Computer Science
ComputerScience,1970-2008-Computer Science-2002.jsonl.gz.gz
Computer Science Mathematics
747a207257cb96813c1ca35a153f9d5a973501a6
Compilability of Abduction

Abduction is one of the most important forms of reasoning; it has been successfully applied to several practical problems such as diagnosis. In this paper we investigate whether the computational complexity of abduction can be reduced by an appropriate use of preprocessing. This is motivated by the fact that part of the data of the problem (namely, the set of all possible assumptions and the theory relating assumptions and manifestations) are often known before the rest of the problem. In this paper, we show some complexity results about abduction when compilation is allowed.


Introduction
Deduction, induction, and abduction [Pei55] are the three basic reasoning mechanisms. Deduction allows drawing conclusions from known facts using some piece of knowledge, so that "battery is down" allows concluding "car will not start" thanks to the knowledge of the rule "if the battery is down, the car will not start". Induction derives rules from the facts: from the fact that the battery is down and that the car is not starting up, we may conclude the rule relating these two facts. Abduction is the inverse of deduction (to some extent [MF96]): from the fact that the car is not starting up, we conclude that the battery is down. Clearly, this is not the only possible explanation of a car not starting up. Therefore, we may get more than one explanation. This is an important difference between abduction and deduction, making the former, in general, more complex.
While deduction formalizes the process of drawing conclusions, abduction formalizes the diagnostic process, which attempts to invert the cause-effect relation by inferring the causes from its observable effects. The example of the car shows such an application: complete knowledge about car would allow finding (i.e., abducing) the possible reasons of why the car is not starting up. The following example shows how abduction can be applied to formalize a diagnostic scenario.
Example 1 While writing a paper with some authors located in another country, you get a set of macros that are used in a nice figure they drew. However, when compiling the .tex file, an incomprehensible error message results. Four explanations are possible: a : the macro has been used with the wrong arguments; p : the package X is required; t : the macro is incompatible with package X; v : the wrong version of TeX has been used.
This scenario can be formalized in logical terms by introducing a variable f to denote the presence of compile errors: since each of the facts above causes f , we know a → f , p → f , etc. Moreover, we know that a package cannot at the same time be required and incompatible with the macros. The following theory T formalize our knowledge.
This theory relates the observed effect (the compile error) with its possible causes (we used the wrong version of TeX, etc.) Therefore, it can be used to find the possible causes: namely, an explanation is a set of facts that logically imply the observed effect. Formally, an explanation is a set of variable that allow deriving the observed effects from the theory T . However, to make sense an explanation has to be consistent with our knowledge, that is, with the theory T .
This example shows that a given problem of abduction may have one, none, or even many possible solutions (explanations). Moreover, a consistent and an implication checks are required just to verify an explanation. These facts intuitively explain why abduction is to be expected to be harder than deduction. This observation has indeed been confirmed by theoretical results. Selman and Levesque [SL90] and Bylander et al. [BATJ89] proved the first results about fragments of abductive reasoning, Eiter and Gottlob [EG95] presented an extensive analysis, and Eiter and Makino have shown the complexity of computing all abductive explanations [EM02]. All these results proved that abduction is, in general, harder than deduction. The analysis has also shown that several problems are of interest in abduction. Not only the problem of finding an explanation is relevant, but also the problems of checking an explanation, or whether a fact is in all, or some, of the explanations are.
A common fact about deduction and abduction is that the knowledge relating facts may be known in advance, while the particular observation may change from time to time. In the example of the car, the fact that the dead battery makes the car not to start is always known, while the fact that the battery is dead may or may not be true. The possible causes of TeX errors are known before a specific error message comes out, etc.
We can therefore assign two different statuses to the knowledge base and to the single facts: while the knowledge base is fixed, the single facts are varying. In the example above, T will always reflect the state of the word, while f is only true when the TeX complains about something. This difference has computational consequences. While the example we have shown here does not present any problem of efficiency, larger and more complex abduction problems result from the formalization of real-world domains. The difference of status of T and the observations can then be exploited. Indeed, since T is always the same, we can perform a preprocessing step on it alone, even before the status of the observations are known. Clearly, we cannot explain an observation we do not know. However, this preprocessing step can be used to perform some computation that would otherwise be done on T alone. As a result, finding a solution might take less time when the observation finally get known.
The idea of using a preprocessing step for speeding-up the solving of abduction problems is not new. For instance, Console, Portinale, and Duprè [CPT96] have shown how compiled knowledge can be used in the process of abductive diagnosis.
Preprocessing part of the input data has also been used in many other areas of computer science, as there are many problems with a similar fixedvarying part pattern. However, the first formalization of intractability with preprocessing is relatively recent [CDLS02]. In this paper, we characterize the complexity of the problems about abductions from this point of view.

