B. Selman and H. Kautz. Knowledge compilation and theory approximation. J. ACM, 43:193--224, 1996.  Home/Search   Document Details and Download   Summary   ACM   TOC   Related Articles   Check

....takes a domain theory and a library of procedure templates as input. The output from TOPS is a modi ed domain theory and a set of procedure instances. We refer to the execution of TOPS as theory compilation, a process that is similar to partial deduction [6] 4] and knowledge compilation [5] [13]. Amphion uses the modi ed theory and executes procedures to synthesize programs to solve problems in the NAIF domain. We refer to this as program synthesis, to differentiate the execution of TOPS from the execution of TOPS created procedures. Axioms captured (implied) by the TOPS created ....

B. Selman and H. Kautz, \Knowledge Compilation and Theory Approximation," JACM, Vol. 43, No. 2, March 1996, pp. 193-224.

....relation M 2 . This decision in turn provides us with an approximate result on deciding whether x is a member of c 1 , based on the following observation: If x is member of a lower bound of c 1 then it is also in c 1 . If x is not member of all upper bounds of c 1 then it is not in c 1 In [Selman and Kautz, 1996] Selman and Kautz propose to use this observation about upper and lower boundaries for theory approximation. We adapt the proposal for defining an approximate classifier M # : I 2 # 0, 1, in the following way: Definition 7.6 (Approximate Re Classification) Let IS 1 = sources ....

....9.2.2 Approximate Terminological Reasoning Closely related to the question of mapping, merging and translation is the use of approximate reasoning techniques. In chapter 7 we presented a specific method for approximating class definitions and concept queries based on the work of Selman and Kautz [Selman and Kautz, 1996]. Beyond this work, there is significant work on approximate reasoning in a logical setting that could be used in terminological reasoning as well. Approximation Techniques Knowledge Compilation In order to avoid complexity at run time, knowledge compilation [Darwiche and Marquis, 2001] aims at ....

Selman, B. and Kautz, H. (1996). Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224.

....provides us with an approximate result on deciding whether x is the result of a query stated in terms of a private ontology, based on the following observation: If x is member of a lower bound of c1 then it is also in . If x is not member of all upper bounds of c1 then it is not in c1 In [15] Selman and Kautz propose to use this observation about upper and lower boundaries for theory approximation. We adapt the proposal for defining an approximate classifier M # that assigns members of shared concepts to private ones in the following way: Definition 4 (Concept Approximation) Let C1 ....

B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, March 1996.

....C # to C is defined by: #glbC . #lubC . c c, otherwise The classification problem can be decided using the semantics of the terminological language introduced in definition 3. In many cases this approximate classification which is based on the ideas described in [Selman and Kautz, 1996] will provide us with a definite result stating that a web page belongs to the class c or not. In the cases where not definite decision is made, we either have to live with incompleteness or we have to extend our vocabulary towards the names of the classes those pages we could not classify belong ....

Selman, B. and Kautz, H. (1996). Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224.

....such a programme. Furthermore, a detailed study of the complexity of nested circumscription in the first order case and restricted fragments (monadic theories, etc) would be interesting. Complementing the results on reasoning complexity, Cadoli et al. 7, 6] Gogic et al. 26] Selman and Kautz [46], Darwiche and Marquis [14, 13] and others have studied representability issues among KR formalisms, considering problems like representing theories in one KR formalism with polynomial resources in another target formalism, such that the set of models or certain inference relations are preserved. ....

B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193-- 224, 1996.

....of conjunctive queries (hybrid with ALN ) using ALN views. Existing work related to approximation in the first order case has dealt with either approximating concept descriptions by simpler concept descriptions ( 8, 2, 7] or with approximating sets of clauses by tractable sets of clauses ([17, 11]) In [8] concept descriptions of ALE (respectively of ALC) are approximated by sequences of simpler concept descriptions of the same description logic, i.e. ALE (respectively ALC) In [2] lower approximations of concept descriptions are defined w.r.t a given description logic and a ....

