T.-H. Ngair, "Convex Spaces as an Order-theoretic Basis for Problem Solving," PhD Thesis, University of Pennsylvania, August 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....elements of Employees. Taking into account that some entries can be nulls, we see that the new consistency condition says that every element of every set in the lower approximation is at least as informative as some element of the upper approximation. Such constructions are called snacks, see [27, 28]. The reason for this name is that they were initially thought of not quite correctly, as we shall show as many sadnwiches , hence snacks. The graphical representation of a snack with two element L is given in Figure 1. The semantics of snacks is defined precisely as the semantics of ....

....satisfy the consistency condition: 8L 2 L 9l 2 L 9u 2 u : u l. Snacks are obtained from scones exactly as mixes are obtained from sandwiches: by using the assumption about keys, additional information is inferred. Thus, the consistency condition is similar to that of mixes. Definition 4 (cf. [27, 28]. Given a poset hA; i, a snack over A is a pair (U; L) where U is a finite antichain, and L = fL 1 ; L k g is a family of finite nonempty antichains which is itself an antichain with respect to v ] A snack is required to satisfy the consistency condition: 8L 2 L 8l 2 L 9u 2 u : u ....

[Article contains additional citation context not shown here]

T.-H. Ngair. "Convex Spaces as an Order-theoretic Basis for Problem Solving", (PhD Thesis), Technical Report MS-CIS-92-60, University of Pennsylvania, 1992.

....following: given a set of databases D1 ; Dn and a query q that can not be answered by using information from one of D i s, approximate the answer to q by using information from all D1 ; Dn . These problems have been investigated and they gave rise to a number of theoretical models [5, 10, 21, 15]. Given a query q, the databases are divided into two groups, one giving the upper approximation to the answer to q (that corresponds to possible information) and the other giving the lower approximation (that corresponds to the definite information) Here we demonstarte how OR SML can be used to ....

T.-H. Ngair. Convex Spaces as an Order-theoretic Basis for Problem Solving, Technical Report MS-CIS-92-60, University of Pennsylvania, 1992.

....given a set of databases D 1 ; D n and a query q that can not be answered by using information from one of D i s, approximate the answer to q by using information from all D 1 ; D n . These problems have been investigated and they gave rise to a number of theoretical models [5, 11, 22, 16]. Intuitively, given a query q, the 13 databases are divided into two groups, one giving the upper approximation to the answer to q (that corresponds to possible information) and the other giving the lower approximation (that corresponds to the definite information) It has been shown in [11, 16] ....

T.-H. Ngair. Convex Spaces as an Order-theoretic Basis for Problem Solving, Technical Report MS-CIS-9260, University of Pennsylvania, 1992.

....become elements of Employees. Taking into account that some entries can be nulls, we see that the new consistency condition says that every element of every set in the lower approximation is at least as informative as some element of the upper approximation. Such constructions are called snacks [29, 31, 19]. The reason for this name is that they were initially thought of not quite correctly, as we shall show as many sandwiches, hence snacks. The graphical representation of a snack with a two element L is given in Figure 1. The semantics of snacks is defined precisely as the semantics of ....

.... U 6= The family of all scones over A is denoted by PP 9 (A) Snacks are obtained from scones exactly as mixes are obtained from sandwiches: by using the assumption about keys, additional information is inferred. Thus, the consistency condition is similar to that of mixes. Definition 4 (cf. [29, 31, 19]. Given a poset hA; i, a snack over A is a pair (U; L) where U is a finite antichain, and L = fL 1 ; L k g is a family of finite nonempty antichains which is itself an antichain with respect to v ] A snack is required to satisfy the consistency condition: 8L 2 L 8l 2 L 9u 2 u : u l ....

[Article contains additional citation context not shown here]

T.-H. Ngair. "Convex Spaces as an Order-theoretic Basis for Problem Solving", (PhD Thesis), Technical Report MS-CIS-92-60, University of Pennsylvania, 1992.

....87] de Kleer 90] and circumscription ( Ginsberg 89] Raiman de Kleer 92] In ModelBased Diagnosis (MBD) de Kleer et al. 90] have characterized diagnoses using PIs. A number of new, and improved algorithms have emerged, e.g. Jackson Pais 90] Kean Tsiknis 90] de Kleer 92] Ngair 92] The algorithm presented here originates from the general SE tree search framework [Rymon 92] First, from that framework, we derive an algorithm which resembles the PI generation algorithm of [Slagle et al. 70] This algorithm is then improved using a standard problem decomposition tactic which ....

....time (CPU sec) vs. Number of Clauses Number of Nodes vs. Number of Clauses Figure 4: Run Time and Number of Nodes of SE HS ( versus SE HSd (2) 6 Other Related Work In [Rymon 92] we present an experiment in which SE HS (without decomposition) is compared to a PHI based PI generation algorithm [Ngair 92] Such comparison is important because Ngair has shown his algorithm to be comparable, and in some instances preferable to other current algorithms such as IPIA [Kean Tsiknis 90] and CLTMS [de Kleer 92] Figure 5 presents some results from this experiment. In [Rymon 92] we also compare SE HS ....

Ngair, T., Convex Spaces as an Order-Theoretic Basis for Problem Solving, Ph. D. Thesis, Department of Computer and Information Science, University of Pennsylvania, 1992.

