P. Patel-Schneider. \A four-valued Semantics for Terminological Logics." Articial Intelligence, 39(2):263-272, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....We have chosen the concept language such that it contains most of the constructs used in kl one systems with the obvious restriction that the interesting inferences such as consistency checking, subsumption checking, and realization are decidable. Of course, taking 1 But see Patel Schneider [Pat89] who uses a four valued semantics to formally describe the behaviour of an algorithm which is incomplete w.r.t. two valued semantics. such a large language means that the complexity of the inference algorithms is relatively high. But KRIS also provides faster algorithms for certain ....

P. Patel-Schneider. \A four-valued Semantics for Terminological Logics." Articial Intelligence, 39(2):263-272, 1989.

....information, which seems to be ubiquitous in applications. To mention just a few areas where subsumption has played an important role we refer to applications in theorem proving ( KC92] in taxonomic reasoning ( ACS91] in semantic query optimization ( CGM90] in terminological logic ([PS89]) or in type subsumption ( AK85] Subsumption also appears in probabilistic reasoning. e.g. given two conditional probability statements 0:70 p(BjA) 0:90 and 0:75 p(BjA) 0:85, the second one clearly subsumes the first one. This is used in the DUCK calculus for probabilistic uncertainty ....

Peter F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38(3):318--352, 1989.

.... algorithm that tests the validity of # # # [32] Relevance description logics based on a 4 valued semantics have already been proposed by PatelSchneider for use in knowledge representation, and have been proven to possess a generally better computational behaviour than their 2 valued analogues[40, 41, 42, 43]. The semantics we adopt departs from Patel Schneider s, whose loss of inferential capabilities is too drastic for the needs of IR: in fact, that semantics sanctions the loss of modus ponens and, in general, of a great deal of conditional reasoning. In addition, the deduction algorithms and the ....

....results concerning 2 valued DLs (see e.g. 8, 13] 11 . If we switch to a 4 valued setting, we need an alternative proof procedure as relation (12) no longer holds. There exists a well known subsumption testing procedure, which is a DL adaption of Levesque s algorithm [32] for entailment [6, 40, 43]. The algorithm performs structural subsumption in a e#cient way, but has several drawbacks. First, it does not work within our semantics. Second, it cannot be used for the instance checking test as this problem is in a higher complexity class than the subsumption problem. Third, it is rather ....

Peter F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38:319--351, 1989.

....by eliminating certain kinds of clauses. Typically, this is done by eliminating certain term constructors and by replacing all clauses which use a corresponding outermost constructor by an equivalent clause using a different outermost constructor. In this way a canonical form is defined, e.g. in [Patel Schneider, 1989] . 3.2.2 An Implementation of the Difference Operation In a description logic with structurally unique RCFs it is relatively straightforward to calculate the difference operation, as it is defined in Definition 2.1. The difference B Gamma A can be calculated by removing all clauses from an RCF ....

....was defined in respect to a canonical form and it was formulated with the help of outermost constructors. Our definition with the help of RCFs is more general, since it does not require the existence of a canonical form. Logics with a structural subsumption have several interesting properties. In [Patel Schneider, 1989] the structural subsumption was an important basis for proving that calculating the subsumption relation was tractable in that logic. However, a structural subsumption does not necessary imply tractability. Either the tractability can be prevented by the complexity of the clauses, or the ....

Patel-Schneider, P. F.: A Four-Valued Semantics for Terminological Logics. Artificial Intelligence 38, 319--351 (1989)

.... subject of theoretical investigation within the field of knowledge representation, but they also find use in practical knowledge based systems (Beck et al. 1989; Borgida, 1992; Borgida Patel Schneider, 1992; Brachman et al. 1983; Cohen Hirsh, 1994b; Devanbu et al. 1991; Mays et al. 1987; Patel Schneider, 1989). 1.1. Classic Classic permits constructing certain quantified descriptions that distinguish a particular subset of a domain I of individuals. Classic descriptions contain primitive symbols which get mapped to arbitrary subsets of I, disjoint primitive symbols which get mapped to mutually ....