B. Selman and H. Kautz, `Knowledge compilation and theory approximation', Journal of the ACM, 43(2), (1996).

....acyclic graphs (such as OBDDs) which we show to include a relatively large number of target compilation languages. 1 Introduction Knowledge compilation has emerged recently as a key direction of research for dealing with the computational intractability of general propositional reasoning [9, 5, 2, 20, 33, 32, 25, 14, 12, 30]. According to this direction, a propositional theory is compiled off line into a target language, which is then used on line to answer a large number of queries in polytime. The key motivation behind knowledge compilation is to push as much of the computational overhead into the off line phase, ....

.... A classical result in knowledge compilation states that it is not possible to compile any propositional formula # into a polysize data structure # such that: # and # entail the same set of clauses, and clausal entailment on # can be decided in time polynomial in its size, unless NP # P poly [33, 5]. This last assumption implies the collapse of the polynomial hierarchy at the second level [19] which is considered very unlikely. We use this classical result from knowledge compilation in some of our proofs of Proposition 3.1, which explains why some of its parts are conditioned on the ....

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B. Selman and H.A. Kautz. Knowledge compilation and theory approximation. Journal of ACM, 43:193--224, 1996.

....of target compilation languages. 1 Introduction Knowledge compilation has emerged recently as a key direction of research for dealing with the computational intractability of general propositional reasoning [Darwiche, 1999a; Cadoli and Donini, 1997; Boufkhad et al. 1997; Khardon and Roth, 1997; Selman and Kautz, 1996; Schrag, 1996; Marquis, 1995; del Val, 1994; Dechter and Rish, 1994; Reiter and de Kleer, 1987] According to this direction, a propositional theory is compiled off line into a target language, which is then used on line to answer a large number of queries in polytime. The key motivation behind ....

....= L 2 indicates that L 1 and L 2 are equally succinct, L 1 L 2 and L 2 L 1 . Dotted arrows indicate unknown relationships. polysize data structure such that: and entail the same set of clauses, and clausal entailment on can be decided in time polynomial in its size, unless NP P poly [Selman and Kautz, 1996; Cadoli and Donini, 1997] This last assumption implies the collapse of the polynomial hierarchy at the second level [Karp and Lipton, 1980] which is considered very unlikely. We use this classical result from knowledge compilation in some of our proofs of Proposition 3.1, which explains why ....

Bart Selman and Henry Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, 1996.

....of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606 8501, Japan. ibaraki i.kyoto u.ac.jp) x Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560 8531, Japan. makino sys.es.osaka u. ac.jp) e.g. [21, 22, 19, 20, 4, 5, 1]. One of these approaches [21, 22] uses a greatest Horn lower bound (also called Horn core [15] which is a maximal Horn theory c , if we view theories as sets of models, and the least Horn upper bound (Horn envelope [15] which is the minimal Horn theory e such that logically ....

....of Informatics, Kyoto University, Kyoto 606 8501, Japan. ibaraki i.kyoto u.ac.jp) x Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560 8531, Japan. makino sys.es.osaka u. ac.jp) e.g. 21, 22, 19, 20, 4, 5, 1] One of these approaches [21, 22] uses a greatest Horn lower bound (also called Horn core [15] which is a maximal Horn theory c , if we view theories as sets of models, and the least Horn upper bound (Horn envelope [15] which is the minimal Horn theory e such that logically implies e . Note that in general, ....

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B. Selman and H. Kautz. Knowledge Compilation and Theory Approximation. Journal of the ACM, 43(2):193-224, 1996. 24

....Horn theory R which includes , i.e. logically implies R. While there is more than one Horn core in general, the Horn envelope is always unique. Semantical and computational issues on Horn cores and the Horn envelope have been studied extensively, and a number of results have been obtained, cf. [21, 16, 2, 17, 18, 3, 4, 10, 1]. The main results of the present paper can be summarized as follows. We present characterizations of the Horn cores and the Horn envelope of a Horn di erence 1 n 2 , which will form a basis of the algorithms discussed in this paper. We either present a polynomial time algorithm or prove ....