Preliminaries
The problem of abduction is formalized by a knowledge base, a set of observations, and a set of possible facts that can explain the observations. In this paper, we are only concerned about propositional logic. Therefore, the knowledge is formalized by a propositional theory T . We usually denote by M the set of observations.
The theory is T must necessarily contain all variables of M, otherwise there would be no way of explaining the observations. In general, the theory T contains other variables as well, describing facts we do not know whether they are true or not. Some of these facts can be taken as part of a possible explanation, while others are can not. Intuitively, when we are trying to establish the causes of an observation, we want the first cause, and not something that is only a consequence of it. In the example of the car, the fact that there is no voltage in the starting engine explains the fact that the car is not starting up, but it is not an acceptable explanation, as it does not tell where the real problem is (the battery). Therefore, the abduction problem is not defined only in terms of the theory and the observation, but also of the set of possible facts (variable) we would accept as first causes of the observation.
Formally, an instance of abduction is a triple H, M, T . The observations are formalized as M, which is a set of variables. T is a propositional theory formalizing our knowledge of the domain. Finally, H is a set of variables; these variables are the ones formalizing facts that we regards as possible first causes.
Abduction is the process of explaining the observation. Its outcome will therefore be a set of facts from which all observations can be inferred. Since we can only use variables of H to form explanations, these will be subsets H ′ ⊆ H. Moreover, an explanation can only be accepted if it is consistent with our knowledge. This leads to the following definition of the possible solutions (explanations) of a given abduction problem H, M, T .
We apply this definition to the running example of the TeX file.
Example 2 The propositional theory of the example shown in the introduc- All non-empty subsets of H implies, together with T , the observation M. However, the subsets containing both p and t are inconsistent with T . Therefore, the set of solution of the problem is given by: This is simply the formal result of our current definition. However, some explanations in this set are not really reasonable: for example, the explanation is {a, t, v} seems overly pessimistic: the macro has been called in the wrong way and a package is required and we used the wrong compiler version.
The set SOL(H, M, T ) contains all explanations we consider possible. However, some explanations may be more likely than others. For example, explanations requiring a large number of assumptions are often less likely than explanations with less assumptions.
Likeliness of explanations is formalized by an an ordering over the subsets of H. Given a specific , the set of minimal solutions is defined as follows.
SOL The ordering is used to formalize the relative plausibility explanations: H ′ ≺ H ′′ means that H ′ is considered more likely to be the "real" cause of the manifestations than H ′′ . The ordering represents the concept of "at least as likely as", thus H ′ ∼ = H ′′ holds if H ′ and H ′′ are equally likely. The definition of SOL formalizes the principle of choosing only the explanations we consider more likely. An implicit assumption of this definitions is that the ordering does not depend on the set of manifestations. We also assume that is a "wellfounded" ordering, that is, any non-empty set of explanations has at least one -minimal element. Therefore, if the set SOL(H, M, T ) is not empty, then min(SOL(H, M, T ), ) is not empty as well.
In this paper we take into account several plausibility ordering. The absence of a preference among the explanations can be formalized as the ordering that is equal to the universal relation, that is, H ′ H ′′ for any pair of sets of variables H ′ and H ′′ .
Besides this no-information ordering, the two simplest and most natural orderings are ⊆-preference, where an explanation H 1 is more likely of H 2 if H 1 ⊆ H 2 , and ≤-preference, where H 1 is preferred to H 2 if it contains less hypothesis, that is, |H 1 | ≤ |H 2 |.
Both these orderings are based on the principle of making as few hypotheses as possible, and by assuming that all hypotheses are equally likely. Two other orderings follows from assuming that the hypotheses are not equally likely: the ⊆-prioritization and the ≤-prioritization.
In particular, we assume that the hypotheses are partitioned into equivalence classes of equal likeliness. Let H 1 , . . . , H m be such a partition. By definition, it holds H 1 ∪ · · · ∪ H m = H and H i ∩ H j = ∅ for each i = j. The instances of the problem of abduction can thus be written as H 1 , . . . , H m , M, T . The set of all assumptions H is implicitly defined as the union of the classes H i . We assume that the hypotheses in H 1 are the most likely, while those in H m are the least likely.
The ⊆-prioritization and ≤-prioritization compare explanations on the basis of their relative plausibility. Namely, the explanations that use hypothesis in lower classes are more likely than explanations using hypothesis in higher classes. This idea, when combined with subset containment, defines the ⊆-prioritization. When it is combined with the cardinality-based ordering, it defines the ≤-prioritization. Formal definition is below.
Penalization is the last form of preference we consider. The idea is to assign weights to assumptions to formalize their likeliness. Explanations with the least total weight are preferred. Weights encodes the likeliness of assumptions: the most high the weight of an assumption, the unlikely it is to be true. To use penalization, the instance of the problem must include, besides H, M, and T , an n-tuple of weights W = w 1 , . . . , w n , where each w i is an integer number (the weight) associated to a variable h i ∈ H. The instance can thus be written W, H, M, T .
The considered orderings are formally defined as follows: Let us consider the use of these orderings on the running example. Prioritization allows for a further refinement of the set of solutions by exploiting the plausibility ordering over the hypotheses. For example, we may assume that the fact that package X is required and that we used the wrong version of the compiler are the two most likely hypotheses. Formally, they will be part of the first set of assumptions H 1 , while the other assumptions will therefore go in H 2 . Formally, the problem instance is now H 1 , H 2 , M, T . Both ⊆-prioritization and ≤-prioritization produce {p} and {v} as the only minimal explanations. This is because all other explanations either have a bigger intersection with H 2 , or an equal intersection with H 2 but a bigger intersection with H 1 .
Finally, penalization requires a weight (an integer number) for each hypothesis. Let us for example use the set of weights 4, 2, 4, 1 associated with the set of hypotheses a, p, t, v}. Since larger weights correspond to less likely hypotheses, we are assuming that our first and third hypotheses (a and t) are the least likely, while p is more likely and v is the most likely. From definition, the explanation {v} is the one having the least weight, and is therefore the only solution of the problem.
The basic problem of abduction is that of finding one or more explanations. However, we have already remarked that none may exist. Therefore, the first problem we consider is the existence one: given an instance of abduction, does an explanation exist? Another related problem is that of verifying, once a set of hypotheses has been found, whether it is really an explanation or not.
Other problems are related to the structure of the explanations. Namely, hypotheses that are in all explanations may considered as "sure" conclusions of the abductive process. On the other hand, hypotheses that are part of some explanations can be regarded as "possible" conclusions.
The formal definition of these questions as decision problems is as follows. Necessity: is h ∈ H in all, and at least one, minimal solution? That is, Dispensability: is h ∈ H such that either there is no solution or there exists one who does not contain h? That is, Dispensability is the converse of the problem of necessity, since an hypothesis h is dispensable if and only if it is not necessary. The problem of dispensability is not of much interest by itself, but is sometimes useful for simplifying the proofs.
Clearly, the ordering does not matter for the problem of existence, since we consider only well-founded orderings: therefore, an explanation exists if and only if a minimal explanation exists. For the other problems, the ordering must be taken into account. Different orderings may lead to different computational properties.
In this paper, we assume that T is a 3CNF formula: this assumption does not cause a loss of generality unless we want to assume that H ∪M = Var(T ).

