Autor's copy.
Efficient Technique for Transformation from Concepts to Boolean Algebra Grzegorz Borowik and Dariusz Nogalski PUBLISHED ON:
2nd Asia-Pacific Conference on Computer Aided System Engineering -- APCASE 2014, South Kuta, Bali, Indonesia, February 10-12, 2014
BIBTEX: @INPROCEEDINGS{APCASE_2014_2_GBorowik, AUTHOR = {Grzegorz Borowik and Dariusz Nogalski}, TITLE = {Efficient Technique for Transformation from Concepts to Boolean Algebra}, BOOKTITLE = {2nd Asia-Pacific Conference on Computer Aided System Engineering -- APCASE 2014}, EDITOR = {Zenon Chaczko and Ford Lumban Gaol and Franz Pichler and Christopher Chiu}, PAGES = {24--27}, YEAR = {2014}, ADDRESS = {South Kuta, Bali, Indonesia}, MONTH = {February 10--12}, ISBN = {978-0-9924518-0-6} }
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Autor's copy.
Efficient Technique for Transformation from Concepts to Boolean Algebra Grzegorz Borowik1, Dariusz Nogalski2 1
2
Warsaw University of Technology, Institute of Telecommunications, Warsaw, Poland Military University of Technology, Cybernetics Faculty, Warsaw, Poland
[email protected],
[email protected]
Abstract. Knowledge Cartography (KC) allows for fast answering of Description Logics (DL) knowledge base queries, but requires expensive preprocessing to represent knowledge in internal representation, i.e. computation of map of concepts as binary signatures in pessimistic time can be exponential. Pre-processing is already part of DL reasoning process and some computations are pre-calculated before user issues query to knowledge base. Using Tableaux no knowledge preprocessing is required and all reasoning is done after user issues query. That's why KC is faster than Tableaux. The discussed in the paper approach makes the KC method faster regarding the generation of binary signatures and signatures rebuilding. It is achieved, enhancing the method by logic synthesis algorithms.
Introduction and significance of the problem Knowledge Representation (KR) is a field in artificial intelligence that focuses on creation of languages and formalisms which allow to capture knowledge about the world. KR typically depicts objects, types of objects, and rules/axioms that describe laws and constraints existing within the world. Another important element of KR is automated reasoning; therefore use of combined term Knowledge Representation and Reasoning (KRR). Reasoning algorithm is a generic purpose algorithm that allows to solve complex problem stated in KR language. In the context of programming paradigms, declarative and imperative computer languages can be distinguished. Imperative language allows to specify algorithm how to solve the problem (e.g. Java procedure which is a sequence of steps to derive a solution). On contrary, declarative languages (e.g. Prolog) allow to specify user goal, and generic reasoning algorithm finds the solution to satisfy the goal [9]. KRR is mainly applied to solve complex decision problems on which traditional solutions (e.g. imperative languages) fail. KRR is a building block of many expert systems and has many applications ranging from medical diagnosis, knowledge management, classification systems [2-4,7], system mediation [8], information fusion [10] to autonomous agents. Description Logics (DL) [1] is a KR formalism to represent Knowledge Bases (KB). DL is a subset of First Order Logic (FOL), limited to two types of predicates (unary
Autor's copy. and binary). Unary predicates allow to represent concepts (types of objects) and binary predicates represent relationship between objects. As opposed to generic set of statements in FOL, DL has decidable decision procedures that make DL applicable in computer systems. DL is a family of languages where each language is determined by the set of allowable logical constructors. Some constructors, however, qualify reasoning to computations of non-polynomial complexity [1]. Knowledge Reasoning in DL is based on the Open World Assumption (OWA) [12]. DL knowledge base consists of two parts: TBox (terminology box) and ABox (assertion box). TBox is a set of axioms – logical formulas that describe relationship between concepts: 𝑀𝑎𝑛 ⊑ 𝑇𝐵𝑜𝑥 𝑃𝑒𝑟𝑠𝑜𝑛 𝑀𝑜𝑏𝑖𝑙𝑒𝑃𝑒𝑟𝑠𝑜𝑛 ≡ 𝑇𝐵𝑜𝑥 𝑃𝑒𝑟𝑠𝑜𝑛 ⊓ ∃𝑖𝑠𝑂𝑤𝑛𝑒𝑟𝑂𝑓. 𝐶𝑎𝑟 ⊓ ∃ℎ𝑎𝑠𝐿𝑖𝑐𝑒𝑛𝑐𝑒. 𝑉𝑎𝑙𝑖𝑑𝐷𝑟𝑖𝑣𝑖𝑛𝑔𝐿𝑖𝑐𝑒𝑛𝑐e
ABox is a set of assertions – logical formulas which describe memberships of particular individuals and binary relationships between individuals: 𝑀𝑎𝑛(𝐽𝑜ℎ𝑛) 𝑖𝑠𝑂𝑤𝑛𝑒𝑟𝑂𝑓(𝐽𝑜ℎ𝑛, 𝐶𝑎𝑟_𝐼𝐷_𝑃𝑂𝐿123) ℎ𝑎𝑠𝐿𝑖𝑐𝑒𝑛𝑐𝑒(𝐽𝑜ℎ𝑛, 𝐿𝐼𝐶_𝐼𝐷_456) 𝑉𝑎𝑙𝑖𝑑𝐷𝑟𝑖𝑣𝑖𝑛𝑔𝐿𝑖𝑐𝑒𝑛𝑐𝑒(𝐿𝐼𝐶_𝐼𝐷_456)
In the above example reasoning system should deduce that John is a MobilePerson.