....and model based representations complement each other with respect to their tractability intractability pro le. Our results on the di erence of Horn theories have applications in di erent contexts. On one hand, we extend the results on propositional knowledge base (KB) approximation in [16, 2, 10, 1], by providing a polynomial time algorithm for computing some Horn core of a KB which is expressible as the di erence of two Horn theories. This should be compared with the general result [2] that computing a Horn core of a KB described by an arbitrary CNF formula is not possible in polynomial ....

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H. Kautz and B. Selman. Knowledge Compilation and Theory Approximation. Journal of the ACM, 43(2):193-224, 1996.

.... Sigma 1 and Sigma 2 , i.e. Sigma = Sigma 1 n Sigma 2 . In general, the resulting theory Sigma is not Horn. Therefore, we consider approximating Sigma by a Horn theory, in order to maintain the desired closure property. Different such approximations are possible, among which Horn cores [11, 12] are quite natural. A Horn theory Pi f0; 1g n is a Horn core (or a Horn greatest lower bound) of a theory Sigma f0; 1g n , if Pi Sigma , i.e. Pi logically implies Sigma , and there is no weaker Pi of this property, i.e. no Horn Pi 0 exists such that Pi ae Pi 0 Sigma . ....

....2 co NP complete P Compute the Horn not polynomial total envelope of Sigma 1 n Sigma 2 time unless P = NP P cores Sigma 1 = f(110)g and Sigma 2 = f(101)g, respectively. Approximating a propositional logic theory by Horn theories (or Horn cores) is used for knowledge compilation in [11]. Because of its theoretical and practical importance, semantical and computational issues on Horn cores have been studied extensively, and a number of results have been obtained, cf. 11, 2, 12, 7, 1] The main contributions of the present paper can be summarized as follows. ffl We present ....

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H. Kautz and B. Selman. Knowledge Compilation and Theory Approximation. JACM, 43(2):193--224, 1996.

....in [18] proposes a prime implicants generation algorithm which is unsound but complete with respect to exact compilation. An analogous strategy has been proposed by Selman and Kautz in the context of Horn approximation for computing all the greatest lower bounds (GLB) of a clausal knowledge base [20]. The purpose of the paper is to introduce a unifying, logic oriented framework which captures the main ideas of these two approaches. Our investigation generalizes and expands in several directions previous results by Cadoli and Schaerf in [2, 17] The framework is based on multi modal logic ....

B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, 1996.

....combined information. In the context of related work, our study was mainly inspired from two currently disjoint research fields. The first is concerned by tractable approaches of knowledge representation. Standard techniques such as language restriction (e.g. 23] or knowledge compilation (e.g. [28]) are ill suited in this setting. Notably, they use classical logic and thus become fragile when contradictions appear within a pool of combined information. Furthermore, the technique of language restriction is not adequate for modeling agents that need at least the expressiveness of first order ....

B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, 1996.

....to knowledge compilation (surveyed in [Cadoli and Donini, 1997] Sometimes we will be interested in theories which are logically equivalent to P I LT ( but which need not include all L T implicates, and can thus be much more concise. We refer to any such theory as a L T LUB of , following [Selman and Kautz, 1996] , see also [del Val, 1995] We ll see one particular application of L T LUBs in our discussion of diagnosis later. 2 Kernel resolution: Review Kernel resolution, described in [del Val, 1999; 2000a] is a consequence finding generalization of ordered resolution. We assume a total order of the ....

B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, 1996.

