1 Introduction
Horn functions constitute a rich and important subclass of Boolean functions and have many applications in artificial intelligence, combinatorics, computer science, and operations research. Furthermore, they possess some nice structural and algorithmic properties. An example of this claim is found on the satisfiability problem (SAT) of formulae in conjunctive normal form (CNF): while it is one of the most famous NPcomplete problems for general Boolean CNFs (Cook [11]; see also: Arora and Barak [1], Garey and Johnson [19]), it can be solved in linear time in the number of variables plus the length of the Horn CNF formula being considered (Dowling and Gallier [16], Itai and Makowsky [23], and Minoux [26]), where length in this context means the number of literal occurrences in the formula (i.e., multiplicities are taken into account).
The problem of finding short Horn CNF representations of Horn functions specified through Horn CNFs has received some considerable attention in the literature since it has an intrinsic appeal stemming from both theoretical and practical standpoints. The same can be said about some special cases, including Horn 3CNFs, i.e., the ones in which each clause has at most three literals.
Two of the common measures considered are the number of clauses and the number of literal occurrences (henceforth, number of literals). The NPhardness of minimizing the number of clauses in Horn CNFs was first proved in a slightly different context of directed hypergraphs by Ausiello et al. [6]. It was later shown to also hold for the case of pure Horn 3CNFs by Boros et al. [8]. The NPhardness of minimizing the number of literals in Horn CNFs was shown by Maier [25]. A simpler reduction was found later by Hammer and Kogan [21]. Boros et al. [8] showed NPhardnes for the pure Horn 3CNFs case.
The attention then shifted to trying to approximate those values. Hammer and Kogan [21] showed that for a pure Horn function on variables, it is possible to approximate the minimum number of clauses and the minimum number of literals of a pure Horn CNF formula representing it to within factors of and , respectively. For many years, this was the only result regarding approximations. Recently, a superlogarithmic hardness of approximation factor was shown by Bhattacharya et al. [7] for the case of minimizing the number of clauses for general Horn CNFs. We provide more details on this shortly.
Another measure for minimum representations of Horn functions concerns minimizing the number of source sides, grouping together all clauses with the same source set. Maier [25] and Ausiello et al. [6] showed that such minimization can be accomplished in polynomial time (see also Crama and Hammer [12]). While this measure is sometimes used in practice, providing reasonably good results, we consider it an important intelectual quest, following Bhattacharya et al. [7], to try to precisely understand the hardness of the other two measures.
In this work, we focus on the hardness of approximating short pure Horn CNF/3CNF representations of pure Horn functions, where pure means that each clause has exactly one positive literal (definitions are provided in Section 2). More specifically, we study the hardness of approximating

the minimum number of clauses of pure Horn functions specified through pure Horn CNFs,

the minimum number of clauses and the minimum number literals of pure Horn functions specified through pure Horn 3CNFs,
when either polinomial or subexponential computational time is available.
Below, we present pointers to previous work on the subject, discuss our results in more details, and mention the main ideas behind them. We then close this section with the organization of the rest of the paper.
Previous Work
The first result on hardness of approximation of a shortest pure Horn CNF representations was provided in Bhattacharya et al. [7]. Specifically, it was shown that unless , that is, unless every problem in NP can be solved in quasipolynomial deterministic time in the size of the input’s representation, the minimum number of clauses of a pure Horn function on variables specified through a pure Horn CNF formula cannot be approximated in polynomial (depending on ) time to within a factor of , for any constant small enough.
This result is based on a gappreserving reduction from a fairly well known network design problem introduced by Kortsarz [24], namely, MinRep and has two main components: a gadget that associates to every MinRep instance a pure Horn CNF formula such that the size of an optimal solution to is related to the size of a clause minimum pure Horn CNF representation of , and a gap amplification device that provides the referred gap. Despite being both necessary to accomplish the result, each component works in a rather independent way.
Inspired by the novelty of their result and by some characteristics of their reduction, we are able to further advance the understanding of hardness of approximation of pure Horn functions. We discuss how we strength their result below.
Our Results and Techniques
Our strengthening of the result of Bhattacharya et al. [7] can be summarized as follows: the hardness of approximation factor we present is stronger, the complexity theoretic assumption we use for polynomial time solvability is weaker, and the class of CNF formulae to which our results apply is smaller. We are also able to derive further nonapproximability results for subexponential time solvability using a different complexity theoretic hypothesis.
