Bylander, T., D. Allemang, M. C. Tanner, and J. R. Josephson: 1991, `The Computational Complexity of Abduction'. Artificial Intelligence 49(1-3), 25--60. Charniak, E.: 1986, `A neat theory of marker passing'. In: Proceedings of the 5th National Conference on Artificial Intelligence (AAAI'86). pp. 584 -- 588.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....or natural language processing to mention but a few. One of the uses of abduction is to obtain explanations for observations, which loosely speaking is accomplished by a kind of reversed modus ponens. There is quite some work on algorithms and complexity of finding abductive explanations (e.g. [4, 8, 9, 11, 43, 46]) Roughly, in a logic based setting, abductive explanations are defined as follows (cf. 34, 46] Given some background knowledge , which is a theory, i.e. a set of sentences in some logic, and a set of observations , which are typically facts, a set of sentences from a set of ....

T. Bylander, D. Allemang, M. C. Tanner, and J. R. Josephson. The computational complexity of abduction. Artif. Intell., 49:25--60, 1991.

....[3,4,5] is developed for abductive reasoning with sound mathematical foundation, there has been a shift in attention from deductive reasoning to abductive reasoning. Abductive reasoning is promising compared to deductive reasoning as it gives all possible explanations for a given problem at hand [6]. The objective of this paper is to add the probabilistic features to the diagnostic problem solving using Realistic Abductive Reasoning Model (or Realistic ARM) 1] so as to explore the explanations in the decreasing order of plausibility. The realistic abductive reasoning model (RARM) is ....

Bylander T, Allenmang D, Tanner M. C, and Josephson J. R, The Computational Complexity of Abduction, Artificial Intelligence, Vol.49, 1991, pp.25 - 60.

....context sets. Proof (sketch) Membership in 4 follows from Lemma 4.3 and Theorem 2.4. Hardness for 4 can be proved similarly as in the proof of Theorem 4.8. 2 5 RELATED WORK AND CONCLUSION There is quite some work on algorithms and complexity of finding abductive explanations (e.g. [3, 6, 7, 8, 35, 37]) which play an important role in many AI problems including diagnosis, planning, or natural language processing. Roughly, a set of facts E is an abductive explanation of an observation O on some background theory T , if E is compatible with T and entails O; further minimality conditions are ....

T. Bylander, D. Allemang, M. C. Tanner, and J. R. Josephson. The computational complexity of abduction. Artif. Intell. , 49:25--60, 1991.

....113, JAPAN Its crucial problem, however, is the slow inference speed. That is, the inconsistency among element hypotheses causes its inference time to grow exponentially with the number of possible element hypotheses. Its computational complex ity has been proven to be NP complete or NP hard [2, 3]. To overcome this problem, many symbolic inference approaches have been proposed. Top down inference starting with the goal, like [4] employing the qSqR approach [5] can focus the search to the useful portion of knowledge for proving the goal. On the other hand, Bottom up and paralleled ....

Bylander,T., Allemang,D., et al.: The Computational Complexity of Abduction, Artif. Intell., Vol.49, 25-60, 1991.

....is defined by the name and domain of the attribute. 3. the diagnostic mapping, here IS is its domain and FS is its value field. As it shows in definition 1, the essence of the diagnostic problem is to find the function . and it is easy to know that (I) SXFS, which has been proofed to be NP hard[8] in general. In the following, we will give eI) an approximately solution. Theoretically we can combine the FS and AS to form a new set ES. Actually, ES has a practical meaning, it is much like an instance set, which can be denoted by the set of attribute fault pair. Definition 2 We define the ....

Bylander T, Allemang D, Tanner M C, Josephson J R. The computational complexity of abduction, In: Brachman R J, Levesque H J, Reiter R, eds. Knowledge Representation. Cambridge, MA: The MIT Press. 1992, 25-60.

....[7,8,9,10] is developed for abductive reasoning with sound mathematical foundation, there has been a shift in attention from deductive reasoning to abductive reasoning. Abductive reasoning generates all the possible explanations which may require further refinement to arrive at appropriate covers [11]. Deductive reasoning though generates only appropriate covers, will not generate those required covers which it would, if the missing information were to be present. Both, the abductive and the deductive reasoning strategies are far from reality. The proposed model for telecommunication network ....