....we present polynomial time algorithms for bounded disjunctions in the formula based case. 1 Introduction Since deduction from a set of propositional clauses is a well known co NP complete problem, different approximation methods for reasoning from a clausal theory have been investigated, e.g. [10, 13, 3, 4, 1]. One of these approaches [10] uses a Horn greatest lower bound (also called Horn core [11] i.e. a maximal Horn theory c , if we view theories as sets of models, and the least Horn upper bound (Horn envelope [11] which is the minimal Horn theory e , i.e. e 0 for any Horn ....

....bounded disjunctions in the formula based case. 1 Introduction Since deduction from a set of propositional clauses is a well known co NP complete problem, different approximation methods for reasoning from a clausal theory have been investigated, e.g. 10, 13, 3, 4, 1] One of these approaches [10] uses a Horn greatest lower bound (also called Horn core [11] i.e. a maximal Horn theory c , if we view theories as sets of models, and the least Horn upper bound (Horn envelope [11] which is the minimal Horn theory e , i.e. e 0 for any Horn theory 0 such that 0 ....

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H. Kautz and B. Selman. Knowledge Compilation and Theory Approximation. JACM, 43(2):193--224, 1996.

....as the complexity is decreased. As an example, starting from a PSPACE complete problem, it could make sense to have just an on line NP complete problem to deal with. 1.1. State of the Art During the last few years many researchers in AI have introduced various forms of knowledge compilation [36, 37] or o# line reasoning [32] Specifically, they focused on the problem of deciding whether a set of propositional clauses logically entails a clause. As an example, Moses and Tennenholtz [32] show that, under some non trivial restrictions on the query language, it is possible to modify o# line the ....

B. Selman and H. A. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43:193--224, 1996.

....that, unless there is a collapse in the polynomial hierarchy, the size of the smallest representation of the least Horn upper bound of a propositional theory is superpolynomial in the size of the original theory. These results are also presented in a di#erent form in the more comprehensive paper (Selman Kautz, 1996). The technique used in the proof has been then used by us and other researchers to prove several other results on the relative complexity of propositional knowledge representation formalisms (Cadoli et al. 1996, 1997, 1999; Gogic et al. 1995) In a recent paper (Cadoli et al. 1996b) we ....

Selman, B., & Kautz, H. A. (1996). Knowledge compilation and theory approximation. Journal of the ACM, 43, 193--224.

....compilation (surveyed in (Cadoli and Donini 1997) Sometimes we will be interested in theories which are logically equivalent to P I LT (#) but which need not in clude all L T implicates, and can thus be much more concise. We refer to any such theory as a L T LUB of #, following (Selman and Kautz 1996), see also (del Val 1995) We ll see one particular application of L T LUBs in our discussion of diagnosis later. After reviewing kernel resolution, we analyze its complexity for one particular exhaustive search strategy. We then present several applications of our results. Finally, we briefly ....

....answers to all or some queries, by replacing a theory by a compiled one with better complexity. However, knowledge compilation faces fundamental limits in the sense that for many query languages it is extremely unlikely that one can guarantee tractable compiled representations of polynomial size (Selman and Kautz 1996; Cadoli and Donini 1997) It seems the only way out of this hurdle is to ensure that the query language has polynomial size. Our analysis of LK consequence finding is a contribution to this goal. Extensions and refinements All of the above should be seen as a quite condensed summary of our ....

B. Selman and H. Kautz. Knowledge compilation and theory approximation. J. ACM, 43(2):193--224, 1996.

.... many applications in AI, ranging from plain query answering to a variety of tasks needed for automating common sense reasoning (see below, and also the survey [23] This can be formally characterized in terms of approximate KBs, which are supposed to improve reasoning over the restricted language [7, 8, 26]. In [7, 8] and more recently in [11] I provide e cient procedures for restricted consequence nding. The latter paper introduces kernel resolution, which despite its novelty is already treated in depth in a very recent survey [23] Recent work [13] includes the characterization of the ....

B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193-224, March 1996.

