, F H ( B ) is covered by H unless F H ( B ) = F H ( B ). Hence X can be chosen to be a minimal set among the sets F H ( B ) for B ∈ B ( H ) with F H ( B ) = F H ( B ). We can find such a set by applying the forward chaining procedure for each body B ∈ B () in both H and H . As forward chaining can be performed in O (||) time [8], the set X in Step 4 can be determined in O (|B ()| · ||) time. The unique minimal true set containing a given false set X is just F h ( X ) which can be determined by using forward chaining (based on ), hence Step 5 can be performed in O (||) time. As the number of bodies B ∈ B ( H ) with F H ( B ) = F H ( B ) decreases in each phase, there are at most |B ()| phases. By the above, the steps within a phase can be performed in O (|B ()| · ||) time. We can conclude that the algorithm terminates after O (|B ()|2 · ||) steps. 2 A short description of the direct algorithm is presented by Procedure BodyMinimal. Procedure BodyMinimal. Input : A pure Horn CNF representation = ( V , C ) of h. Output: A body-minimal representation of h.
E := ∅ H := ( V , E ) := ∅ while ∃ B ∈ B( H ) : F H ( B ) = F H ( B ) do X := argmin{ F H ( B ) : B ∈ B( H ), F H ( B ) = F H ( B )} Y := F H ( B ) E := E ∪ {( X , v ) : v ∈ Y − X } 8 := ∧ v ∈Y \ X ( X → v ) 1 2 3 4 5 6 7
9 end 10 Output .
The proofs of Theorems 2 and 4 imply the following, somewhat surprising result. Theorem 5. Let h be a pure Horn function given by a pure Horn CNF representation . Then there exists a body-minimal pure Horn representation such that B () ⊆ B ().
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Proof. Let X be a set determined in Step 4 of the algorithm and let B ∈ B ( H ) be a body for which X = F H ( B ). Such a body exists according to the proof of Theorem 4. Then in Step 6 of the algorithm hyperedges {( B , v ) : v ∈ Y − B } could be added to H instead of {( X , v ) : v ∈ Y − X } where Y = F H ( X ). Indeed, for every v ∈ Y − X , ( B , v ) is a valid hyperedge and covers every set that is covered by ( X , v ), proving the theorem. 2 4. Guigues–Duquenne basis In [11], a canonical body-minimal representation of pure Horn functions has been introduced called as Guigues–Duquenne basis (GD basis). Algorithms for determining the GD basis of a pure Horn function h given by an arbitrary pure Horn CNF were proposed in [3,7]. In what follows, we show that our algorithm also finds the GD basis. The uniqueness of the GD basis lies in its saturation, a notion that has been introduced already in [2,4]. A pure Horn CNF representation = ( V , C ) is called right-saturated if for every clause ( B → v ) ∈ C we have ( B → v ) ∈ C for every v ∈ F ( B ) \ B, and is called leftsaturated if B 1 ⊂ B 2 for ( B 1 → v 1 ), ( B 2 → v 2 ) ∈ C implies v 1 ∈ B 2 . Finally, is saturated if it is both left- and rightsaturated. These definitions can be naturally extended to directed hypergraphs: H = ( V , E ) is right-saturated if ( B , v ) ∈ E implies ( B , v ) ∈ E for every v ∈ F H ( B ) \ B, and H is left-saturated if ( B 2 ⊂ B 1 ) for ( B 1 , v 1 ), ( B 2 , v 2 ) ∈ E implies v 2 ∈ B 1 . Finally, H is saturated if it is both leftand right-saturated. It is easy to check that is left- or right-saturated if and only if H is left- or right-saturated, respectively. For