Bylander T, Allenmang D, Tanner M. C, and Josephson J. R, The Computational Complexity of Abduction, Artificial Intelligence, Vol.49, 1991, pp.25-60.

....RR 1843 01 08 Proof (sketch) Membership in P 4 follows from Lemma 4.3 and Theorem 2.4. Hardness for P 4 can be proved similarly as in the proof of Theorem 4.8. 2 5 Related Work and Conclusion There is quite some work on algorithms and complexity of finding abductive explanations (e.g. [3, 6, 7, 8, 34, 36]) which play an important role in many AI problems including diagnosis, planning, or natural language processing. Roughly, a set of facts E is an abductive explanation of an observation O on some background theory T , if E is compatible with T and entails O; further minimality conditions are ....

T. Bylander, D. Allemang, M. C. Tanner, and J. R. Josephson. The computational complexity of abduction. Artif. Intell., 49:25--60, 1991.

....hypothesis generation by classification had a computational complexity that was linear in the number of hypotheses. Goel and Bylander similarly analyzed the computational properties of structured matching [28] see Section 2. 4) Bylander et al. showed in a series of papers which culminated in [7] that many of the strategies used in the construction of our abductive assembler (see Section 2.3) had attractive computational properties, explaining how knowledge of the right type can help solve problems in acceptable time even though the general abductive problem was NP complete. We now had a ....

....algorithms to modeling knowledge and methods by which tasks are decomposed and subtasks are accomplished. The fact that we do not start with a uniform normative algorithm does not mean that we cannot be precise about the behavior of systems built in the Task Structure framework. Bylander et al. [7] and Goel et al. 30] are examples of analyses in which the role of specific types of knowledge in producing good computational properties is studied within the general framework of the task structure view. For example, Goel shows why classification is an attractive method, if knowledge in the ....

Bylander, T., Allemang, D., Tanner, M.C., Josephson, J.R.: The computational complexity of abduction. Artificial Intelligence, 49(1991):25-60, 1991

....can reach some goal without causing contradictions. If contradictions can occur, abduction must create worlds: maximal consistent sets of beliefs. If multiple such worlds can be generated, then a BEST assessment operator selects the preferred world(s) For more details on abduction, see [2, 10]. For an example of abduction, see below. 3 investor confidence company profits trade deficit corporate spending wages restraint domestic inflation sales public confidence account balance current foreign sales Figure 1: An economics ....

T. Bylander, D. Allemang, M.C. M.C. Tanner, and J.R. Josephson. The Computational Complexity of Abduction. Artificial Intelligence, 49:25--60, 1991.

....determining whether actually observed findings can be predicted using the causal relation. Subsequent work has yielded several algorithms to compute set covering diagnoses efficiently in practical applications [29,38,44] although this type of diagnostic reasoning is known to be NP hard in general [5]. Experimental studies of set covering theory and its variants have been performed by several researchers [21,34,41] The formal aspects of diagnosis employing causal knowledge have also been studied, using logic as the primary tool [11,13,30,32] In the logical theory of abductive diagnosis, ....

....or compute, the exponential number of function values of an evidence function e; it suffices to provide only part of them explicitly. Any algorithm for diagnosis using an evidence function of the form discussed in the previous section, without simplifying assumptions, will be intractable. In [5], in which the complexity of algorithms for abductive diagnosis is analysed, it is therefore assumed that the specification of a domain theory is polynomial in the sum of the cardinalities of the sets # and # . A partial specification of an evidence function e consists of a restriction of e, ....

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T. Bylander. D. Allemang, M.C. Tanner, J.R. Josephson, The computational complexity of abduction, in: R.J. Brachman, H.J. Levesque, R. Reiter (Eds.), Knowledge Representation, The MIT Press, Cambridge, MA, 1992, pp. 25--60.

....is often willing to prefer as simplest solutions as possible. In particular, solutions should be nonredundant, i.e. an acceptable solution must not contain another solution properly. The property of subset minimality is the most commonly applied criterion for acceptable abductive solutions, cf. [58, 74, 9, 41]. For example, for the observation light off in the above example, fbulb broken; power failureg would be a solution (under both variants of inference) The simpler solution bulb broken (or power failure as well) is clearly preferable. Besides subset minimality, smallest size of solutions is a ....