Complexity and Compilability
The basic complexity classes of the polynomial hierarchy [Sto76,GJ79], such as P, NP, coNP, etc., are assumed known to the reader. We denote by C, C ′ , etc. arbitrary classes of the polynomial hierarchy. The length of a string x ∈ Σ * is denoted by ||x||.
We summarize some definitions and results proposed to formalize the online complexity of problems [CDLS02]. In computational complexity, problems whose solution can only be yes or no are the most commonly analyzed. Such problems are called decision problems. Any such problem can be formalized as set of strings, those whose solution is yes. For example, the problem of propositional satisfiability (deciding whether a formula is satisfiable or not) is characterized by the set of the strings that represent exactly all satisfiable formulae.
The strings that compose the set associated to a problem represent the possible problem instances that produce a positive solution. Problems like abduction, however, have instances that can be naturally broken into two parts: one part is known in advance (T and H) and one part is only known at run-time (M). Therefore, the instances of such problems are better encoded as pairs of strings. Therefore, a problem like abduction is formalized by a set of pairs of strings, rather than a set of strings. We define a language of pairs S as a subset of Σ * × Σ * .
The difference between the first and second element of a pair is that some preprocessing time can be spent on the first string alone. This is done to the aim of solving the problem faster when the second string comes to be known. While our final aim is to reduce the running time of this second phase, some constraints have to be put on the preprocessing phase. Namely, we impose its result to be of polynomial size. Poly-size function are introduced to this purpose: a function f from strings to strings is called poly-size if there exists a polynomial p such that, for all strings x, it holds ||f (x)|| ≤ p(||x||). An exception to this definition is when x represents a natural number: in this case, we impose ||f (x)|| ≤ p(x). Any polynomial function is polysize, but not viceversa. Indeed, a function g is poly-time if there exists a polynomial q such that, for all x, g(x) can be computed in time less than or equal to q(||x||). Clearly, the running time also bounds the size of the output string; on the other hand, even a function requiring exponential running time can produce a very short output. The definitions of polysize and polytime function extend to binary functions as usual.
Using the above definitions, we introduce a new hierarchy of classes of languages of pairs, the non-uniform compilability classes [CDLS02], denoted by ;C, where C is a generic uniform complexity class, such as P, NP, coNP, or Σ p 2 .
Definition 1 ( ;C classes, [CDLS02]) A language of pairs S ⊆ Σ * × Σ * belongs to ;C iff there exists a binary poly-size function f and a language of pairs S ′ ∈ C such that, for all x, y ∈ S, it holds: x, y ∈ S iff f (x, ||y||), y ∈ S ′ Clearly, any problem whose time complexity is in C is also in ;C: just take f (x, ||y||) = x and S ′ = S. Some problems in C however belongs to ;C ′ with C ′ ⊂ C; for example, some problem in NP are in ;Π p . These are in fact the problems we are most interested, as the preprocessing phase, running on x only, will produce f (x), which allows solving the problem in polynomial time. This is important if these problems cannot be solved in polynomial time without the preprocessing phase (e.g., they are NP-complete).
The class ;C generalizes the non-uniform class C/poly -i.e., C/poly ⊂ ;C -by allowing for a fixed part x. We extend the definition of polynomial reduction to a concept that can be used with these classes.
Definition 2 (Non-uniform comp-reduction) A non-uniform comp-reduction is a triple of functions f 1 , f 2 , g , where g is polytime and f 1 and f 2 are polysize. Given two problems A and B, A is non-uniformly comp-reducible to B (denoted by A ≤ nucomp B) iff there exists a non-uniform comp-reduction f 1 , f 2 , g such that, for every pair x, y it holds that x, y ∈ A if and only if f 1 (x, ||y||), g(f 2 (x, ||y||), y) ∈ B.
These reductions allows for a concept of hardness and completeness for the classes ;C.
Definition 3 ( ;C-completeness) Let S be a language of pairs and C a complexity class. S is ;C-hard iff for all problems A ∈ ;C we have that The hierarchy formed by the compilability classes is proper if and only if the polynomial hierarchy is proper [CDLS02, KL80, Yap83] -a fact widely conjectured to be true.
Informally, ;NP-hard problems are "not compilable to P". Indeed, if such compilation were possible, then it would be possible to define f as the function that takes the fixed part of the problem and gives the result of compilation (ignoring the size of the input), and S ′ as the language representing the on-line processing. This would implies that a ;NP-hard problem is in ;P, and this implies the collapse of the polynomial hierarchy. In general, a problem that is ;C-complete for a class C can be regarded as the "toughest" problem in C, in the assumption that preprocessing the fixed part is possible.
While ;C-completeness is adequate to show the compilability level of a given reasoning problem, proving it requires finding a nucomp reduction. We show a technique that let us reuse, with simple modifications, the polytime reductions that were used to prove the usual (uniform) hardness of the problem. Namely, we present sufficient conditions allowing for a polynomial reduction to imply the existence of a nucomp reduction [Lib01].
Let us assume that we know a polynomial reduction from the problem A to the problem B, and we want to prove the nucomp-hardness of B. Some conditions on A should hold, as well as a condition over the reduction. If all these conditions are verified, then there exists a nucomp reduction from * A to B.
Definition 4 (Classification Function) A classification function for a problem A is a polynomial function Class from instances of A to nonnegative integers, such that Class(y) ≤ ||y||.
Definition 5 (Representative Function) A representative function for a problem A is a polynomial function Repr from nonnegative integers to instances of A, such that Class(Repr(n)) = n, and that ||Repr(n)|| is bounded by some polynomial in n.