Where we do address the problem? Most commonly used in DL reasoning algorithm is the FOL Tableaux. However, this paper addresses an alternative approach to DL reasoning, namely Knowledge Cartography (KC) [6]. KC reduces each DL problem to Boolean logic algebra over binary vectors. The idea is based on the fact that DL TBox concepts are interpreted as sets over domain of disclosure. Relationship among the sets (intersection, disjointness, etc.) can be shown on the Venn diagram (map of concepts). Each TBox concept (set) is a sum of atomic regions on the map. This observation allows to define a binary vector for each concept. Thus, DL TBox operations (sum, intersection, other) can be reduced to respective operations (disjunction, conjunction, etc.) over binary signatures algebra. The task of reducing DL knowledge base to Boolean logic representation reduces itself to CNF-satisfiability [6,11] which is main bottleneck of KC.
Autor's copy. Methodology scheme The main goal of this paper is an efficient transformation of concepts in DL to Boolean representation. The paper proposes enhancing the task of transformation with logic synthesis algorithms, such as Boolean function complement algorithm and Shannon expansion algorithm [3,5]. This work is motivated by previous works. The algorithms mentioned had been used for calculating data tables' reducts [4] and data discretization [2], successfully reducing computation time. This optimization has made the calculations efficient and applicable to real-life examples. Especially, this method and its software implementation have proven to be effective for data mining in medicine [2,4]. Hence, the idea of using the logic synthesis algorithms in KRR. The method proposed is based on transformation of CNF to DNF by computation of the complement of a given Boolean function in CNF. It can be performed using the Shannon expansion algorithm recursively. The recursion continues until, at each leaf of the recursion tree, a unate function or one of special cases is encountered [3,5]. The special cases require only trivial computations. For unate functions we use fast complementation algorithm [3]. The recursion process creates a binary tree and the final result is calculated performing the special merging procedure for the two subtrees below each node [3,5]. Such methodology is possible due to the fact that a Boolean expression can be reduced to monotonic CNF, which can be represented by a binary matrix. Thus, the transformation process of CNF into DNF is reduced to the calculation of minimum column covers of this matrix [3].
Summary Summarizing, KC allows for fast reasoning, however, adding/removing new concepts requires rebuilding the signatures. On the contrary, using Tableaux one can easily add or remove concepts, however, the reasoning is complex. It's due to the fact that in KC some computations are pre-processed – executed during generation of the map of concepts [6] and Tableaux makes complete case analysis each time. It's still in favor of KC because adding new concepts is rare in comparison to making a decision, and this is particularly important in real-time systems. Some work on optimization of map generation has already been undertaken, i.e. tree fusion [6]. The novel approach proposed in this paper makes the KC method faster regarding generation of the map of concepts as binary signatures as well as signatures rebuilding. It is achieved, enhancing the method by logic synthesis algorithms. These algorithms are known, however, have not been previously used or have rarely been used in the field of KRR. It is mainly due to the lack of knowledge of methods and algorithms of logic synthesis and therefore they are skipped and not used by specialists of data mining [2].
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