.... been discussed by Denecker and De Schreye [16] and by Baral and Gelfond [4] For related complexity results in the area of logic programming and nonmonotonicreasoning, cf. 11, 71, 10, 73] The complexity of abduction in a set covering based rather than logic based framework has been analyzed in [9], where polynomial and NP complete abductive problems are discussed. Most recently, our work in this paper has been complemented by complexity results for abduction from logic programs in the presence of functions [47] 8 Conclusion Abduction from logic programs has become a topic of growing ....

T. Bylander, D. Allemang, M. Tanner, and J. Josephson. The Computational Complexity of Abduction. Artificial Intelligence, 49:25--60, 1991.

....discussed above immediately apply. The reflexive closure makes it possible to enter defects as observed findings, which are interpreted as already established defects, yielding a slight extension to the theory treated above. In the set theoretical formalisation of diagnosis by Bylander et al. [Bylander et al. 1992], an effects function e is used to represent both the knowledge base and the method of diagnostic problem solving. In contrast to the theory by Peng and Reggia, the function e can be used to represent diagnostic interactions among defects, because the assumption that e(D) is the union of function ....

.... et al. 1985] This system can also be described in terms of the set covering theory of diagnosis, although several aspects of the system go beyond the theory, such as the representation of interactions among particular antibody reactions, requiring a generalisation of the set covering theory [Bylander et al. 1992] (See also Section 5) Peirce is a domain independent tool that generalised on the techniques used in RED [Punch III et al. 1990] In [Tuhrim et al. 1991] an expert system for the diagnosis of brain lesions, based on the set covering theory of diagnosis is described. Of the systems mentioned ....

T. Bylander. D. Allemang, M.C. Tanner and J.R. Josephson (1992). The computational complexity of abduction. In Knowledge Representation (R.J. Brachman, H.J. Levesque and R. Reiter, eds.), 25--60. Cambridge, Massachusetts: The MIT Press.

....On the other hand, applying a particular notion of diagnosis to solve a diagnostic problem implies that a particular (diagnostic) meaning is given to the associated evidence function. The framework, which is inspired by the work on abductive diagnosis by Reggia et al. 9] and Bylander et al. [2]) differs in several respects from the diagnostic frameworks based on logic [4] 7] 10] 11] Firstly, the logical notions of diagnosis proposed in the literature have been designed in close connection with specific domain models, such as causal models or models of structure and behaviour. In ....

T. Bylander. D. Allemang, M.C. Tanner and J.R. Josephson (1992). The computational complexity of abduction. In Knowledge Representation (R.J. Brachman, H.J. Levesque and R. Reiter, eds.), pp. 25--60. Cambridge, Massachusetts: The MIT Press.

....by determining whether actually observed ndings can be predicted using the causal relation. Subsequent work has yielded several algorithms to compute set covering diagnoses eciently in practical applications [29,38,44] although this type of diagnostic reasoning is known to be NP hard in general [5]. Experimental studies of set covering theory and its variants have been performed by several researchers [21,34,41] The formal aspects of diagnosis employing causal knowledge have also been studied, using logic as the primary tool [11,13,30,32] In the logical theory of abductive diagnosis, ....

....or compute, the exponential number of function values of the evidence function e; it suces to provide only part of them explicitly. Any algorithm for diagnosis using an evidence function of the form discussed in the previous section, without simplifying assumptions, will be intractable. In [5], in which the complexity of algorithms for abductive diagnosis is analysed, it is therefore assumed that the speci cation of a domain theory is polynomial in the sum of the cardinalities of the sets and . A partial speci cation of an evidence function e consists of a restriction of e, denoted ....

[Article contains additional citation context not shown here]

T. Bylander. D. Allemang, M.C. Tanner and J.R. Josephson, The computational complexity of abduction, in: R.J. Brachman, H.J. Levesque and R. Reiter, eds., Knowledge Representation (The MIT Press, Cambridge, Massachusetts, 1992) 25-60.

....diagnosis problems by abductive inference. Pirri and Pizzuti [PP90] have also combined the diagnosis problem with the stable model semantics which is one of the semantics of logic programming. Konolige [Kon92] has introduced a causal theory, and compared it with abduction. Bylander et al. [BATJ91] has formulated abduction in order to analyze the computational complexity of abduction for propositional logic and for the diagnosis problem. In computational linguistics, Hobbs et al. HSME88] has introduced abduction in order to interpret natural language. Stickel [Sti91] has also investigated ....