Patel-Schneider, P. F. (1989). A four-valued semantics for terminological logics. Artificial Intelligence, 38, 319--351.

....has been described; in [32] a fuzzy extension of propositional logic within a four valued semantics has been described and [25] briefly formalises a fourvalued fuzzy DL. The rational behind the combination of fuzzy semantics and four valued semantics is twofold. In [26] a four valued semantics [3,6,23,28] has been proposed for DLs in order to enforce a notion of logical entailment in which premises (i.e. the knowledge base) need be relevant to conclusions (i.e. the query) to a stronger extent than classical material logical implication does. The logic specified in [26] is still insu#cient for ....

Peter F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38:319--351, 1989.

....of the problem solver, clearly the efficiency of the KR system is important. Recent attempts to deal with the intractability of such systems have generally fallen under two headings: limited languages (e.g. Patel Schneider 1984, Borgida et al. 1989] and limited inference (e.g. Frisch 1988, Patel Schneider 1989]) In the former, what can be expressed in the knowledge base (KB) is restricted (sometimes severely) to guarantee that queries can be answered in more or less reasonable time. In the latter, restrictions like avoiding chaining or four valued interpretations yield limited conclusions, albeit from ....

Patel-Schneider, P. F. [1989], "A Four-Valued Semantics for Terminological Logics", Artificial Intelligence 38, 1989, 319-351.

....retrieval have led us to incorporate tautological entailment in our DL. Technically, we have done this by switching from a two to a four valued semantics. Logics of this kind have already been used in KR, and have shown a better computational behaviour than their two valued analogues (see e.g. [5]) In semantical terms, the interpretation function of I now maps every concept (resp. role) into a function from Delta I (resp. Delta I Theta Delta I ) to the powerset of ft; fg, and every object into Delta I . The interpretation of a concept C can best be understood as consisting of a pair ....

....KB Sigma entails an assertion ff (written Sigma j= 4 ff) iff all interpretations satisfying Sigma also satisfy ff. It can be proven that j= 4 is a subset of j= which guarantees the soundness of the entailment relation wrt the two valued semantics. Also, unlike other four valued logics such as [5], it licenses a restricted form of modus ponens, which we call modus ponens on roles, whose consequence is that f(8R:C) a 1 ) R(a 1 ; a 2 )g j= 4 C(a 2 ) and f(8R:C 1 ) a) 9R:C 2 ) a)g j= 4 (9R:C 1 uC 2 ) a) Also, j= 4 avoids the paradoxes of material implication; i.e. in our logic (1) not ....

P. F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38:319-- 351, 1989.

.... tests if ff fi holds [Levesque, 1984] Relevance TLs, i.e. four valued TLs based on some relevance logic, have already been used in Knowledge Representation and proven to have a generally better computational behaviour than their two valued analogues [Levesque, 1984; Patel Schneider, 1987a; Patel Schneider, 1989] For all these reasons, we view tautological entailment as a major source of inspiration for incorporating a form of relevance into the inference relation of ALMIR. In particular, the four valued semantics of ALMIR is a variation of tautological entailment, in which the implication relation has ....

Peter F. Patel-Schneider. A fourvalued semantics for terminological logics. Artificial Intelligence, 38:319--351, 1989.

.... is instance checking and amounts to verify whether the individual a is an instance of the concept C with respect to the knowledge base Sigma , i.e. in symbols Sigma j= C(a) Originally, four valued DLs have been proposed in order to develop expressively powerful DLs with tractable subsumption [21, 22, 23]. It has been shown that the four valued subsumption relation captures an important subset of twovalued subsumption relationships. More recently, DLs and their four valued versions have been proposed in the area of (multimedia) Document Retrieval (DR, for short) 12, 17, 18, 19, 26] In this ....