....long the complexity is decreased. As an example, starting from a PSPACE complete problem, it could make sense to have just an on line NP complete problem to deal with. 1. 2 State of the Art During the last few years many researchers in AI have introduced various forms of knowledge compilation [33, 34] or off line reasoning [31] Specifically, they mainly focused on the problem of deciding whether a set of propositional clauses logically entails a clause. As an example, in [31] Moses and Tennenholtz show that, under some non trivial restrictions on the query language, it is possible to modify ....

B. Selman and H. A. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43:193--224, 1996.

....in the affirmative, by proving such results for (at most) k Horn satisfiability, the natural parameterized version of Horn satisfiability studied in the previous chapter. This problem is also of practical interest in Artificial Intelligence, mainly in connection to theory approximation [14]. The results can be summarized as follows: 1. For an unbounded k = k(n) the threshold phenomenon is essentially the one from the uniform case k(n) n. In particular there exists a rescaled parameter that makes the graphs of the limit probabilities superimpose (Theorem 3.1) 2. For any ....

H. Kautz and B. Selman. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, 1996.

....counterexample) or f j= h but g 6j= h (a positive counterexample) When it answers No it also supplies the counterexample. We note that EnEQ(f;Q) actually checks the equivalence of f and g relative to Q. In fact, EnEQ(f;Q) checks the equivalence of the least upper bounds of g and f in Q. See (Selman and Kautz, 1996; Khardon and Roth, 1996) for definitions and discussion of least upper bounds. Familiarity with this concept is not needed in the rest of this paper. Below we define more oracles that we need to distinguish from the ones defined above in the on line learning scenario. We hence identify the ones ....

Selman, B. and H. Kautz. 1996. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224.

....range restricted and Horn) Proof: The proof adapts the technique of McKinsey (1943) to the current setting. Let T = 8X;C 1 C 2 : C s and assume that C = C 1 is not Horn, namely it has j 1 positive literals, so that C = P 1 : Pm Pm 1 : Pm j . Define j Horn Strengthening (Selman Kautz, 1996) clauses for C each including one of the positive literals of C, so that for 1 i j, C i = P 1 : Pm Pm i . We claim that for some i, T j= d C i and therefore T can be rewritten as T = 8X;C i C 2 : C s . In this way all the non Horn clauses of T can be replaced with Horn ....

....is not Horn then it will find a Horn expression which is as close to it as possible. In fact we show that the algorithm is a Learn to Reason algorithm (Khardon Roth, 1997) with respect to the class H(P ) This is formalised using the notion of least upper bounds that were introduced by Selman and Kautz (1996) and discussed by various authors (e.g. Frazier Pitt, 1993; Khardon Roth, 1996; Del Val, 1996) Definition 6. Let G; H be classes of first order expressions over the signature P . An expression T 2 H is the least upper bound of G 2 G in H, if (1) G j= T , and (2) for all T 0 2 H such that G ....

Selman, B., & Kautz, H. (1996). Knowledge compilation and theory approximation.

....used in an on line phase to answer multiple queries. The main value of such compilation is that most of the computational overhead is shifted into the off line phase, which is amortized over all on line queries. One of the key approaches for compiling propositional theories has been proposed in [ 7 ] Here, a propositional theory is compiled in an off line phase into a Horn theory, which is used in an on line phase to answer multiple queries. As it is not always possible to compile a propositional theory into a Horn theory, the propositional theory is generally compiled into two Horn ....

....to DNNF theories. Our goal in this section is two fold. First, to prove that every propositional theory can be expressed in DNNF. Second, to provide an algorithm for this purpose. The following theorem is the key to proving that every propositional theory can be converted into DNNF. Theorem 7 Let Delta 1 and Delta 2 be two propositional sentences in DNNF. Let Delta be the sentence W fi ( Delta 1 j fi) Delta 2 j fi) fi, where fi is an instantiation of all atoms shared by Delta 1 and Delta 2 . Then Delta is in DNNF and Delta is equivalent to Delta 1 Delta 2 . Here is a ....

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Bart Selman and Henry Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, March, 1996.