....explanation. However, we can regard their abduction as the extension of Poole s abduction [Ino92] Konolige [Kon92] has investigated the relationship between abduction and the diagnosis problem by introducing a causal theory. We can regard it as the extension of Poole s abduction. Bylander et al. [BATJ91] have introduce the another framework of abduction for propositional logic. They have extended the symbol of logical implication to the causal relation, and analyze the computational complexity of abduction and the diagnosis problem. We can also regard it as the extension of Poole s abduction ....

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Bylander, T., Tanner, M. C., Allemang, D. and Josephson, J. R.: The computational complexity of abduction, Artificial Intelligence 49, 25--60, 1991.

....limited to solutions that are acceptable according to some criteria. Different concepts of acceptable solutions will be considered below. There are several variants and refinements of the set covering approach to abduction. For instance, a plausibility order may be attached to subsets of H, cf. [6]. Furthermore, it may be useful to require that an abduction problem is independent, i.e. the function e satisfies e(X) S h2H e(fhg) or that an abduction problem is monotonic, i.e. e satisfies 8X;Y H : X Y ) e(X) e(Y ) 6] The set covering model is best suited when the relationships ....

.... a plausibility order may be attached to subsets of H, cf. 6] Furthermore, it may be useful to require that an abduction problem is independent, i.e. the function e satisfies e(X) S h2H e(fhg) or that an abduction problem is monotonic, i.e. e satisfies 8X;Y H : X Y ) e(X) e(Y ) [6]. The set covering model is best suited when the relationships between causes and effects are simple such that they can be easily made explicit in the form of a function. Probabilistic abduction models the hypotheses in H and the manifestations in M as events. In addition to structural ....

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T. Bylander, D. Allemang, M. Tanner, and J. Josephson. The computational complexity of abduction. Artificial Intelligence, 49:25--60, 1991.

.... reasoning to identify classes of limited inference that can be performed efficiently (e.g. see [ Frisch and Allen, 1982 ] Brachman and Levesque, 1984 ] Patel Schneider, 1985 ] Dowling and Gallier, 1984 ] Levesque, 1988 ] Selman and Levesque, 1989 ] McAllester, 1990 ] Bylander et al. 1991 ] Kautz and Selman, 1991 ] This work has covered a wide band of the complexity spectrum but none that meets the strong tractability requirement discussed above. Most results stipulate polynomial time complexity, restrict inference in implausible ways (e.g. by excluding chaining of rules) ....

T. Bylander, D. Allemang, M. C. Tanner, J. R. Josephson. The computational complexity of abduction. Artificial Intelligence, 47(13) , 25--60.

....confine ourselves to inferences of particular statements, which is no real restriction, since our topic is abductive inference anyway. 5 A GENERAL MODEL OF ABDUCTIVE INFERENCE In this section we introduce a general model of abductive inference incorporating hypothesis assessment that is based on [Bylander et al. 1991] , but also closely related to [Peng and Reggia, 1989] Since this model cannot be implemented directly (it would require too much storage space) we have to look for simplifications, which finally lead us to probabilistic networks. 5.1 Formal Definition We start by giving a formal definition of ....

....may be used with several different sets D obs and therefore it is reasonable to consider a general representation that is independent of D obs . However, it has to be admitted that without a set of observations to be explained, there is no problem to be solved, and therefore, in contrast to [Bylander et al. 1991] , we chose to add it to the definition. Before we can proceed further, we have to say a few words about the interpretation of subsets of the sets D all and H all , for instance, the interpretation of D obs . Given a set D D all , we assume that all statements contained in D must hold, i.e. D ....

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T. Bylander, D. Allemang, M. C. Tanner and J. R. Josephson. The computational complexity of abduction. Artificial Intelligence 49: 25--60. North-Holland, Amsterdam, 1991.

.... reasoning having fruitful applications in a number of areas such diverse as model based diagnosis [46, 45, 16] speech recognition [30] maintenance of database views [34] and vision [12] Various formalizations of abductive reasoning have been proposed, among which set covering based approaches [43, 7] and logic based approaches [46, 45, 15, 16] are well known. These two types basically di er A short and preliminary version of this paper appeared in the Proceedings of the Fourteenth International Joint Conference on Arti cial Intelligence (IJCAI 95) C. Mellish ed, pp. 870 876, AAAI Press, ....