In more details, for a pure Horn function on variables, we show that unless , the minimum number of clauses in a prime pure Horn CNF representation of and the minimum number of clauses and literals in a prime pure Horn 3CNF representation of cannot be approximated in polynomial (depending on ) time to within factors of even when the inputs are restricted to pure Horn CNFs and pure Horn 3CNFs with clauses, for some small constant . It is worth mentioning that for some constant in this case.
After that, we show that unless the Exponential Time Hypothesis introduced by Impagliazzo and Paturi [22] is false, it is not possible to approximate the minimum number of clauses and the minimum number of literals of a prime pure Horn 3CNF representation of in time , for some , to within factors of for some small constant . Such results hold even when the inputs are restricted to pure Horn 3CNFs with clauses, for some small constant . Furthermore, we also obtain a hardness of approximation factor of under slightly more stringent, but still subexponential time constraints. We would like to point out that our techniques leave open the problem of determining hardness of approximation factors when computational time is available. We conjecture, however, that constant factor approximations are still not possible in that case.
The main technical component of our work is a new gappreserving reduction^{1}^{1}1Our reduction is actually approximationpreserving, but we do not need or rely on such characteristic. See Vazirani [29] or Williamson and Shmoys [30] for appropriate definitions. from a graph theoretical problem called LabelCover (see Section 3 for its definition) to the problem of determining the minimum number of clauses in a pure Horn CNF representation of a pure Horn function. We show that our reduction has two independent parts: a core piece that forms an exclusive component (Boros et al. [9]) of the function in question and therefore, can be minimized separately; and a gap amplification device which is used to obtain the hardness of approximation factor. It becomes clear that the same principle underlies Bhattacharya et al. [7]. The hardness of approximation factor comes (after calculations) from a result of Dinur and Safra [15] on the hardness of approximating certain LabelCover instances.
We then introduce some local changes into our reduction that allow us to address the case in which the representation is restricted to be a pure Horn 3CNF. Namely, we introduce extra variables in order to cubify clauses whose degree is larger than three, that is, to replace each of these clauses by a collection of degree two or degree three clauses that provide the same logical implications. This is done for two families of high degree clauses and for each family we use a different technique: a linkedlist inspired transformation that is used on the classical reduction from SAT to 3SAT instances (Garey and Johnson [19]), and a complete binary tree type transformation. The latter type is necessary to prevent certain shapes of prime implicates in minimum clause representations that would render the gapamplification device innocuos. From this modified reduction, we are also able to derive in a straightforward fashion a hardness result for determining the minimum number of literals of pure Horn functions represented by pure Horn 3CNFs.
At this point we should mention that our reduction is somewhat more complicated than the one given in Bhattacharya et al. [7]. While we could adapt their reduction and obtain the same hardness of approximation factor we present in the case of pure Horn CNF representations (as based in our Lemma 14 we can argue that the LabelCover and the MinRep problems are equivalent), the more involved gadget we use is paramount in extending the hardness result for the pure Horn 3CNF case. Loosely speaking, the simple form of their reduction does not provide enough room to correctly shape those prime implicates in minimum pure Horn 3CNF representations that we mentioned addressing in ours. In this way, the extra complications are justified.
Finally, using newer and slightly different results on the hardness of approximation of certain LabelCover instances (Moshkovitz and Raz [27], Dinur and Harsha [14]) in conjunction with the Exponential Time Hypothesis [22], we are able to show (also after calculations) that it might not be possible to obtain constant factor approximations for the minimum number of clauses and literals in pure Horn 3CNF representations.
Outline
The remainder of the text is organized as follows. We introduce some basic concepts about pure Horn functions in Section 2 and present the problem on which our reduction is based in Section 3. The reduction to pure Horn CNFs, its proof of correctness, and the polynomial time hardness of approximation result are shown in Section 4. In Section 5, we extend that result to pure Horn 3CNF formulae and address the case of minimizing the number of literals. In Section 6 we show that subexponential time availability gives smaller but still superconstant hardness of approximation factors. We then offer some final thoughts in Section 7. A short conference version of this article appeared in Boros and Gruber [10].
2 Preliminaries
In this section we succinctly define the main concepts and notations we shall use later on. For an almost comprehensive exposition, consult the book on Boolean Functions by Crama and Hammer [12].