....is a homomorphism. The uniqueness of f follows from (1) Since f (j (x; x) f(x; x) 2f(x; x) f(x; x) we have f ffi j = f . The theorem is proved. 2 3.3 The snack powerdomain Snacks were introduced by Peter Buneman. They were studied by Teow Hin Ngair in his dissertation [22] and characterized by Puhlmann [27] as free distributive bisemilattices [8, 24] Since Pulhmann s proof is not very complicated and since it exploits an unusual presentation of the equational theory, for the sake of completeness I prove the characterization theorem here. At the end of this ....

....selecting the kind of sets that can be approximated. In many domain theory motivated works [6, 16, 17, 27] and also [14] it is assumed that the maximal elements of a domain are complete descriptions and sets of those are approximated. Others suggest that any set can be approximated, for example, [19, 22]. The idea of the second approach comes from consideration of recursive types. For instance, solving the domain equation D = C Theta D corresponding to a simple recursive record type would give us a domain whose maximal elements are infinite sequences of maximal elements of C and it is unlikely ....

[Article contains additional citation context not shown here]

T.-H. Ngair. Convex Spaces as an Order-theoretic Basis for Problem Solving, Technical Report MS-CIS-92-60, University of Pennsylvania, 1992.

....obtain mixes from sandwiches. In the new approximation that we obtain, the condition expressing consistency is much stronger: now every element of every set in the lower approximation is at least as informative as some element of the upper approximation. Such constructions are called snacks, see [20, 21]. The reason for this name is that they were initially thought of not quite correctly, as we shall show as many sandwiches , hence snacks. The graphical representation of a snack with a two element L is given in Figure 1. The semantics of snacks is defined precisely as the semantics of ....

....] L j if i 6= j. The consistency condition is 8L 2 L 9l 2 L 9u 2 u : u l. Snacks are obtained from scones exactly as mixes are obtained from sandwiches: by using the assumption about keys, additional information is inferred. Thus, the consistency condition is similar to that of mixes, and snacks [20, 21] can be defined as pairs (U; L) where U is a finite antichain, and L = fL 1 ; L k g is a family of finite nonempty antichains which is itself an antichain with respect to v ] The consistency condition for snacks is 8L 2 L 8l 2 L 9u 2 u : u l. One can see that there are three main ....

[Article contains additional citation context not shown here]

T.-H. Ngair. "Convex Spaces as an Order-theoretic Basis for Problem Solving". Technical Report MS-CIS-92-60, University of Pennsylvania, 1992.

....of the or sets, thus reducing the size of the normal form. In [16] a logical language was used, and we would like to see if the idea can be worked out for our languages. There are various sophisticated order theoretic models of partial information in databases sandwiches [6] mixes [11] snacks [19, 28], scones and salads [19, 22] They are used when a real world situation can be approximated from below and above by information in a database. These structures enjoy universality properties and therefore can be incorporated into the programming language syntax [22] We plan to investigate the ....

T.-H. Ngair. Convex Spaces as an Order-theoretic Basis for Problem Solving, Technical Report MS-CIS-92-60, University of Pennsylvania, 1992.

....given a set of databases D 1 ; D n and a query q that can not be answered by using information from one of D i s, approximate the answer to q by using information from all D 1 ; D n . These problems have been investigated and they gave rise to a number of theoretical models [5, 11, 22, 17]. Intuitively, given a query q, the databases are divided into two groups, one giving the upper approximation to the answer to q (that corresponds to possible information) and the other giving the lower approximation (that corresponds to the definite information) It has been shown in [11, 17] ....

T.-H. Ngair. Convex Spaces as an Order-theoretic Basis for Problem Solving, Technical Report MS-CIS92 -60, University of Pennsylvania, 1992.

....of the or sets, thus reducing the size of the normal form. In [16] a logical language was used. We would like to see if the idea can be worked out for our languages. There are various sophisticated order theoretic models of partial information in databases sandwiches [6] mixes [10] snacks [31, 30], and their generalizations [31, 22] They are used when a real world situation can be approximated from below and above by information in a database. These structures enjoy universality properties and therefore can be incorporated into the programming language syntax [22] We have recently shown ....

T.-H. Ngair, "Convex Spaces as an Order-theoretic Basis for Problem Solving," PhD Thesis, University of Pennsylvania, August 1992.

....involves relating the semantic entailment j= for the Horn clause formulas OE to the least fixed point of the closure operators OE. This can be done through the use of a minimal model for a collection of Horn clauses. Details sufficient to construct a proof of Theorem 22 can be found in [5] and [11]. 41 Table 4: Converting a Horn Clause OE to a Closure Operator OE. function OE(F) b) if b 6= a then F(b) else F(a) u (fold( u , map(F) S) f;g) endif where OE is a Horn clause such that S is the set of its premises and a is its conclusion. It is worth noting how Theorem 22 ....

T-H. Ngair. Convex Spaces as an Order-Theoretic Basis for Problem Solving. PhD thesis, University of Pennsylvania, 1992.

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T.-H. Ngair, "Convex Spaces as an Order-theoretic Basis for Problem Solving," PhD Thesis, University of Pennsylvania, August 1992.

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