A Computational Model for Inference Chains in Expert Systems

Transcription

1 A Comptational Moel for Inference Chains in Expert Systems József Sziray Department of Informatics Széchenyi University Egyetem tér, H-926 Győr Hngary Abstract: This paper eals with the calclations performe in the reasoning process of rle-base expert systems, where inference chains are applie. It presents a logic moel for representing the rles an the rle base of a given system. Also, the fact base of the same expert system is involve in the logic moel. The propose eqivalent representation manifests itself in a logic network. After that, a for-vale logic algebra is introce. This algebra is se for the calclations where forwar chaining is carrie ot. Next, the notion of line-vale jstification is escribe. This operation is applie in the backwar chaining process, also on the base of the previosly introce for-vale logic. The paper escribes two exact algorithms which serve for the forwar an backwar chaining processes. These algorithms make it possible to be implemente by a compter program, reslting in an efficient inference engine of an expert system. The achieve reslt enhances the reliability an sability of the intelligent software systems which is extremely important in embee environments. I. INTRODUCTION The application area of embee systems an the relate economical an reliability reqirements imply a specific harware-software strctre that is significantly ifferent from the resorces available in moern high-en systems. The relatively low processing performance, small memory space an the safety prescriptions have reslte in varios architectral properties an programming soltions. In case of real-time safety-critical systems the reaction time for the external events is a key isse. It means that the spee of the calclations is a critical factor. On the other han, the same applies to the memory consmption []. In many cases, artificial intelligence is realize within the frames of expert systems. This approach has gaine a wie-sprea se in controlling railway stations, angeros chemical processes, airplane flights, meical systems, etc. These applications are eqally relate to safety-oriente systems. As known, the most common form of storing knowlege in expert systems is the se of rles. It means that the knowlege base (long-term memory) consists of rles an facts. The other component of sch an expert system is the inference engine which is the most important factor for a sccessfl operation. An inference engine sally works in a fixe manner, for example, it col be esigne as either ata riven (i.e., forwar reasoning or forwar chaining) or goal riven (i.e., backwar reasoning or backwar chaining), however, most of the moern systems may well se both ways of reasoning [2]-[4]. The major concern relate to the inference processes is their excessive comptational amont. The algorithmic complexity erives from the fact that the task to be solve belongs to the so-calle NP-complete problems. As known, NP-complete problems have a comptational complexity for which there exists no pper bon by a finite-egree polynomial of the problem size. It means actally that the nmber of the comptational steps is finite, bt npreictable [5]-[7]. Here the problem size can be expresse by the nmber of rles in the knowlege base. De to the escribe featres of the comptations, the exection spee of the software is a crcial factor. This paper eals with the calclations performe in the reasoning process of rle-base expert systems, where inference chains are applie. It presents a logic moel for representing the rles an the rle base of a given system. Also, the fact base of the same expert system is involve in the logic moel. The propose eqivalent representation manifests itself in a logic network. After that, a for-vale logic algebra is introce. This algebra is se for the calclations where forwar chaining is carrie ot. Next, the notion of line-vale jstification is escribe. This operation is se in the backwar chaining process, also on the base of the previosly introce for-vale logic. The paper escribes two exact algorithms which serve for the forwar an backwar chaining processes. These algorithms make it possible to be implemente by a compter program, reslting in an efficient inference engine of an expert system.