....DRAT hierarchy is used to classify abstract relation symbols in the domain theory. For each relation symbol classified, a procedure is selected from the DRAT library of procedures. TOPS augments the procedure by pre computing terms using partial deduction and knowledge compilation [Komorowski 92, Selman and Kautz 96] Axioms captured (implied) by the TOPS procedure are removed from the domain theory. Remaining axioms that contain terms in the language of the theory of the procedure are separated relative to this language. The six steps below outline the TOPS algorithm. 1) Each abstract relation symbol is ....

B. Selman and H. Kautz, "Knowledge Compilation and Theory Approximation ", JACM, Vol. 43, No. 2, March 1996, pp. 193-224.

.... The idea to represent knowledge in one language which is then translated into another language where all the inferences are made plays a key role in Selman and Kautz s work on knowledge compilation, which is their approach to reach computational efficiency in knowledge representation systems (cf. SK96] It has been noted in [LB85] that there exists a trade off between tractability and expressiveness in knowledge representation systems. In general, computationally efficient representation systems are either obtained by restricting the expressive power of the knowledge representation language ....

B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):97--142, 1996.

....24 98, September 1998. Copyright c fl 1999 by the authors 2 INFSYS RR 1843 99 02 1 Introduction Since deduction from a set of propositional clauses is a well known co NP complete problem, different approximation methods for reasoning from a clausal theory Sigma have been investigated, e.g. [14, 18, 3, 4, 1]. One of these approaches [14] uses a greatest Horn lower bound (also called Horn core [15] i.e. if we view theories as sets of models, a maximal Horn theory Sigma c Sigma, and the least Horn upper bound (Horn envelope [15] which is the minimal Horn theory Sigma e Sigma such that ....

.... by the authors 2 INFSYS RR 1843 99 02 1 Introduction Since deduction from a set of propositional clauses is a well known co NP complete problem, different approximation methods for reasoning from a clausal theory Sigma have been investigated, e.g. 14, 18, 3, 4, 1] One of these approaches [14] uses a greatest Horn lower bound (also called Horn core [15] i.e. if we view theories as sets of models, a maximal Horn theory Sigma c Sigma, and the least Horn upper bound (Horn envelope [15] which is the minimal Horn theory Sigma e Sigma such that Sigma logically implies Sigma e ....

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H. Kautz and B. Selman. Knowledge Compilation and Theory Approximation. Journal of the ACM, 43(2):193-- 224, 1996.

....counterexample) or f j= h but g 6j= h (a positive counterexample) When it answers No it also supplies the counterexample. We note that EnEQ(f;Q) actually checks the equivalence of f and g relative to Q. In fact, EnEQ(f;Q) checks the equivalence of the least upper bounds of g and f in Q. See (Selman and Kautz, 1996; Khardon and Roth, 1994b) for definitions and discussion of least upper bounds. Familiarity with this concept is not needed in the rest of this paper. These are, of course, not all the oracles that can be defined, but rather those oracles that will be used in the rest of this paper. For example, ....

Selman, B. and H. Kautz. 1996. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, March.

....A correct answer is also precise if also KB j= We can also consider relative measure of precision: For a fixed KB, we say that 1 is more precise than 2 if if KB j= 2 ) 1 ; in fact, this induces a partial ordering. Imprecise Logical Reasoners: Horn Approximation reasoners [SK96] are imprecise logical reasoners. Recall that, while inference from a general propositional theory is NP hard, there are efficient algorithms for reasoning from a Horn theory. Now note that we can always bound a nonHorn theory Sigma by a pair of Horn theories h S; W i, where S j= Sigma j= W , ....

Bart Selman and Henry Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 1996.

....CA 94720, U.S.A. Martha Sideri mss aueb.gr Athens University of Economics and Business Patission Street, Department of Informatics Athens 104 34, Greece Abstract Approximating a general formula from above and below by Horn formulas (its Horn envelope and Horn core, respectively) was proposed by Selman and Kautz (1991, 1996) as a form of knowledge compilation, supporting rapid approximate reasoning; on the negative side, this scheme is static in that it supports no updates, and has certain complexity drawbacks pointed out by Kavvadias, Papadimitriou and Sideri (1993) On the other hand, the many frameworks and ....