.... extensions are excluded and, for the latter problems, if default theories are in addition normal [27, 59] Cases of lower complexity and tractable fragments were identi ed in [35, 58] The complexity of logic based abduction from classical theories has been analyzed in [57, 6, 22] cf. [7] for a comprehensive analysis of the set covering approach) Basically, abduction has the same complexity as default reasoning, and bears P 2 complete and P 2 complete reasoning problems. 3 Formalizing default abduction In this section, we describe a basic formal model for abduction ....

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T. Bylander, D. Allemang, M. Tanner, and J. Josephson. The computational complexity of abduction. Articial Intelligence, 49:25-60, 1991.

....utilized abduction as the problem solving strategy for natural language understanding was created by Hobbs et al. 1990) They use predicate logic as their form of knowledge representation and abduce over the propositions and predicates. Their abductive inference is computationally intractable [Bylander et al. 1990]. Abduction has been implemented in many ways. In general, the hypothesis that explains a datum that no other hypothesis can explain must be included in the composite; it is essential to the explanation. Next, the hypothesis that explains a datum better than other hypothesis and with greater ....

T. Bylander, D. Allemang, M. Tanner and J. Josephson. The Computational Complexity of Abduction. Tech. Report. OSU.

....are composed of one observable literal and few if any achievable literals. As such, the number of minimal tests generated as abductive explanations is unlikely to be exponential in the number of observables and achievables. Selman (Selman, 1990) Levesque (Levesque, 1989) and Bylander et al. (Bylander et al. 1991) have all defined classes of tractable abductive reasoning problems. There are some gaps in the complexity results that need to be filled in to deal fully with test generation, however from the existing results we can gain some insight into what makes test generation problems tractable, or for ....

....forbidden pairs. In this instance, the forbidden pairs are mutually incompatible assumptions drawn from our assumption set. It would appear that if we got rid of the problem of forbidden pairs, that the complexity problem would be resolved. This indeed appears to be the case. Bylander et al. (Bylander et al. 1991) define the class of independent abduction problems. This class of problems has a polynomial time algorithm for finding an explanation, if one exists. The trick is to get rid of Selman s forbidden pairs to ensure that no assumptions are mutually incompatible in the one instance and to then ....

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T. Bylander, D. Allemang, M. Tanner and J. Josephson (1991). The computational complexity of abduction, Artificial Intelligence Journal 49:25--60.

....such as the diagnosability of the system under investigation. Introduction The definition of frameworks for characterizing diagnosis attracted a lot of attention in the model based reasoning community (see e.g. chapt. 2 of (Hamscher, Console, de Kleer 1992) or (Brusoni et al. 1998; Bylander et al. 1991; de Kleer, Mackworth, Reiter 1992; Lucas 1998) These frameworks have been used to provide a semantics for diagnostic problem solving and to analyze its properties, e.g. its computational complexity. Moreover, they have been used to analyze properties of the system to be diagnosed, e.g. ....

Bylander, T.; Allemang, D.; Tanner, M.; and Josephson, J. 1991. The computational complexity of abduction.

....a middle ground. Approximate reasoning is interesting for several reasons. First of all, most AI problems (tasks) are hard problems in term of their complexity measure. Planning, diagnosis and configuration are all examples of AI tasks for which even simple varieties are already intractable (e.g. [Bylander et al. 1991]) Therefore it is necessary to look for cheaper but approximate solutions instead of intractable precise solutions. Secondly, it depends on the particular application of the problem type (e.g. design, diagnosis) whether a precise solution is actually needed or whether an approximate solution ....

T. Bylander, D. Allemang, M. Tanner, and J. Josephson. The computational complexity of abduction. Artificial Intelligence, 49:25--60, 1991.

....derives from the happy circumstance that a diagnosis can be computed eciently. A single, for example, subset maximal diagnosis D that disregards no relevant disorder, can be computed in polynomial time, where the original abductive problem, in which alternative diagnoses are computed, is NP hard [Bylander et al. 1992]. Still, a disadvantage of this adapted abductive model of diagnosis is that a disorder may be ruled out by the assumption of the absence of certain ndings, included in the set F n , which for certain medical domains might place too much emphasis on assumed negative ndings. Doing away with the ....

Bylander, T, Allemang, D, Tanner, MC and Josephson, JR, 1992. \The computational complexity of abduction", In: RJ Brachman, HJ Levesque and R Reiter (eds.), Knowledge Representation, pp 25-60. The MIT Press, Cambridge, Massachusetts.