sake of completeness, we prove that pure Horn functions have a unique saturated representation. Theorem 6. A pure Horn function has a unique saturated representation. Proof. Assume indirectly that the pure Horn function h has two different saturated representations H 1 = ( V , E1 ) and H 2 = ( V , E2 ). Let ( B , v ) be a hyperedge in the symmetric difference of E1 and E2 with | B | being minimal. Without loss of generality, assume that ( B , v ) ∈ E1 . Then B∈ / B ( H 2 ) as otherwise H 2 is not right-saturated. As B ∈ Fh , there exists a hyperedge ( B , w ) ∈ E2 covering B. By the choice of ( B , v ), we have ( B , u ) ∈ E1 , thus H 1 is not left-saturated, a contradiction. 2 Now we show that the algorithm determines the GD basis. Theorem 7. The output of Procedure BodyMinimal is the GD basis of h. Proof. By Theorem 6, it suffices to show that the output of the algorithm is both left- and right-saturated. Let H = ( V , E ) denote the directed hypergraph constructed by the algorithm and let X be a body of . That is,
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X = F H ( B ) for some B ∈ B ( H ) where H denotes the hypergraph constructed by the algorithm before considering X in Step 5. As H represents , F H ( X ) = F H ( X ). Indeed, F H ( X ) is the unique smallest true set in T that contains X while F H ( X ) is the unique smallest true set in T H containing X , hence they must coincide. Similarly, F H ( B ) = F H ( B ). But H is a subhypergraph of H , hence F H ( B ) = X implies F H ( B ) = F H ( X ). Concluding these observations, we get F H ( X ) = F H ( B ). Step 7 of the algorithm ensures that ( X , v ) ∈ E for v ∈ F H ( X ) \ X . Thus H , and in turn are indeed right-saturated. Now consider two clauses ( X 1 → v 1 ) and ( X 2 → v 2 ) of such that X 1 ⊂ X 2 . By Theorem 2, X 1 and X 2 must be independent false sets, hence there exists a true set Y ∈ T H separating X 1 and X 2 , that is, X 1 ⊂ Y ⊂ X 2 . As T H and T H coincide, H may contain only hyperedges not covering any true set in T H , hence v 1 ∈ X 2 as required. Thus H , and in turn are left-saturated. 2 5. Edge-minimal representations While trying to give a good approximation algorithm for finding an edge-minimal representation, we came to the following interesting result which may be useful in further examinations. Theorem 8. Assume that H 1 = ( V , E1 ) and H 2 = ( V , E2 ) are two hypergraph representations of a pure Horn function h. Then there exists a hypergraph representation H = ( V , E ) of h such that |E | ≤ |E1 |, H( H ) ⊆ H( H 1 ) and B ( H ) ⊆ B ( H 2 ). Proof. We may assume that for every hyperedge in E1 there exists a false set covered only by that hyperedge, as otherwise the hyperedge could be simply deleted. Our proof is based on the following lemma. Lemma 9. For every hyperedge ( B , v ) ∈ E1 such that B ∈ / B( H 2 ), there exists a body B ∈ B( H 2 ) such that ( V , E1 − ( B , v ) + ( B , v )) is also a representation of h. Proof. Let ( B , v ) ∈ E1 be a hyperedge of H 1 such that B∈ / B ( H 2 ) and let M ⊆ V be an inclusionwise maximal set such that ( V , E1 − ( B , v ) + ( M , v )) is also a representation of h. We distinguish two cases. Case 1: M ∈ B ( H 2 ). In this case B = M satisfies the requirements of the lemma.