A mapping is called a Boolean function on propositional variables. Its set of variables is denoted by . A literal is a propositional variable (positive literal) or its negation (negative literal).
An elementary disjunction of literals
(1) 
with is a clause if . The set of variables it depends upon is and its degree or size is given by . It is customary to identify a clause with its set of literals.
Definition 1.
A clause as in (1) is called pure or definite Horn if . For a pure Horn clause , the positive literal is called its head and is called its subgoal or body. To simplify notation, we sometimes write simply as or as the implication .
Definition 2.
A conjunction of pure Horn clauses is a pure Horn formula in Conjunctive Normal Form (for short, pure Horn CNF). In case every clause in has degree at most three, the CNF is a 3CNF. A Boolean function is called pure Horn if there is a pure Horn CNF formula , that is, if for all .
Let be a pure Horn CNF representing a pure Horn function . We denote by
the numbers of clauses and literals of , respectively. We say that is a clause (literal) minimum representation of if () for every other pure Horn CNF representation of . With this in mind, we define
Problem 3.
The clause (literal) pure Horn CNF minimization problem consists in determining () when is given as a pure Horn CNF. Similar definitions hold for the pure Horn 3CNF case.
A clause as in (1) is an implicate of a Boolean function if for all it holds that implies . An implicate is prime if it is inclusionwise minimal with respect to its set of literals. The set of prime implicates of is denoted by . It is known (cf. Hammer and Kogan [20]) that prime implicates of pure Horn functions are pure Horn clauses. A pure Horn CNF representing is prime if its clauses are prime and is irredundant if the pure Horn CNF obtained after removing any of its clauses does not represent anymore. Let us note that a clause minimun representation may involve non prime implicates, though it is always irredundant. As Hammer and Kogan [20] pointed out any Horn CNF can be reduced in polynomial time to an equivalent prime and irredundant CNF. In the sequel we shall assume all CNFs considered, including the clause minimum ones, to be prime and irredundant.
Let and be two clauses and be a variable such that , , and and have no other complemented literals. The resolvent of and is the clause
and and are said to be resolvable. It is known (e.g. Crama and Hammer [12]) that if and are resolvable implicates of a Boolean function , then is also an implicate of . Naturally, the resolvent of pure Horn clauses is also pure Horn.
A set of clauses is closed under resolution if for all , . The resolution closure of , , is the smallest set closed under resolution. For a Boolean function , let .
Let us note that the set of all implicates of a Horn function may, in principle, contain clauses involving arbitrary other variables, not relevant for . To formulate proper statements one would need to make sure that such redundancies are also handled, which complicates the formulations. To avoid such complications, we focus on in the sequel, which is completely enough to describe all relevant representations of .
Definition 4.
Let be a pure Horn CNF representing a pure Horn function and let . The Forward Chaining of in , denoted by , is defined by the following algorithm. Initially, . As long as there is a pure Horn clause in such that and , add to . Whenever a variable is added to , we say that the corresponding clause was trigged.
The result below is pivotal in our work. It tells us that we can make inferences about a pure Horn function using any of its pure Horn CNF representations.
Lemma 5 (Hammer and Kogan [21]).
Two distinct pure Horn CNFs and represent the same pure Horn function if and only if , for all . Consequently, we can do Forward Chaining in , which we denote by , through the use of any of ’s representations.
The following definitions and lemma concerning exclusive sets of clauses are useful when decomposing and studying structural properties of Boolean functions.
Definition 6 (Boros et al. [9]).
Let be a Boolean function and be a set of clauses. is an exclusive set of clauses of if for all resolvable clauses it holds that: implies and .
An example of an exclusive set of clauses is given by the set of pure Horn implicates of a Horn function: it is not hard to see that if a resolvent is a pure Horn clause, then both the resolvable clauses must also be pure Horn.
Definition 7 (Boros et al. [9]).
Let be an exclusive set of clauses for a Boolean function and let be such that . The Boolean function is called the component of .
The following claim justifies the use of “the” in the previous definition.
Lemma 8 (Boros et al. [9]).
Let , , such that and let be an exclusive set of clauses. Then and in particular also represents .
The above lemma is a particularly useful and important tool in our work. Loosely speaking, it means that once we are able to identify an exclusive component of a function , we can separately study . Moreover, we can draw conclusions about using any of its representations (even alternate between distinct representations as convenient) and then reintegrate the acquired knowledge into the analysis of .