....can be reduced to instance checking in ALC. This property holds for most DLs presented in the literature and in particular it will hold for all DLs we will consider in this paper. 3 Four valued Description Logics The four valued semantics for ALC is based on a four valued semantics similar as in [23], but we will also present a different choice for the semantics of the (8) constructor. As usual, the four truth values are the elements of 2 ft;fg , the powerset of ft; fg, i.e. ft; fg, fg, ftg and ffg. These values are known as contradiction, unknown, true and false, respectively. An ....

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Peter F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38:319--351, 1989.

....Clearly, modelling multimedia documents requires much more. However, we do think that providing these closures is a necessary step towards the definition of an adequate logic for multimedia retrieval. In fact, we are currently working on a number of extensions as ffl a Relevance logic based DL [13] with the aim to (i) capture a tight connection in meaning between query and the information that are retrieved in response to it; ii) ruling out undesiderable inferences as paradoxes of logical implication and reasoning by cases; iii) dealing with multimedia document s content descriptions ....

Peter F. Patel-Schneider, `A four-valued semantics for terminological logics', Artificial Intelligence, 38, 319--351, (1989).

.... [MacGregor, 1988] classic [Brachman et al. 1989; Borgida et al. 1989] and sb one [Kobsa, 1989] and the formal properties of these systems were investigated [Schmolze and Israel, 1983; Brachman and Levesque, 1984; Patel Schneider, 1986; Levesque and Brachman, 1987; Nebel, 1988; Schild, 1988; Patel Schneider, 1989a; PatelSchneider, 1989b; Schmidt Schau , 1989; Schmidt Schau and Smolka, 1990; Nebel and Smolka, 1990; Nebel, 1990; Donini et al. 1990; Hollunder et al. 1990] When studying the above mentioned papers, one notes that terminological cycles are usually ignored or explicitly excluded. ....

....language for which subsumption is decidable in the acyclic case but undecidable when terminological cycles are permitted. Usually subsumption algorithms are specified over concept descriptions only, assuming that all atomic concepts are primitive (see, for instance, Levesque and Brachman, 1987; Patel Schneider, 1989a; Schmidt Schau and Smolka, 1990; Hollunder et al. 1990] This is sufficient as long as the terminology is acyclic because in this case we can expand all nonprimitive concepts by their definitions until the concept descriptions contain only primitive atomic concepts. 11 In [Nebel, 1990] it ....

Peter F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38(3):319--351, April 1989. Terminological Cycles 34

....time computations. They started an inquiry concerning the trade off between expressiveness and tractability of representation Attributive Description Formalisms : and the Rest of the World 7 formalisms, which led to a number of analyses of different terminological logics [Nebel, 1988; Patel Schneider, 1989a; Schmidt Schau and Smolka, 1991] However, only recently, terminological logics that are maximally expressive and still tractable have been identified [Donini et al. 1991a] using the constraint solving technique introduced in [Schmidt Schau and Smolka, 1991] Another open problem was whether ....

Peter F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38(3):319--351, April 1989.

....subsumption algorithm. Subsequently, researchers concentrated on developing ways to circumvent this complexity trap by radically restricting the expressiveness (as exercised, e.g. in krypton [11] and classic [8] by using weaker semantics that permit tractable subsumption determination [38], or by restricting the inferential capabilities in a pragmatic way leading to sound and fast but incomplete systems, such as back [31] and loom [29] Only recently, complete algorithms for a number of powerful term description languages for which the subsumption problem is NP hard have been ....

P. F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38(3):319--351, Apr. 1989.

....propositional conjunctions are learnable. In this paper, we study a midpoint on the spectrum of conjunctive languages called description logics [ Woods and Schmolze, 1992; Borgida et al. 1989; Patel Schneider, 1984; Brachman et al. 1983; Edelman and Owsnicki, 1986; Bergamaschi et al. 1988; Patel Schneider, 1989 ] Description logics (DLs) form a family of knowledge representation languages that allow one to ascribe descriptions to individuals and then reason about these descriptions. Each description corresponds to a wellformed formula in predicate calculus, and thus DLs are subsets of predicate ....