Definition 6 (Extension Function) An extension function for a problem
A is a polynomial function from instances of A and nonnegative integers to instances of A such that, for any y and n ≥ Class(y), the instance y ′ = Exte(y, n) satisfies the following conditions: Let us give some intuitions about these functions. Usually, an instance of a problem is composed of a set of objects combined in some way. For problems on boolean formulas, we have a set of variables combined to form a formula. For graph problems, we have a set of nodes, and the graph is indeed a set of edges, which are pairs of nodes. The classification function gives the number of objects in an instance. The representative function thus gives an instance with the given number of objects. This instance should be in some way "symmetric", in the sense that its elements should be interchangeable (this is because the representative function must be determined only from the number of objects.) Possible results of the representative function can be the set of all clauses of three literals over a given alphabet, the complete graph over a set of nodes, the graph with no edges, etc.
Let for example A be the problem of propositional satisfiability. We can take Class(F ) as the number of variables in the formula F , while Repr(n) can be the set of all clauses of three literals over an alphabet of n variables. Finally, a possible extension function is obtained by adding tautological clauses to an instance.
Note that these functions are related to the problem A only, and do not involve the specific problem B we want to prove hard, neither the specific reduction used. We now define a condition over the polytime reduction from A to B. Since B is a problem of pairs, we can define a reduction from A to B as a pair of polynomial functions r, h such that x ∈ A if and only if Definition 7 (Representative Equivalence) Given a problem A (having the above three functions), a problem of pairs B, and a polynomial reduction r, h from A to B, the condition of representative equivalence holds if, for any instance y of A, it holds: The condition of representative equivalence can be proved to imply that the problem B is ;C-hard, if A is C-hard [Lib01].

Compilability of Abduction: No Ordering
In this section we analyze the problems of existence of explanation, explanation verification, relevance, and necessity, for the basic case in which no ordering is defined. Formally, we want to determine whether the complexity of the problems related to SOL(H, M, T ) decrease thanks to the preprocessing step on H and T .
We first give an high-level explanation of the method we use to prove the incompilability of the considered problems. We begin by applying the method to the problem of existence of explanations, and then we used it for verification, relevance and necessity.