The Forward Chaining procedure provides us with a convenient way of identifying exclusive families for pure Horn functions, as stated in the next lemma. This result appeared recently in Boros et al. [8]. As we make explicit use of it in the analysis of our construction, we decided to include its proof below for completeness.
Lemma 9 (Boros et al. [8]).
Let be a prime pure Horn CNF representing the function , let be such that , and define the set
Then is an exclusive family for .
Proof.
Let be as specified in the lemma’s statement and suppose there are clauses such that but . By the definitions of and of resolution, all but one of the variables in must belong to and that variable, say , is precisely the variable upon which and are resolvable, that is, occurs as head in one of those clauses. Now, since for the same input the forward chaining outcome is independent of the pure Horn representation of , it follows that , a contradiction. ∎
When showing hardness results, we make use of the standard asymptotic notations: , , , , and . Namely, bigoh, bigtheta, bigomega, littleoh or omicron, and littleomega, respectively. For definitions and examples of use, consult e.g. the book by Cormen et al. [13].
We say a function (or ) is quasilinear if and it is nearly linear if for some constant small enough; it is quasipolynomial or superpolynomial if ; and it is subexponential if .
3 The LabelCover Problem
The LabelCover problem is a graph labeling promise problem formally introduced in Arora et al. [2] as a combinatorial abstraction of interactive proof systems (twoprover oneround in Feige et al. [17] and Feige and Lovász [18], and probabilistically checkable in Arora and Motwani [4] and Arora and Safra [5]
). It comes in maximization and minimization flavors (linked by a “weak duality” relation) and is probably the most popular starting point for hardness of approximation reductions. In this section, we introduce a minimization version that is best suited for our polynomial time results. Later in Section
6, when dealing with subexponential time results, we shall mention its maximization counterpart.Definition 10.
A LabelCover instance is a quadruple , where is a bipartite graph, and are disjoint sets of labels for the vertices in and , respectively, and is a set of constraints with each being a nonempty relation of admissible pairs of labels for the edge . The size of is equal to .
Definition 11.
A labeling for is any function assigning subsets of labels to vertices. A labeling covers an edge if for every label there is a label such that . A totalcover for is a labeling that covers every edge in . is said to be feasible if it admits a totalcover.
Following Arora and Lund [3], a way to guarantee that a LabelCover instance is feasible is by imposing an extra condition on it, namely, that there is a label such that for each edge , there is a label with . In this way, a labeling assigning to each vertex in and the set to each vertex in is clearly a totalcover. However, all LabelCover instances that we shall use are, by construction, guaranteed to be feasible. Therefore, we shall not dwell on such imposition and shall consider only feasible LabelCover instances in the sequel.
Definition 12.
For a totalcover of , let with . The cost of is given by and is said to be optimal if is minimum among the costs of all totalcovers for . This minimum value we denote by .
Observe that the feasibility of implies that , for any totalcover . Also, without loss of generality, we can assume that has no isolated vertices as they do not influence the cost of any labeling.
We now give an example of a LabelCover instance . Let be a set of Boolean variables and let be a formula in CNF such that each clause of depends on variables of (as in a variation of the satisfiability problem in which each clause has exactly literals). For a clause and a variable , we write whenever depends on .
The bipartite graph is constructed from as follows. Let have a vertex for every occurrence of a variable in , and let have a vertex for every clause . Let denotes the set of vertices corresponding to the variable , and define
that is, each vertex is connected to all occurrences of all variables in the clause .
Define the labelsets as and . For an edge , assume that and that is the th variable in , and define
where and .
Now, it is not hard to see that in this case, there is a totalcover with if and only if is satisfiable. Notice that choosing establishes the NPcompleteness of the problem of deciding if an optimal totalcover for a given LabelCover instance has cost equal to one.
The above example was adapted from Dinur and Safra [15]. Their original version is used in the proof of Theorem 17. In that context however, is a nonBoolean satisfiability instance produced by a probabilistic checkable proof system and the labelsets involved are larger (see Remark 18 below).
Definition 13.
A totalcover is tight if , i.e., if for every , it holds that .
Lemma 14.
Every LabelCover instance admits a tight, optimal totalcover.
Proof.
Suppose as in Definition 11 is a minimally nontight, optimal totalcover for . Hence, there is a such that . Let and define a new labeling where for all and . Note that and that every edge for is covered (for is a totalcover). Moreover, clearly . Hence, is an optimal totalcover for in which , contradicting the minimality of . The result thus follows. ∎
Notation 15.