P. F. Patel-Schneider. A fourvalued semantics for terminological logics. Artificial Intelligence, 38:319--351, 1989.

....(domain C) is interpreted as the left cylindrification of the set C I , while (range C) is interpreted as the right cylindrification of C I . An example of an operator not captured in Peirce algebra is a generalized version of the atleast operator, called the fillers operator, defined in Patel Schneider (1989). iii) Peirce algebras are easy to use. Even the most complicated terminological constructs in U Gamma have straightforward algebraic translations (as summarized in Table 1) Unlike the model theoretic interpretation the algebraic interpretation of terminological expressions is free of ....

Patel-Schneider, P. F. (1989), A four-valued semantics for terminological logics, Artificial Intelligence 38, 319--351.

.... concept description languages that permit tractable subsumption determination of concept descriptions, exercised, for example, in the terminological subcomponents of kandor [23] krypton [4] meson [8] and recently classic [1, 2] Another interesting approach, pursued by Patel Schneider [22], employs a weaker semantics to achieve tractability of subsumption determination of concept descriptions for quite expressive concept description languages. All of these efforts tried to achieve the goal of forging a powerful system out of tractable parts [15, p. 89] and all of them have ....

....in this language is co NP complete is presented in Section 3. In Section 4, reasons why the theoretical worst case behavior occurs only seldomly in practice are analyzed. Finally, conclusions are discussed in Section 5. 1 Called frame description language in [3] and terminological logic in [22]. Terminological Reasoning 3 2 Terminological Representation and Reasoning The terminological representation language that will be analyzed in this paper, called TL, is a subset of the terminological language of krypton presented in [4] 2 Starting with a finite set A of atomic concepts ....

P. F. Patel-Schneider, A four-valued semantics for terminological logics, Artificial Intelligence 38(3) (1989) 319--351.

....sd expression is equivalent to (not self) For this we need to establish that e ne 0 = e 0 . It follows by R11, R9 and R21 that e ne 0 = e ne 0 = ene 0 = e 0 = e 0 = e 0 . As a final example (which involves the all construct) consider the following expression from Patel Schneider [1990, p. 14] 4.28) and Persons (and (all has child Lawyers) all (restrict has child Lawyers) Doctors) v (and Persons (all has child Doctors) This is quite complicated. It is also not clear how to read this in English. Patel Schneider himself translates the subsumed term by the class of ....

Patel-Schneider, P.F. [1989a]. A Four-Valued Semantics for Terminological Logics. Artificial Intelligence 38, 319--351.

....faced by this approach is to describe to the user the incompleteness. As we have seen, operational definitions are relatively difficult for DLs. Other kinds of semantic specification techniques have been proposed for this purpose, including non standard denotational semantics such as those in [Patel Schneider 1989a, Borgida and Patel Schneider 1992] or proof theoretic axiomatizations, such as in [Borgida 1992a, Borgida 1992b] ffl Providing an escape hatch in the language. It is possible to introduce one or more constructors in the language whose semantics are opaque for subsumption reasoning, but can ....

Patel-Schneider, P. F., "A four-valued semantics for terminological logics ", Artificial Intelligence 38 (1989) 319--351.

....a computationally tractable form of reasoning. In his works [20, 21, 22] he proved that inference in some MV logics can be performed more easily than in classical logic. Following these results, MV logics have been used as tools for designing tractable reasoning systems by Patel Schneider in [23, 24] and by the present authors in [4, 5, 6] The goal of systems of this kind is to provide a meaningful balance between complexity of inference and completeness of deduction. The gain of efficiency of these systems must therefore be compared to a general loss of inferential power. In this paper we ....

....is in that all polynomial cases (both the new ones and those already known) are obtained by means of two simple algorithms, sharing a common method. The MV logics we consider in this work have already been used as the basis for more complex systems, such as the four valued terminological logic [23] and the four valued (decidable) first order logic [24] presented by PatelSchneider. Simple propositional generalizations are also used in the approximation schemata introduced by the present authors in [4, 5, 6] The results shown in this paper can be used for relaxing syntactic restrictions ....