2 II. FUNDAMENTAL CONCEPTS In a rle-base system, any rle consists of two parts: the IF part, calle the anteceent (premise or conition) an the THEN part, calle the conseqent (conclsion or action). The basic syntax of a rle is: IF <anteceent> THEN <conseqent>. In general, a rle can have mltiple anteceents joine by the keywors AND (conjnction), OR (isjnction), or a combination of both. Negation of an anteceent is also allowe. In this case the NOT operator is se. For example, IF the spill is liqi AND the spill ph < 6 AND the spill smell is vinegar THEN the spill material is acetic aci. Forwar chaining is an inference metho where rles are matche against facts to establish new facts, finally reaching a conclsion. In case of backwar chaining the system starts with what it wants to prove, an tries to establish the facts it nees to prove the initial fact. The necessarily applie components of the reasoning process constitte the so-calle inference chain. The knowlege base consists of the set of rles (rle base), an the set of facts (fact base), where the rle base is permanent, while the fact base contains an initial set of facts epening on the actal task to be solve, an it changes in accorance with the concrete reasoning process. The existent expert systems bil p the knowlege base in a sal ata-base strctre, an their inference engine applies an exhastive search throgh all the rles ring each cycle. The aim of the search is to fin the appropriate rles for which the anteceents or the conseqents satisfy the actal conitions. As a conseqence of this process, systems with a large set of rles (over rles) can be slow, an ths they may be nsitable for real-time applications, especially in the fiel of embee systems [3]. In the following a novel knowlege representation which is base on Boolean algebra an logic networks will be introce. On this base, a for-vale logic system is introce. This new moel reslts in a significantly more efficient inference processing than the classical one. The comptational improvement is estimate to be at least two orers of magnite, which is e to the reqire small memory space an fast operations in the logic omain. III. THE USE OF BOOLEAN ALGEBRA AND LOGIC NETWORKS The relationships of Boolean algebra can also be se for the rle-base systems. As it is well-known, Boolean logic involves two vales: (false) an (tre), where the following three basic operations are se: logic AND (enote by the mltiplication point ( ), logic OR (enote by the aition sign (+), an logic NOT (enote by an apostrophe scceeing the actal variable). For instance, A means the negation of A. It can be easily seen that the logic conitions within the rles can irectly be sbstitte by the corresponing Boolean operations an logic gates [7]. As an example let s consier the following set of rles, where the facts are enote by capital letters: IF C AND D THEN L, IF NOT E THEN K, IF L OR K THEN P, IF E AND M THEN Q. The Boolean escription of the above rles is the following: L = C D, K = E, P = L + K, Q = E M. These for rles can be represente by for logic gates: two AND gates, one NOT gate, an one OR gate. Now, if we connect the inpts an otpts of these gates in accorance with the ientical letters, the logic network of Figre will be obtaine. It shol be note here that in case of a simple irect rle, for example, IF U THEN V, its corresponing Boolean form will be V = U. This relation is represente by a YES gate which oes not moify its inpt vale.

3 C D L P K E M Q Fig.. The logic network for the rle base IV. THE USE OF A FOUR-VALUED LOGIC SYSTEM As known, the original Boolean algebra is base on a two-vale logic, i.e., on an. These are calle etermine vales as well. If a fact is tre in the inference process, then its logic variable will have the vale, if it is false then its vale is. However, as far as the general algebraic treatment of rle bases is concerne, it reqires more than these two vales. It can be prove that the nmber of necessary an sfficient vales is for. It means that in aition to an, two more vales are to be involve. These are as follows: ) The inifferent or on t care logic vale:. It is interprete in sch a way that the network line which carries this vale can take on either or freely, withot inflencing the comptational reslts. 2) The nknown logic vale:. In this case we have not any knowlege abot the concrete logic vale (, or ) of the network line carrying. The treatment of the for vales can be extene to the basic Boolean operations. This extension is smmarize in the trth tables of Table, below: TABLE. TRUTH TABLES OF THE FOUR-VALUED LOGIC SYSTEM AND OR NOT