For being a LabelCover instance as in Definition 10, define , , , , , for , and set .
Problem 16.
For any , a LabelCover instance has covering promise if it falls in one of two cases: either there is a tight, optimal totalcover for of cost , or every tight, optimal totalcover for has cost at least . The problem is a promise problem which receives a LabelCover instance with covering promise (also known as a
promise instance) as input and correctly classify it in one of those two cases.
Notice the behavior of is left unspecified for nonpromise instances. Therefore, any answer is acceptable in such case. Due to this characteristic, the problem is also referred as a gapproblem with gap in the literature.
The result below is the basis for the polynomial time hardness we shall exhibit.
Theorem 17 (Dinur and Safra [15]).
Let be any constant in and with . There are LabelCover instances with coveringpromise such that it is NPhard to distinguish between the cases in which is equal to or at least .
The closer to the above constant gets, the larger the hardness of approximation factor becomes. Therefore, from now on we shall consider that is fixed to a value close to .
Remark 18.
Every LabelCover instance produced by Dinur and Safra’s reduction is feasible, has coveringpromise , and satisfies the following relations: , , , , and , for as specified in Notation 15. It is then immediate that each such instance has size .
We now introduce a refined version of the LabelCover definitions, in which the vertices in the sets and have their own copies of the labelsets and , respectively. We then show that all structural and approximation properties are preserved in this new version.
Definition 19.
Let be a feasible LabelCover instance and consider the sets for each vertex , and for each vertex . Also, define the sets , , and , with
The quadruple is called a refinement of .
It is clear that for each vertex , for each vertex , for each edge , , and that a labeling for is a mapping such that for each vertex , and , for each vertex . Furthermore, the remaining definitions and concepts can be adapted in a straight forward fashion, and the size of a refined instance is also .
Lemma 20.
For any , there is a onetoone cost preserving correspondence between solutions to the problem and to its refined version.
Proof.
It is easy to see that is a (tight) totalcover for the problem if and only if is a (tight) totalcover for its refinement, where for every , and for every (or if the totalcovers are tight). Furthermore it is clear that , in any case. ∎
4 Reduction to pure Horn CNFs and a Polynomial Time Hardness Result
Our first reduction starts with a LabelCover instance as input and produces a pure Horn CNF formula , which defines a pure Horn function . The driving idea behind this reduction is that of tying the cost of tight, optimal totalcovers of to the size of clause minimum prime pure Horn CNF representations of .
With this in mind, let be a LabelCover instance (in compliance with Theorem 17 and Definition 19), and let and be positive integers to be specified later. Both and will be used as (gap) amplification devices. For nonnegative integers , define .
Associate propositional variables with every label , and with every edge , every label and every index . Let , for indices , be extra variables, and consider the following families of clauses:
where as before, is the open neighborhood of the vertex .
Definition 21.
Let us call and the canonical pure Horn CNF formulae defined, respectively, by the families of clauses (a) through (d) and by all the families of clauses above. Let and be, in that order, the pure Horn functions they represent.
The construction presented above can be divided into two parts. The families of clauses appearing in , namely, clauses of type (a) through (d), form an independent core since the function is an exclusive component of the function (as we shall show). This core can be analysed and minimized separately from the remainder, and its role is to reproduce the structural properties of the LabelCover instance. In more details, clauses of type

correspond to the constraints on the pairs of labels that can be assigned to each edge;

will assure that edges are covered, enforcing the matching of the labels assigned to all the neighbors of vertex ;

assure that if an edge can be covered in a certain way, then it can be covered in all legal ways — thus implying that it is not necessary to keep track of more than one covering possibility for each edge in clause minimum prime representations;

translate the totalcover requirement and reintroduce all the labels available ensuring that if a totalcover is achievable, so are all the others; this reintroduction of labels is paramount to the proper functioning of the reduction as explained below.
The family of clauses occurring in , namely, the clauses of type (e), constitutes the second part of the construction. These clauses have the role of introducing an initial collection of labels, which sole purpose is to help achieve the claimed hardness of approximation result. The intended behavior is as follows.
Consider initially that . It is known that given any subset of the variables of as input, the Forward Chaining procedure in any pure Horn CNF representation of will produce the same output (cf. Lemma 5). In particular, for the singleton , the output in will be the set with all the variables of , and so will be the output obtained in any clause minimum prime pure Horn CNF representing .