....(1986) in recent years there has been increasing interest in computational models for rapid approximate reasoning, based on a vivid (that is to say, conducive to efficient deductions) representation of knowledge. One important proposal in this regard has been the knowledge compilation idea of Selman and Kautz (1991, 1996), whereby a propositional formula is represented by its optimal upper (relaxed) and lower (strict) approximations by Horn formulas the corresponding Horn formulas are called in the present paper the Horn envelope and the Horn core of the original formula. The key c fl1998 AI Access Foundation ....

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Selman, B., & Kautz, H. A (1996). Knowledge Compilation and Theory Approximation.

....) Gamma (i.e. it is Horn) Proof: The proof adapts the technique of McKinsey (1943) to the current setting. Let T = 8X; C 1 C 2 : C s and assume that C = C 1 is not Horn, namely it has j 1 positive literals, so that C = P 1 : Pm Pm 1 : Pm j . Define j Horn Strengthening (Selman Kautz, 1996) clauses for C each including one of the positive literals of C, so that for 1 i j, C i = P 1 : Pm Pm i . We claim that for some i, T j= d C i and therefore T can be rewritten as T = 8X; C i C 2 : C s . In this way all the non Horn clauses of T can be replaced with Horn ....

....not Horn then it will find a Horn expression which is as close to it as possible. In fact we show that the algorithm is a Learn to Reason algorithm (Khardon Roth, 1997) with respect to the class HD (P ) Gamma . This is formalised using the notion of least upper bounds that were introduced by Selman and Kautz (1996) and discussed by various authors (e.g. Frazier Pitt, 1993; Khardon Roth, 1996; Del Val, 1996) Definition 5 Let G; H be classes of first order expressions over the signature P . An expression T 2 H is the least upper bound of G 2 G in H, if (1) G j= T , and (2) for all T 0 2 H such that G ....

Selman, B., & Kautz, H. (1996). Knowledge compilation and theory approximation. Journal of the ACM, 43 (2), 193--224.

....al. 1995; Gogic et al. 1995) c) propositional belief revision: Cadoli, Donini, Schaerf 1995) 4. Methods for approximate compilation (many of them with experimental results) a) Definitions: i. Horn lower upper bounds: Selman Kautz 1991) ii. General lower upper bounds: Kautz Selman 1991; Selman Kautz 1996; del Val 1995) iii. Bootstrapped approximate KC: Schrag 1996) b) Properties (computational, semantical) Kautz Selman 1992; Cadoli 1993; del Val 1995) 5. Others: Levesque 1986; Borgida et al. 1989) vivid reasoning) Schrag Crawford 1996) experimental evaluation) Gogic, Papadimitriou, ....

Selman, B., and Kautz, H. A. 1996. Knowledge compilation and theory approximation. J. of the ACM. In press.

....the goal is to transform a knowledge base into a vivid form. However, in order to be feasible, the output of the off line preprocessing must have polynomial size wrt the fixed part. State of the Art. In the last years many researchers in AI have introduced various forms of knowledge compilation [17, 18] or off line reasoning [15] Specifically, they have focused on the problem of deciding whether a set of clauses in a propositional language (the knowledge base) logically entails a clause (the query) If no distinction between fixed and variable part is made, this problem is coNP complete. In ....

B. Selman and H. A. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 1996. In press.

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B. Selman and H. Kautz, Knowledge compilation and theory approximation, Journal of the ACM 43(2) (1996) 193--224.