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P. F. Patel-Schneider. A four-valued semantics for terminological logic. Artificial Intelligence Journal, 38:319--351, 1989.

....of the computational cliff they had identified. It took seven years to come up with an exhaustive answer. Between 1987 and 1992, a number of results concerning complexity and decidability of subsumption for different description logics where published [Levesque and Brachman, 1987; Nebel, 1988; Patel Schneider, 1989b; Schmidt Schau , 1989; Hollunder et al. 1990; Schmidt Schau and Smolka, 1991; Donini et al. 1991b; Schild, 1991; Donini et al. 1992] but the boundary between polynomial and NP hard description logic appeared still fuzzy. Finally, Donini et al. 1991a] were able to provide an exhaustive ....

....to correct but incomplete methods. This means that the method only draws logically valid conclusions but may miss some. In order to be able to characterize the behavior of such methods formally, one can try to capture the incompleteness declaratively using a weaker semantics. For example, Patel Schneider [1989a] uses a fourvalued semantics for characterizing a subsumption relation that is polynomial but weaker than the standard one. Levesque and Lakemeyer use a similar technique to weaken the general logical implication for propositional belief logics [Levesque, 1984; Lakemeyer, 1987] Other approaches ....

Peter F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38(3):319--351, April 1989.

....there are nontrivial tractable fragments of the language that include modal formulas. Interestingly, a 4 valued semantics was also used in a different context in order to deal with computational complexity; PatelSchneider defined a 4 valued terminological logic with tractable subsumption [PS89]. 8 Standard World Validity Logical omniscience arises from considering knowledge as truth in all possible worlds. In the approach of this paper, we modify logical omniscience by changing the notion of truth. In this section, we consider the impossible worlds approach, where we modify logical ....

P. F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38:319--351, 1989.

....taxonomy, and filtering by testing each retrieved object against the query concept. 4 Algorithmic Considerations Although we have talked about computational services, we haven t given algorithms which do the actual computations. However, instead of specifying inference algorithms (see, e.g. [27, 37, 41, 46, 56]) we will investigate the computational properties of the problems. As we have seen in the previous section, subsumption determination is the central operation in a TBox 6 Proofs for this and the following theorems and propositions can be found in [37] system. This point is reinforced by ....

.... co NP hard [34] ALC (C u C 0 ) C t C 0 ) C, 8R: C) 9R: C) PSPACE complete [56] with (9 n R) and (9 n R) still in PSPACE, with feature logic [57] added still decidable [20] C u C 0 ) 8R: C) 9 n R) 9 n R) R u R 0 ) Rj C ) polynomial for four valued semantics [46] R (R u R 0 ) R, R ffi R 0 ) undecidable [54] kl one (C u C 0 ) 8R: C) R # R 0 ) R ffi R 0 ) undecidable [55] slightly weaker result: 49] Table 1: Complexity of subsumption in term forming languages The formal meaning of these concept description operators can be ....

[Article contains additional citation context not shown here]

P. F. Patel-Schneider. A four-valued semantics for terminological logics. Artificial Intelligence, 38(3):319--351, Apr. 1989.

....a very small language in comparison with other DLs, and its implementation appeared to be evidently correct. To avoid such problems, it is necessary to produce convincing demonstrations that the algorithm is correct; several such proofs have in fact already appeared in the DL literature (e.g. (Patel Schneider, 1987; Hollunder Nutt, 1990; Donini et al. 1991) albeit only for languages that have not seen use in practical applications. 1.2 Outline The classic 1 2 system is a reasoner based on a moderately complicated DL. It is being used in commercial (Wright et al. 1993) and prototype applications at AT T, and is made ....

Patel-Schneider, P. F. (1989a). A four-valued semantics for terminological logics. Artificial Intelligence, 38 (3), 319--351.

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Patel-Schneider, P. F., "A Four-Valued Semantics for Terminological Logics," Artificial Intelligence, Vol. 38, No. 3, April, 1989, pp. 319--351.

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