The reintroduction of labels performed by the family of clauses (d) may allow for some clauses of type (e) to be dropped without incurring in any loss. In slightly more details, as long as a subset of the family of clauses (e) introduces enough labels so that the Forward Chaining procedure in is able to eventually trigger the family of clauses (d), the remaining clauses of type (e) can be dismissed. All the missing labels will be available by the end of the procedure’s execution. It is not hard to see at this point that subsets of retained clauses of type (e) and totalcovers of the LabelCover instance in which the reduction is based are in onetoone correspondence.
Now, supposing that a clause minimum prime pure Horn CNF representation of resembles the canonical form , we just have to compensate for the number of clauses in to obtain a distinguishable gap that mimics the one exhibited by the LabelCover (as a promise) problem. This is achieved by making the gap amplification parameter which the clauses of type (e) depend upon large enough.
However, in principle, there is no guarantee that a clause minimum prime pure Horn CNF representation of , say , resembles or that has any clause of type (e) whatsoever. It may be advantageous to to have prime implicates where , for , occurs in their subgoals or prime implicates with variables other than , for , occurring as heads. Furthermore, the number of prime implicates in involving might simply not depend on the number of labels. Indeed, if as we are supposing, whenever the number of edges turns out to be strictly smaller than , the cost of an optimal totalcover for , it would be advantageous for to have prime implicates of the form , for . This not only breaks the correspondence mentioned above, but it renders the gap amplification device innocuous and the whole construction useless.
We manage to overcome the above difficulty throughout a second amplification device, the parameter which the clauses of type (a) through (d) depend upon. As we shall prove in Lemma 27, setting (which is strictly larger than the total number of labels available in — cf. Notation 15 and Definition 19) allows us to control the shape of the prime implicates involving variables in prime pure Horn clause minimum representations of : they will be precisely some of the clauses of type (e). Moreover, after showing that the function is an exclusive component of the function , we shall see that we do not need to concern ourselves with the actual form of clause minimum prime pure Horn CNF representations of . Therefore, in a sense, the canonical form has indeed a good resemblance to a clause minimum prime pure Horn CNF representing , and the intended behavior is achieved in the end.
4.1 Correctness of the CNF Reduction
In this subsection, we formalize the discussion presented above. We will constantly use the canonical representations and to make inferences about the functions and they respectively define, and such inferences will most of the time be made throughout Forward Chaining. We start with some basic facts about and .
Lemma 22.
Let and be as above and let , , , , , and be as in Notation 15. It holds that the number of clauses and variables in are, respectively,
In , those numbers are, respectively,
Proof.
For denoting the number of clauses of type in , simple counting arguments show that the equalities
hold. For the number of variables, just notice there are variables , variables , variables , and variables . To conclude the proof, just remember that the only difference between and is the absence of the family of clauses of type (e) in the latter. ∎
Considering the bounds for , , , , and provided in Remark 18, the above result immediately implies that as long as the quantities and are polynomial in , namely, the number of vertices in , the construction of from can be carried out in polynomial time in . More meaningfully, it can be carried out in polynomial time in , the number of variables of .
We now establish the pure Horn function as an exclusive component of . This structural result allows us to handle in a somewhat blackbox fashion. Specifically, as we shall see briefly, it is not required of us to precisely know all the properties and details of a clause minimum representation of . We can mainly concentrate on the study of the prime implicates that might involve the variables in that are not in , namely, the variables , for .
Lemma 23.
The function is an exclusive component of the function . Consequently, can be analysed and minimized separately.
Proof.
Let be the set of variables occurring in . By definition, these are the variables the function depends upon. Since no clause in has head outside , it is immediate that is closed under Forward Chaining in . As represents , Lemma 9 then implies that is an exclusive family for . This gives that is an exclusive component of (cf. Definition 7) and therefore, that is an exclusive component of . Now, using Lemma 8, we obtain a proof of the second claim as wished. ∎
With some effort, it is possible to prove that is a clause minimum prime pure Horn CNF representation of . For our proofs however, a weaker result suffices.
Lemma 24.
Let be a clause minimum prime pure Horn CNF representation of . We have .
Proof.
The upper bound is by construction. For the lower bound, observe that each variable of appears no more than times as a head. As in any clause minimum representation of they must appear as head at least once, the claim follows. ∎
The next lemma is a useful tool in showing whether two different representations of the pure Horn function are equivalent.