....On the other hand, applying a particular notion of diagnosis to solve a diagnostic problem implies that a particular (diagnostic) meaning is given to the associated evidence function. The framework, which is inspired by the work on abductive diagnosis by Reggia et al. 9] and Bylander et al. [2]) di ers in several respects from the diagnostic frameworks based on logic [4, 7, 10, 11] Firstly, the logical notions of diagnosis proposed in the literature have been designed in close connection with speci c domain models, such as causal models or models of structure and behaviour. In ....

T. Bylander. D. Allemang, M.C. Tanner and J.R. Josephson (1992). The computational complexity of abduction. In Knowledge Representation (R.J. Brachman, H.J. Levesque and R. Reiter, eds.), pp. 25-60. Cambridge, Massachusetts: The MIT Press.

....discussed above immediately apply. The re exive closure makes it possible to enter defects as observed ndings, which are interpreted as already established defects, yielding a slight extension to the theory treated above. In the set theoretical formalisation of diagnosis by Bylander et al. [Bylander et al. 1992], an e ects function e is used to represent both the knowledge base and the method of diagnostic problem solving. In contrast to the theory by Peng and Reggia, the function e can be used to represent diagnostic interactions among defects, because the assumption that e(D) is the union of function ....

.... et al. 1985] This system can also be described in terms of the set covering theory of diagnosis, although several aspects of the system go beyond the theory, such as the representation of interactions among particular antibody reactions, requiring a generalisation of the set covering theory [Bylander et al. 1992] (See also Section 5) Peirce is a domain independent tool that generalised on the techniques used in RED [Punch III et al. 1990] In [Tuhrim et al. 1991] an expert system for the diagnosis of brain lesions, based on the set covering theory of diagnosis is described. Of the systems mentioned ....

T. Bylander. D. Allemang, M.C. Tanner and J.R. Josephson (1992). The computational complexity of abduction. In Knowledge Representation (R.J. Brachman, H.J. Levesque and R. Reiter, eds.), 25-60. Cambridge, Massachusetts: The MIT Press.

....based in predicates and extensions of predicate logic. Here we note a large gap, as far as rhematic knowledge is concerned. Normally, the emphasis is on dicent knowledge using symbolic propositions only. In general terms, abduction is not explored, with a few exceptions (Ram Leake, 1991; Bylander et.al. 1991), mainly due to computational complexity problems (Bylander et.al. 1991) Recently, contributions to the study of arguments came from the field known as computational intelligence (Zurada et.al, 1994) Three main areas can be distinguished: fuzzy logic and fuzzy systems, neural networks and ....

....note a large gap, as far as rhematic knowledge is concerned. Normally, the emphasis is on dicent knowledge using symbolic propositions only. In general terms, abduction is not explored, with a few exceptions (Ram Leake, 1991; Bylander et.al. 1991) mainly due to computational complexity problems (Bylander et.al. 1991). Recently, contributions to the study of arguments came from the field known as computational intelligence (Zurada et.al, 1994) Three main areas can be distinguished: fuzzy logic and fuzzy systems, neural networks and evolutive computing. It is interesting to note the similarity among these ....

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25-60.

....as a result of the experience of an expert. For many problems which are completely specifiable it is not possible to find an efficient algorithmic solution. Such problems are easy to specify but it is not necessarily possible to derive an efficient algorithm from these specifications (cf. [32], 110] domainspecific heuristics or domain specific inference knowledge is needed for the efficient derivation of a solution. In simple terms this means that analysis is not simply interested in what happens, as in conventional systems, but also with how and why [31] One must not only ....

T. Bylander, D. Allemang, M.C. Tanner, and J.R. Josephon, The Computational Complexity of Abduction, Artificial Intelligence 49, 1991.

....of ground clauses of L. A clause based strong a . p. is that of finding a primitive explanation F A of an observation O W such that l F, that is F is consistent with T S, l O, l F is subset minimal. The last point of the definition embodies a selection criterion for good explanations that in [Allemang 91] is called parsimony . It prevents the choice, as explanations, of sets of clauses containing a proper subset that itself constitutes a valid explanation. If A is restricted to be a set of atomic sentences (clauses made of a single, positive literal) and if S= then this clause based a. p. ....

Tom Bylander, D. Allemang, M. C. Tanner and J. R. Josephson, The computational complexity of abduction, Artificial Intelligence, 49,25-60, 1991.