/ B ( H 2 ). Note that M is a false set, hence there Case 2: M ∈ exists a hyperedge h = ( B , v ) ∈ E2 covering M. Let Y be the forward chaining closure of B , and let B = M ∪ Y . If v ∈ Y , hyperedge ( B , v ) is valid and covers all sets covered by ( M , v ). This means that H = ( V , E1 − ( B , v ) + ( B , v )) is also a representation of h. Assume now that v ∈ / Y . We claim that H = ( V , E1 − ( B , v ) + ( B , v )) is a representation of h, contradicting the maximality of M. To prove this, it suffices to show that hyperedge ( B , v ) covers all false sets that are covered only by ( M , v ) in E1 − ( B , v ) + ( M , v ). Let F be such a false set and assume that it is not covered by ( B , v ), that is,
Y F . Define F := F ∩ Y . By B ⊆ M ⊆ F and B ⊆ Y , we have B ⊆ F . As Y is the forward chaining closure of B , F is a false set. Hence H 1 has a hyperedge covering F . As Y is a true set, this hyperedge has its body in F and head in Y − F , contradicting to our original assumption that F is covered only by ( M , v ), thus concluding the proof of the lemma. 2 We can apply the lemma for each body in B ( H 1 ) −
B( H 2 ), thus the theorem follows. 2
A surprising corollary of the theorem, which also appeared in [1] in a completely different context, is as follows. Corollary 10. Every pure Horn function h has a representation which is both edge-minimal and body-minimal. Proof. Let H 1 and H 2 be edge minimal and body minimal representations of h, respectively. By applying Theorem 8 to H 1 and H 2 , the resulting representation H is both edge and body minimal. 2 Finally, the next corollary may serve as a starting point for approximating edge-minimal representations. Corollary 11. Every pure Horn function h has an edge-minimal representation which is the subset of the GD-basis. Proof. Let H 1 be an edge-minimal representations of h and let H 2 denote the GD basis. As H 2 is right-saturated, the hypergraph provided by Theorem 8 is a subhypergraph of H 2 . 2 Acknowledgements The authors were supported by the MTA-ELTE Egerváry Research Group and by the Hungarian Scientific Research Fund – OTKA, No. K109240. References [1] K. Adaricheva, J.B. Nation, On implicational bases of closure systems with unique critical sets, Discrete Appl. Math. 162 (2014) 51–69. [2] M. Arias, J.L. Balcázar, Query learning and certificates in lattices, in: Algorithmic Learning Theory, Springer, 2008, pp. 303–315. [3] M. Arias, J.L. Balcázar, Canonical Horn representations and query learning, in: Algorithmic Learning Theory, Springer, 2009, pp. 156–170. [4] M. Arias, A. Feigelson, R. Khardon, R.A. Servedio, Polynomial certificates for propositional classes, Inf. Comput. 204 (5) (2006) 816–834. [5] G. Ausiello, A. D’Atri, D. Sacca, Minimal representation of directed hypergraphs, SIAM J. Comput. 15 (2) (1986) 418–431. [6] A. Bhattacharya, B. DasGupta, D. Mubayi, Gy. Turán, On approximate Horn formula minimization, in: ICALP, Part 1, in: Lect. Notes Comput. Sci., vol. 6198, 2010, pp. 438–450. ˇ K. Makino, A combinatorial min–max theorem [7] E. Boros, O. Cepek, and minimization of pure-Horn functions, in: International Symposium on Artificial Intelligence and Mathematics, 2016. [8] W.F. Downling, J.H. Gallier, Linear-time algorithms for testing the satisfiability of propositional Horn formulae, J. Log. Program. (1984) 267–284. [9] T. Eiter, T. Ibaraki, K. Makino, Bidual Horn functions and extensions, Discrete Appl. Math. 96 (1999) 55–88.
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[10] G. Gallo, G. Longo, S. Pallottino, S. Nguyen, Directed hypergraphs and applications, Discrete Appl. Math. 42 (2) (1993) 177–201. [11] J.-L. Guigues, V. Duquenne, Familles minimales d’implications informatives résultant d’un tableau de données binaires, Math. Sci. Hum. 95 (1986) 5–18. [12] H.K. Büning, T. Lettmann, Propositional Logic: Deduction and Algorithms, Cambridge University Press, 1999.
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[13] D. Maier, Minimum covers in relational database model, J. ACM 27 (4) (1980) 664–674. [14] M. Thakur, R. Tripathi, Linear connectivity problems in directed hypergraphs, Theor. Comput. Sci. 410 (27) (2009) 2592–2618.