The Method
The problem of deciding whether there exists an explanation for a set of manifestations is Σ p 2 -hard [EG95]. Therefore, there exists a polynomial reduction from another Σ p 2 -hard problem to this one. In order to prove it is also ;Σ p 2 -hard we can show that the other problem has the three functions, and the reduction satisfies the condition of representative equivalence. Unfortunately, this is not the case. As a result, we have to look for another reduction.
Such a reduction should be as simple as possible. In general, the more similar two problems are, the easier it is to find a reduction. What is the Σ p 2 -hard problem that is the most similar to the problem of existence of explanation? Clearly, the problem itself is the most similar to itself.
The theorem of representative equivalence is indeed about a reduction between two problems A and B, but it does not forbid using the same problem: it only tells that, if we have a reduction from an arbitrary Σ p 2 -hard problem A to B, satisfying representative equivalence, then B is ;Σ p 2 -hard. Nothing prevent us from choosing A = B. This technique can be formalized as follows: • show that there exists a classification, representative, and extension functions for the problem B; • show that there exists a reduction from B to B satisfying representative equivalence.
The most obvious reduction from a problem to itself is the identity. In our case, however, identity does not satisfy the condition of representative equivalence. As a result, we have to look for another reduction.
Before showing the technical details of the reductions used, we point out an important feature of this technique. Since the condition of representative equivalence tells that B is ;C-hard if A is C-hard, using A = B we prove that B is ;C-hard whenever B is C-hard. This result holds even if a precise complexity characterization of B is not known. For example, if we only know that B is in Σ p 2 , but do not have any hardness result, we can still conclude In order to simplify the following proofs, we denote with Π(X) the set of all distinct clauses of length 3 on a given alphabet X = {x 1 , . . . , x n }. Since the theory T is in 3CNF by assumption, we have that T ⊆ Π(V ), where V is the set of variables appearing in T .

Existence of Solutions
In order to define a reduction from the problem of existence of solutions to itself, we first consider the function f from abduction instances to abduction instances defined as follows: In these formulae, X denotes the alphabet of T , while C and D are sets of new variables in one-to-one correspondence with the clauses in Π(H ∪ X). Note that, by definition, T is a subset of Π(H ∪ X). 1. We prove that The same holds for any d i ∈ R. This proves that S ′ ∩ (C ∪ D) ⊇ R.
Since R contains either c i or d i for any i, the same holds for S ′ . No other variable in C ∪D can be in S ′ , otherwise S ′ would be inconsistent with T ′ , which contains the clauses ¬c i ∨ ¬d i .
2. Let S ′ be an element of SOL( H ′ , M ′ , T ′ ). We prove that S = S ′ \(C ∪ D) ∈ SOL( H, M, T ). The point proved above shows that, for each i, S ′ contains either c i or d i , depending on whether γ i ∈ T . As a result: As a result, S ∪T is consistent because the above formula is. Moreover, since the above formula implies M, and each variable in C ∪ D appears only once, it also holds S ∪ T |= M. As a result, S is a solution of H, M, T .
3. Let S ∈ SOL( H, M, T ), and let The claim is thus proved.
This lemma shows that any abduction instance can be converted into another one in which the set H and the theory T only depends on the number of variables of the original instance. This reduction can be used to build a reduction satisfying the condition of representative equivalence. The extension function is also easy to give. For example, we may add to T a set of tautologies with new variables. We are now able to show a reduction satisfying the condition of representative equivalence. Let i be the reduction defined as follows. The following theorem is a consequence of the fact that i satisfies the condition of representative equivalence.

Verification
We We define two functions f ′ and g ′ c to be similar to the functions f and g c of the last section, except for the addition of a candidate explanation H a .
These functions can be composed to generate a function that satisfies representative equivalence. This way, we prove the nucomp-hardness of the problem of verification.

Relevance, Dispensability, and Necessity
We make the following simplifying assumption: given an instance of abduction H, M, T , where H = {h 1 , . . . , h m }, the problem is to decide whether the first assumption h 1 is relevant/dispensable/necessary. Clearly, the complexity of these problems is the same, as we can always rename the variables appropriately.
Theorem 3 The problems of relevance and dispensability with no ordering is ;Σ p 2 -hard, while necessity is ;Π p 2 -hard.
Proof. By Lemma 1 and Lemma 2, i( H, M, T ) is a reduction from the problem of relevance to the problem of relevance. Indeed, for any H a ⊆ H, the set

Compilability of Abduction: Preferences
In this section, we consider the problems of verification, relevance, and necessity when the ordering used is either ≤ or ⊆. These orderings have in common the fact that the instance of an abduction problem is simply a triple H, M, T , whereas the orderings of the next section employee classes of priority or weights that are part of the instances. The problem of existence is the same as with no ordering, as these orderings are well founded.

Some General Results
We give some general results about the problem of abduction in the case in which an ordering on explanation is given. In order to keep results as general as possible, we consider an arbitrary ordering satisfying the following natural conditions. We determine the compilability of abduction with preference in the same way we did in the case of no ordering: we show that the function i is a polynomial reduction from the problems of abduction to themselves, and that it satisfies the condition of representative equivalence. To this aim, we need the analogous of Lemma 1 and Lemma 2.