....multiple world test suite generation is abduction [36] and abduction is slow. Selman Levesque show that even when only one abductive explanation is required and the theory is restricted to be acyclic, then abduction is NP hard [103] Bylander et al. make a similar pessimistic conclusion [13]. Computationally tractable abductive inference algorithms (e.g. 13,36] typically 255 make restrictive assumptions about the nature of the theory or the available data. Such techniques are not applicable to arbitrary theories. Therefore, it is reasonable to question the practicality of ....

....is slow. Selman Levesque show that even when only one abductive explanation is required and the theory is restricted to be acyclic, then abduction is NP hard [103] Bylander et al. make a similar pessimistic conclusion [13] Computationally tractable abductive inference algorithms (e.g. [13,36]) typically 255 make restrictive assumptions about the nature of the theory or the available data. Such techniques are not applicable to arbitrary theories. Therefore, it is reasonable to question the practicality of multiple world test suite generation for medium to large theories. Hence the ....

T. Bylander, D. Allemang, M.C. M.C. Tanner, and J.R. Josephson. The Computational Complexity of Abduction. Artificial Intelligence, 49:25--60, 1991. 1395

....mel true. In total, this results in n 2 =4 n=2 1 calls to test. Hence the worst case running time of a mel is Omega Gamma N 2 ) Algorithm a mel is worst case running time of Theta(N 2 ) 2 The recursive descent method employed by a mel is certainly not a new observation. Bylander et al. [3] show this algorithm for certain abduction problems, closely related to the MEL problem here, and they show that it is O(N 2 ) to find an answer. 3.3 Finding all MELs It is still not clear, however, what is the best search strategy for finding all the MELs of a set (with respect to a monotonic ....

....; A Gamma j= C. The direction C 2 SAT hS; T i 2 enumerate (MEL, n 1) follows in a similar manner. 2 The above theorem was devised independently to address the MFS problem. It should be noted that a similar theorem for a given class of abduction problems is presented by Bylander et al. in [3]. The proof of that theorem follows in a very similar manner. The reason we restrict the domain of MEL in the above theorem to pairs hS; T i such that jT j t (jSj) is to show that the MEL enumeration problem is intractable with respect to the size of the input set S, not just with respect to the ....

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T. Bylander, D. Allemang, M. C. Tanner, and J. R. Josephson. The computational complexity of abduction. Artificial Intelligence, 49:25--60, 1991.

....observations of the entity being modeled in the KB. Based on work by Feldman Compton [Feldman et al. 1989] a general validation framework based on the HT4 abductive inference engine has been developed. Elsewhere [Menzies, 1996a] we have given an overview of abductive research [O Rourke, 1990,Bylander et al. 1991,Eshghi, 1993,Selman Levesque, 1990,Poole, 1990b,Menzies, 1996a] Here, we offer an approximate characterisation of abduction as the search for consistent subsets of some background theory that are relevant for achieving some goal. If multiple such subsets can be generated, then a BEST ....

Bylander, T., Allemang, D., M.C. Tanner, M., & Josephson, J. (1991). The Computational Complexity of Abduction. Artificial Intelligence, 49:25--60.

....2. 5 Practicality Abduction has a reputation of being impractically slow [8] Selman Levesque show that even when only one abductive explanation is required and T is restricted to an acyclic theories, then abduction is NP hard [42] Bylander et al. make a similar pessimistic conclusion [3]. In practice these theoretical restrictions may not limit application development. Ng Mooney report reasonable runtimes using a beam search proof level assessment operator [29] Figure 1 shows the average runtime in sections for executing HT4 using worlds level assessment (BEST 4 ) over 94 ....

T. Bylander, D. Allemang, M.C. M.C. Tanner, and J.R. Josephson. The computational complexity of abduction. Artificial Intelligence, 49:25--60, 1991.

....one may wish to maximize some preference criterion, defined over all such sets. For example, one that minimizes risk or cost, maximizes the likelihood of success, etc. Presented at this level of abstraction, this problem resembles many abduction problems, e.g. abductive explanation and diagnosis [129, 119, 11, 28, 97]. Set Covering [78, 75, 16] is an abstract mathematical problem which was previously used to formalize abductive reasoning (e.g. 129, 4] and which I next use to formalize the means selection sub task. Definition 4.2.1 Goal Procedure Mapping Let goals def = f g i g n i=1 be a set of goals, ....