Lemma 25.
For all indices , it holds that .
Proof.
It is enough to show that , for a fixed . The inclusion would be false if there existed a label such that . As for every label , is a clause in , this cannot happen. Hence, the inclusion holds, implying the claim. ∎
The next couple of lemmas deal with the structure of prime implicates involving the variables , for . The first states a simple, but useful fact which is valid in any representation of . The second proves how the amplification device depending on the parameter shapes those prime implicates in clause minimum representations of to the desired form: , with .
Lemma 26.
A variable , for some index , is never the head of an implicate of . Moreover, every prime implicate of involving is quadratic.
Proof.
The first claim is straight forward as all implicates of can be derived from by resolution, and is not the head of any clause of . By Lemma 25, is an implicate of for all . Since is a pure Horn function, the claim follows. ∎
Lemma 27.
Let . In any clause minimum prime pure Horn CNF representation of , the prime implicates involving the variables have the form , for all indices , and for some labels .
Proof.
Let be a clause minimum prime pure Horn CNF representation of , with being a clause minimum pure Horn CNF representation of . According to Lemma 26, all prime implicates of involving the variables are quadratic. So, for all indices and all indices define the sets
Our goal is to show that the chosen value for the parameter forces all the sets to be simultaneously empty and consequently, that all the prime implicates involving the variables in clause minimum pure Horn CNF representations of have the claimed form. We shall accomplish this in two steps.
Let . We first show that if a set for some index , then for all indices , simultaneously.
Claim 1.
All clauses of type (d) have the same body. Therefore, during the execution of the Forward Chaining procedure from , either they all trigger simultaneously or none of them do. The reason for them not to trigger is the absence of some variable , with and , in the Forward Chaining closure from , i.e, .
Proof.
A simple inspection of the families of clauses shows that Claim (1) holds. ∎
Now, for each index , let be the collection of clauses of types (a), (b), and (c) that depend on .
Claim 2.
It holds that
for all indices , with .
Proof.
Notice that the families of clauses (a), (b), and (c) are completely symmetric with respect to the indexing variable . Moreover, for , the clauses indexed by do not interfere with the clauses indexed by during an execution of the Forward Chaining procedure. In other words, variables depending upon do not trigger clauses indexed by , and viceversa. These two properties, symmetry and non interference, proves Claim (2). ∎
Claim 3.
If there is a variable , with and , such that
then
Moreover, this implies that .
Proof.
The symmetry and non interference properties of families of clauses (a), (b), and (c) also justifies the first part of Claim (3). To see it, just notice that were the claim to be false, the prime implicates in would be trigging clauses involving the variable in an execution of the Forward Chaining procedure. Since , this cannot happen. The second part follows immediately from the validity of the first part together with the fact that represents . ∎
To finish the first step, notice that since Claim (3) is valid for any , Claim (2) implies that if for some index , then for all indices , simultaneously.
For the second step, suppose that for all indices . We then have that
that is, is strictly larger than the number of all available labels in . This implies that the following pure Horn CNF
has fewer clauses than (or more precisely, it implies that ).
Now, since that is a (clause minimum) representation of the exclusive component , and that the set of clauses makes all available labels reachable by Forward Chaining from , it follows that . Furthermore, the change in clauses did not influence the Forward Chaining procedure from any other variable (other than ), and thus for all variables . Thus, Lemma 25 implies that is a representation of .
We then have that is a shorter representation for , contradicting the optimality of . Therefore, the sets for all indices . As the above arguments do not depend on any particular value of , they can be repeated for all of them. ∎
The next property we can show more generally for any prime and irredundant CNF of .
Definition 28.
Let and let be a prime and irredundant pure Horn CNF representation of . For each , consider the set
and define the function given by for vertices and for vertices .
The next three lemmas provide important properties of the functions above.
Lemma 29.
Let be as in the above Definition. For all indices and vertices , it holds that .
Proof.
Let be as in Definition 28 and suppose indirectly that the claim is false, that is, there is an index and a vertex such that .
During the proof, recall that the chosen value for the parameter implies, according to Lemma 27, that all prime implicates of involving the variable must have the form , with .
Let and define the expression
It is enough to show that , that is, that is also a representation of (cf. Lemma 25). Suppose that is not the case. Since and differ only in the clause , it must be the case that
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