Lemma 3 If is a meaningful ordering, it holds:
We use the result of Lemma 1. Namely, since all solutions of f ( H, M, T ) coincide on C ∪ D, these variables are irrelevant thanks to the fact that is meaningful.
Formally, we have: This proves the claim.
We can also prove the analogous of Lemma 2. These lemmas can be used to prove incompilability of abduction when an irredundant and meaningful ordering is used.

Verification
We consider the problem of verifying whether a set of assumptions is a minimal explanation according to the orderings ≤ and ⊆. More generally, we prove the following theorem for any meaningful and irredundant ordering.
Theorem 4 If is a meaningful and irredundant ordering, verifying whether a set of assumptions is a minimal explanation is ;C-hard for any class C for which the problem is C-hard.
Proof. The same classification, representative, and extension functions used for the case of no ordering can be used for this case as well.
Let now consider the functions f ′ and g ′ c . From Lemma 3 and Lemma 4 it follows that they are reductions from the problem of verification to itself. Moreover, their composition i ′ satisfies representative equivalence.

Relevance, Dispensability, and Necessity
We make the following simplifying assumption: given an instance of abduction H, M, T , where H = {h 1 , . . . , h m }, the problem is to decide whether the first assumption h 1 is relevant/dispensable/necessary. There is no loss of generality in making this assumption. as we can always rename the variables appropriately.
Theorem 5 If is a meaningful and irredundant ordering, then the problems of relevance/dispensability/necessity are ;C-hard for any class C of the polynomial hierarchy for which they are C-hard.
Proof. From Lemma 3 and Lemma 4, it follows that the reduction i is a reduction from the problems of relevance/dispensability/necessity to themselves, if is meaningful and irredundant, and it also satisfies representative equivalence.
Since ⊆ and ≤ are meaningful irredundant orderings, their complexity implies their compilability characterization.

Compilability of Abduction: Prioritization and Penalization
We consider the cases in which the ordering over the explanations is defined in terms of a prioritization. The instances of the problem are different from those of the previous section, since H is replaced by a partition of assumptions H 1 , . . . , H m . In the cases of ≤-prioritization and ⊆-prioritization, the induced ordering is meaningful and irredundant. However, the results on meaningful irredundant ordering cannot be directly applied because, in Theorem 4 and Theorem 5, we assumed that the instances have the form H, M, T , while now they have the form H 1 , . . . , H m , M, T . Therefore, we have to find new classification, representative, and extension functions.
We first consider the problem of verification, and prove its nucomphardness. Then, we move to the problems of relevance, dispensability, and necessity. As for the case of ≤-preference and ⊆-preference, we employee a sort of normal form, in which the assumption we check is the first one.

Verification
First of all, we show the classification, representative, and extension functions for the problem of verification. The instances of the problem include a "candidate explanation" H a . These functions can be easily proved to be valid classification, representative, and extension functions. What is missing is a reduction from the problem of verification to itself satisfying the condition of representative equivalence.
To this extent, we use two functions f ′′ and g ′′ c that are similar to f and g c , respectively. In particular, f ( H 1 , . . . , H m , H Besides the partition of the assumptions, this is exactly the function used in Lemma 1. As a result, we have that: Since is a meaningful irredundant ordering, the same property holds replacing SOL with SOL . The last step is to define a function g ′′ c similar to g c . This is done as follows. Theorem 6 The problem of verification for any prioritization based on a meaningful and irredundant ordering is ;C-hard for any class C for which it is C-hard.
Proof. The composition of i ′′ of f ′′ and g ′′ c is a reduction satisfying representative equivalence.
i ′′ ( H 1 , . . . , H m , H a ,  The fact that is a reduction from the problem of verification to itself easily follows from the fact that both f ′′ and g ′′ c are. Moreover, the result of f (g ′′ Class( H 1 ,...,Hm ,Ha,M,T ) ( H 1 , . . . , H m , H a , M, T )) is an instance in which the number of classes of assumption, of variables in each class, and the number of other variables, all coincide with the class of the original instance. The function f ′′ produces an instance in which everything but M and H a depends only on these numbers. As a result , the function i produces an instance in which everything but M and H a depends on the class of the original instance only. As a result, this function i ′′ is a reduction from the problem of verification to itself, satisfying representative equivalence, which implies the incompilability of the problem.