....in these notes. Finally, we will discuss work that attempts to circumvent the complexity of reasoning with a propositional theory by translating, or compiling it so that subsequent queryanswering is faster. This work is covered in the paper Knowledge Compilation and Theory Approximation , by Selman and Kautz (1996), which discusses techniques for approximating a general propositional theory by a set of Horn theories, is included in these notes. We will also examine equivalence preserving compilation techniques, as developed by del Val (1994) and Marquis (1995) 2 Overview of Satisfiability Testing We ....

Selman, B., and Kautz, H., Knowledge Compilation and Theory Approximation. Journal of the ACM, 43(2) (1996) 193-224.

....Fig. 1. The circled models are M 0 0 , which is the closure of the example set of models M 0 . only if its models contain M : M models ( Sigma) The upper bound with the fewest models is called the Horn approximation of M , and corresponds to the notion of the least upper bound defined in [21,22]. The Horn approximation of any set of models is unique up to logical equivalence. It is useful for our purposes to develop an alternative but equivalent modeltheoretic characterization of a Horn approximation. We begin by defining the intersection of a pair of models as the model that assigns ....

....the query is false in [0101] 1000] 0000] There is, however, a more sophisticated way of evaluating queries on the set of characteristic models, that does yield an efficient sound and complete algorithm. Our approach is based on the idea of a Horn strengthening , which we introduced in [21,22]. Definition 8 Horn strengthening A Horn clause CH is a Horn strengthening of a clause C iff CH is a Horn clause, CH subsumes C, and there is no other Horn clause that subsumes C and is subsumed by CH . Another way of saying this is that a Horn strengthening of a clause is generated by ....

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B. Selman and H. Kautz, Knowledge compilation and theory approximation J. ACM , in press.

....knowledge base updates and revisions; as was pointed out in [ Eiter and Gottlob, 1992; Gogic et al. 1994 ] none of the known formalisms supports efficient changes. Gogic et al. 1994 ] propose a tractable revision mechanism using the theory approximation technique of [ Selman and Kautz, 1991; Selman and Kautz, 1996 ] Incidentally, change has its own expressiveness aspect (which changes in the set of models can be expressed, and how succinctly ) which is not at all understood at present. In this paper we find that the representational succinctness criterion (3) above can tell us interesting and unexpected ....

....H: 8. CNF ;H , H ;CNF Proof: Let M be a Horn set of models. We will prove that shortest CNF representing M must be a Horn formula. Suppose the shortest CNF representing M contains a non Horn clause C. Then there is a Horn clause CH C that can be entailed from M (see [ Selman and Kautz, 1991; Selman and Kautz, 1996 ] for the proof) and we can put CH in the CNF instead of C (the set of models will not change) This gives an even shorter formula representing M ; contradiction. 9. DNF ;CM The proof is given in the full version of the paper. Note that since CM cannot represents all possible sets of models, we ....

B. Selman, H.A. Kautz. Knowledge compilation and theory approximation. Journal of ACM, to appear.

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B. Selman and H. Kautz. Knowledge compilation and theory approximation. J. ACM, 43:193--224, 1996.

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B. Selman and H. Kautz, `Knowledge compilation and theory approximation ', JACM, 43(2), 193--224, (1996).

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Kautz, H. and Selman, B. (1996). Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193-224.

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B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, March 1996.

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B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, March 1996.

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Selman, B., and Kautz, H., Knowledge compilation and theory approximation, J.ACM 43,2 (1996), 193-224.

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B. Selman and H. A. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43:193--224, 1996.

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B. Selman and H.A. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2): pp. 193-224, 1996.

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B. Selman and H.A. Kautz. Knowledge compilation and theory approximation. Journal of the Association for Computing Machinery, 43:193--224, 1996.

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B. Selman and H. Kautz, Knowledge compilation and theory approximation, Journal of the ACM, 43(2) (1996) 193--224.

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B. Selman and H. Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193--224, March 1996.

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Bart Selman and Henry Kautz. Knowledge compilation and theory approximation. Journal of the ACM, in press, 1995.

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Bart Selman and Henry Kautz. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193-- 224, March 1996.

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