Bylander, T., Allemang, D., Tanner, M. C., and Josephson, J., The Computational Complexity of Abduction. Artificial Intelligence 49, (1-3), pp. 25-60, 1991.

....is proportional to the number of paths and the number of world defining assumptions; i.e. EQ 3 :O(jPj jENVj) O(X F jENVj) 4.3 In Practice The exponential complexity of abductive validation is not surprising. Abductive validation is a variant on abduction and abduction is known to be NPhard [BAMTJ91]. Nevertheless, it has been shown that abductive validation is practical for many real world theories such as certain fielded expert systems and theories from neuroendocrinology [Men96b] Feldman Compton [FCS89] followed by Menzies [MC97] have shown that abductive validation engines could ....

T. Bylander, D. Allemang, M.C. M.C. Tanner, and J.R. Josephson. The Computational Complexity of Abduction. Artificial Intelligence, 49:25--60, 1991.

....model which can explain the largest subset of known behaviour. Given a model with E edges, then there are 2 E possible subsets; i.e. the number of subsets varies exponentially with model connectivity. For more formal proofs of the NP hard nature of abduction, see (Selman and Levesque 1990; Bylander, Allemang et al. 1991). An interesting feature of abduction is that this worst case behaviour is often the usual case: most known abductive inference engines exhibit exponential runtimes for real world inputs, even for sophisticated algorithms (e.g. the ATMS (Selman and Levesque 1990) Hence, many of the articles in ....

....the articles in (O Rourke 1990) are concerned with heuristic optimisations of abduction. Eshghi report a class of polynomial time abductive inference problems, but this class of problems requires at least a non cyclic and or graph. Eshghi 1993) Bylander reports techniques for tractable abduction (Bylander, Allemang et al. 1991), but many of these techniques (e.g. rule out knowledge to cull much of the search space) are not applicable to hypothetical models in poorly measured domains (e.g. neuroendocrinology) Early versions of our generalised test used a basic chronological backtracking approach (i.e. no memoing) that ....

Bylander, T., D. Allemang, M.C. Tanner and J.R. Josephson (1991). The Computational Complexity of Abduction.

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Tom Bylander, Dean Allemang, Michael C. Tanner, and John R. Josephson. The computational complexity of abduction. Artificial Intelligence, 49:25--60, 1991.

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Tom Bylander et al. The computational complexity of abduction. In John R. Josephson and Susan G. Josephson, editors. Abductive Inference: Computation, Philosophy, Technology.Cam- bridge: Cambridge University Press, 1994.

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Tom Bylander et al. The computational complexity of abduction. In John R. Josephson and Susan G. Josephson, editors. Abductive Inference: Computation, Philosophy, Technology. Cambridge: Cambridge University Press, 1994.

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Bylander T, Allenmang D, Tanner M. C, and Josephson J. R, "The Computational Complexity of Abduction", Artificial Intelligence, 49, 1991, 25-60

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Bylander T, Allenmang D, Tanner M. C, and Josephson J. R, The Computational Complexity of Abduction, Artificial Intelligence, Vol.49, 1991, pp.25-60.

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T. Bylander, D. Allemang, M. C. Tanner, and J. R. Josephson. The computational complexity of abduction. Artif. Intell. , 49:25--60, 1991.

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T. Bylander, D. Allenmang, M. C. Tanner and J. R. Josephson, The Computational Complexity of Abduction, Artificial Intelligence, 49, pp.25-60, 1991.

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T. Bylander, D. Allemang, M.C.M.C. Tanner, and J.R. Josephson. The Computational Complexity of Abduction. Artificial Intelligence, 49:25-60, 1991.

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T. Bylander. D. Allemang, M.C. Tanner and J.R. Josephson (1992). The computational complexity of abduction. In Knowledge Representation (R.J. Brachman, H.J. Levesque and R. Reiter, eds.), 25-60. Cambridge, Massachusetts: The MIT Press.

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Bylander, Tom, Dean Allemang, Michael C. Tanner, and John R. Josephson. \The Computational Complexity of Abduction." Articial Intelligence, 49, pp.25-60, May 1991.

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T. Bylander, M.C. Allemang, M.C. Tanner and J.R. Josephson, "The computational complexity of abduction" Artificial Intelligence, 49, 25-60, (1991).

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