Relevance and Necessity
We restrict the problems to the case the assumption we want to check for relevance/dispensability/necessity is the first variable of H 1 . The problems have the same complexity of the general ones (in which the assumption can be an arbitrary one.) This, however, cannot be proved with a simple renaming of the variables, as we did for the case of preference.
Theorem 7 Let be a meaningful and irredundant ordering. It holds: Proof. We first give an informal sketch of the proof. The set of solutions (with no ordering) of the first and the second instances only differ because the explanations for the second instances must contain either s or both h i j and t.
The explanations of the second instances are first compared on the assumptions in H 1 , . . . , H m , and then on {s, t}. Therefore, the ordering for the second instance is a refinement of the ordering of the first one. Namely, a minimal solution of the second instances is either a minimal solution of the first one plus s, or a minimal solution of the first one plus t. However, the latter is a solution only if it contains h i j . Therefore, the presence of a solution containing h i j in the first instance is equivalent to the presence of a solution for the second instance containing t.
The formal proof is as follows. Let H, M, T be the first instance and H ′ , M ′ , T be the second one.
This can be proved as follows. First, since S ′ ∪ T ′ is consistent, it follows that S ′ ∪ T is consistent as well (because T ⊆ T ′ ), which also implies that (S ′ \{s, t}) ∪ T is consistent.
Let us now prove that (S ′ \{s, t}) ∪ T |= M. By assumption, we have S ′ ∪ T ′ |= M ′ . The following chain of implications leads to the claim. First of all, S ∪ {t} is solution of the second instance. What is left to prove is its minimality. This is also easy: removing t leads to a set of assumptions which does not explain v. If removing some variable from S leads to another solution, then S is not minimal.
. The "if" direction is easy. Let us assume that S is a minimal solution of the first instance. Then S ∪ {s} is a solution of the second one. Let us prove that it is minimal. We cannot remove variables from S, otherwise S would be not minimal. As a result, the only other possible explanations that can be preferred are S ∪ {t} and S ∪ ∅. None of them is a solution, because they do not imply u.
It is now possible to prove the claim. If there exists a minimal solution of the first instance containing h i j , then there exists a minimal solution of the second one containing t. On the other hand, if no minimal solution contains h i j , then all corresponding minimal solutions of the second instances contains s, which means that t is in none of them. Therefore, relevance and necessity of h i j on the first instance are equivalent to relevance and necessity, respectively, of t in the second instance.
As a result of this theorem, we can assume that relevance or dispensability are evaluated w.r.t. the first variable in H 1 . In order to prove that these problems are not compilable, we give a classification, representative, and extension function. where r = |V ar(T )\(H 1 ∪ · · · ∪ H m )| Given these three functions, all is needed is a reduction from the problem of relevance to itself satisfying representative equivalence. The function i ′′ cannot be used only because the instance it deals with contains the set of assumptions H a . However, removing this part of the instance both from its argument and its result, we obtain a reduction with the right properties. We can thus conclude that the problems of relevance, dispensability, and necessity are incompilable.
Theorem 8 Let be a meaningful and irredundant ordering. The problems of relevance, dispensability, and necessity for the problem of prioritized abduction are ;C-hard for any class C of the polynomial hierarchy for which these problems are C-hard.
As a result, we easily obtain the compilability properties of the problem of prioritized abduction using the orderings ⊆ and ≤.
Theorem 9 Relevance and dispensability are ;Σ p 3 -hard if ⊆ is used, and ;∆ p 3 -hard if ≤ is used instead.
The compilability of relevance and dispensability in the case of penalization is an easy consequence of the last theorem, as relevance with ≤ (prioritized) can be directly translated (using a nucomp reduction) to relevance with penalization.
Corollary 2 Relevance and dispensability are ;∆ p 3 -hard, in the case of penalization.

The Horn Case
The Horn case can be dealt with using the same technique of the general case. Since, however, only Horn clauses are allowed, each time we use Π(H ∪ X), which contains all clauses of three literals over H ∪ X, we have to replace it with the Π H (H ∪ X) that contains all Horn clauses of three literals over the set H ∪ X. The reductions we used employ clauses ¬c i ∨ ¬d i and ¬c i ∨ γ i , which are Horn if γ i is Horn. The reduction used in Theorem 7 also involves Horn clauses only. Therefore, all results holding for the general case hold for the Horn case as well. Namely, all problems about Horn clauses are ;C-hard for the same classes C they are C-hard. An important feature of reduction from the same problem is that it allows proving nucomp-hardness result even for a restriction of the problem, provided that these reduction do not transform an instance into a non-valid one (e.g., unless an Horn instance is mapped into a non-Horn one.) The even more restricted case of definite Horn clauses, however, cannot be dealt with in the same manner. Indeed, the clauses ¬c i ∨ ¬d i , are not definite. Some problems, however, becomes polynomial, in this case. Namely, all problems in the case of no order are polynomial, as well as necessity for ⊆-preference. We only show that a reduction for the case of ≤-preference. As before, the problem is that of checking whether h 1 is in an explanation of minimal size of H, M, T . Since h 1 is part of H, we regard H, M, T as being the instance of the problem. The classification, representative, and extension functions are as usual (tautologies are definite Horn clauses.) The reduction we use is based on the following function f , where n = |H|.
The idea is simply that of replicating each variable c j i for n + 1 times. This way, if S ⊆ H, then a clause γ i holds in S ∪ T only if S contains all clauses c j i . The reduction is based on the following two facts: Let us first assume that S is an explanation of H, M, T , that is, is therefore an explanation because the latter set is indeed M ′ . The set C ′ is not relevant to this part of the proof.
Let us now assume that S ′ = S ∪ {c j i | γ i ∈ T } ∪ C ′ is an explanation of f ( H, M, T ). Since |C ′ | ≤ n, then S ′ does not contain all c j i for any i. Therefore, all clauses that are not in T contains at least an unassigned c j i in S ′ ∪ T ′ . Therefore, these clauses are cannot be used to derive a single literal in M ′ . As a result, S ∪ {c j i | γ i ∈ T } ∪ T ′ |= M ′ . This is equivalent to S ∪ T |= M, that is, S is an explanation of H, M, T .
This lemma can be used to relate the minimal explanations of the two instances. Proof. The lemma above implies that S is an explanation if and only if S ∪ {c j i | γ i ∈ T } is an explanation, as this is the case of C ′ = ∅. Let us now prove that the minimality of these two explanations coincide.
Let us first assume that S is a minimal explanation. We prove that S ′ = S ∪ {c j i | γ i ∈ T } is a minimal explanation of H ′ , M ′ , T ′ . By the lemma above, S ′ is an explanation; we have therefore only left to prove that it is of minimal size. Assume that S ′′ is another explanation of H ′ , M ′ , T ′ . By construction, S ′′ contains {c j i | γ i ∈ T }. Therefore, S ′′ can be smaller than S ′ only if S ′′ \{c j i | γ i ∈ T } is smaller than S ′ \{c j i | γ i ∈ T }. Since the latter coincide with S ′ ∩ H, whose size is bounded by n, we have that |S ′′ \{c j i | γ i ∈ T }| ≤ n. Therefore, S ′′ can be written as S ′′ = S ′′′ ∪ {c j i | γ i ∈ T } ∪ C ′ with |S ′′′ ∪ C ′ | < |S|. The latter inequality implies |C ′ | < n: by the lemma above, S ′′′ would be an explanation of H, M, T . Since |S ′′′ ∪ C ′ | < |S|, then |S ′′′ | < |S|, that is, S is not be minimal.
Let us now assume that S ′ = S ∪ {c j i | γ i ∈ T } is a minimal explanation of H ′ , M ′ , T ′ , and prove that S is a minimal explanation of H, M, T . Assume, indeed, that S ′′ is a smaller explanation of H, M, T . By the lemma above, S ′′ ∪{c j i | γ i ∈ T } would then be an explanation of H ′ , M ′ , T ′ smaller than S ′ .
The reduction can be defined as for the Horn case, by taking into account the fact that the original instance H, M, T may not have any explanation. Such a reduction ratifies the condition of representative equivalence, thus proving that problems about ≤-preference are ;C-hard whenever they are C-hard. Similar reductions can be defined for the other orderings.

Conclusions
In this paper, we have shown that logic-based abduction cannot be simplified by preprocessing the theory T and the hypotheses H. In particular, this result holds for various kinds of explanation orderings, and also for the Horn restriction. These results have been proved using the technique of representative equivalence [Lib01]; since reductions are from a problem to itself, they prove that a problem is "compilability-hard" for any class for which it is hard. In other words, we did not prove that a problem is hard for some class, but rather that it complexity decreases thanks to preprocessing. Using these "self-reductions" allows for proving such a result even if the complexity of the problem is not known. For example, we prove that a preprocessing step does not simplify the problem of finding a minimal explanation for any ordering that is both meaningful and irredundant. The complexity of this problem is not known for all such orderings; moreover, it depends on the ordering itself.
The technique we used to prove that "preprocessing does not simplify abduction", being based on complexity classes at last, should however not be considered as implying that preprocessing is not useful for speeding up solving of abduction problems. Indeed, as for any result based on the theory of NP-completeness, this conclusion only holds as a worst-case result. In other words, it does not tell that no instance can ever by made simpler by preprocessing, but simply that any preprocessing procedure necessarily has some hard instances that are not simplified. In a sense, our result is more positive than it appears, as it tells that a worst-case exponential on-line algorithm is reasonable, given than no worst-case polynomial one exists.
Compilability results based on hardness and reductions have consequences similar to complexity results based on the theory of NP-completeness: they tell that, since no worst-case polynomial algorithm can solve the problem, alternative directions have to be considered. Approximation is one example: the preprocessing phase may result in some data structure that allows a better (or faster) approximation of the best abductive explanations. Another possible direction is that of incomplete compilation, in which the preprocessing phase produces a result that is only useful in some cases, but not always. Another common solution to hard-to-compile problems is that of generating a worst-case exponential preprocessing result. This approach is especially useful if part of the result can be used, as we can then try to generate it and use only the part we can store. All these alternative approaches, however, only make sense when the impossibility of preprocessing the problem into a polynomial problem has been proved. This is the practical impact of our hardness results.
Finally, compilability has been proved to be related to expressibility of logical formalisms, that is, their ability of representing information in little space [CDLS00]. Logical-based abduction formalisms could then be characterized by the set of abductive problems they are able to express. Compilation classes (and not complexity ones) have been proved useful to this aim.