4 Next we are going to show how these tables are se for forwar an backwar chaining. To reach this goal, consier the rle base above an the logic network belonging to it (see Figre ). Let the initial set of facts be as follows: T = {A, B, C, D, E, G, H}. A. The forwar chaining procere In or representation, the forwar chaining is performe in the following way: Step : C = an D =, since they both are in the fact base, which reslts in L =, so L is place in the fact base. Step 2: E =, becase E is in the fact base, from which it follows that K =, bt L = alone implies P =, so P is place in the fact base. Step 3: The fact base oes not contain M. In or logic system it can be interprete as M =. Thogh E =, e to the nknown vale of M, this is not sfficient to imply Q =. It means that Q =, ths Q cannot be place in the fact base. Here the final conclsion of the forwar chaining was that fact P is tre alone. The above comptational procere can also be calle forwar tracing of the logic vales. It means that we calclate the otpt vales of the logic gates with knowlege of the gate-inpt vales. B. The backwar chaining procere The same way as before, the backwar chaining procere involves backwar tracing of the logic vales throgh the network. Now the inpt vales of a gate have to be etermine with knowlege of the actal gateotpt vale. In this case the goal is to jstify that a primary otpt vale is, i.e., a selecte fact is tre. It reqires a sccessive ecision process which is also calle line-vale jstification [9] [2]. As known, line-vale jstification is a procere with the aim of sccessively assigning inpt vales to the logic elements in sch a way that they are consistent with each previosly assigne vale. (This concept is an axiliary calclation process for jstifying an initial set of logic vales in a network, first applie in the so-calle D-algorithm, for two-vale logic [9].) The backwar tracing of the logic vales can also be performe in accorance with the for-vale trth table. However, this principle iffers in some points from the forwar tracing. In case of forwar tracing, the otpt vale of a gate is to be calclate with knowlege of the inpt vales at the gate. If the inpts are given then they etermine the otpt nambigosly. On the other han, for a given otpt vale at a gate not only one inpt combination can be assigne, there may be more than one possible choices. If two-inpt gates are consiere, the possible choices are smmarize in Figre 2 an Figre 3. If the nmber of inpts at a gate were more than two, it wol increase the nmber of choices, bt wol not case any ifference in principle. No soltion exists Fig. 2. Backwar tracing choices for an AND gate

5 No soltion exists Fig. 3. Backwar tracing choices for an OR gate When performing this process the following viewpoints have to be taken into consieration: Only the etermine logic vales, an, have to be trace back, i.e., these vales are to be jstifie at the gate inpts. The vale of nees no jstification, so it is nnecessary to trace it back. The otpt vale of is jstifie only by the inpt vales of. Since oes not reqire jstification, it is worth assigning the minimm nmber of etermine vales to the gate inpts, while leaving the others at the vale of. In this logic system, the etermine vales an are consistent only with. This fact is to be taken into consieration when a network line has alreay a previosly assigne vale, an another vale is reqire at the same line. Whenever a contraiction, i.e., inconsistency occrs, we have to make a new choice or change the last possible ecision. In or example (Figre ), the comptations procee as follows: Step : The proof of P = : At first let K = an L =, which are the minimally necessary assignments. Now from K = it follows that E =, which is a contraiction, for E is in the fact base, so E = hols. Step 2: We have to moify or previos ecision: Now let L = an K =. In this way L = is jstifie by C = an D =, withot any contraiction. Step 3: It is nnecessary to trace back the vale K =, since the inifferent vale oes not reqire jstification. So the proof of P has been finishe. Step 4: The proof of Q = : This conition reqires that both inpts to the AND gate be, i.e., E = an M =. Since E is a member of the fact base, E = hols. However, M is missing from the fact base, which means that M =. In this case it is impossible to jstify (prove) that Q =. V. CONCLUDING REMARKS This paper has presente a logic moel which is irectly applicable for inference chains in expert systems. Both forwar an backwar reasoning can be performe on the base of the moel. In comparison with the conventionally organize knowlege bases, the calclations sing this for-vale logic can avantageosly be organize an carrie ot in embee compting systems e to the following reasons: The storage reqirement of the for logic vales at the network lines is negligible: only two bits are necessary an sfficient for coing them.

6 Comptations among logic vales are ab ovo fast an efficient. This fact manifests itself especially when bit-level implementation is applie. The ata-base strctre of a logic network is comparatively simple. Only the gate types an the inpt-otpt connections of the gates are to be encoe an store. The forwar an backwar tracing are carrie ot irectly on this network strctre. VI. REFERENCES [] Massa, A.J., Embee Software Development with ecos, Prentice-Hall, Inc., USA, 23. [2] Waterman, D.A., A Gie to Expert Systems, Aison-Wesley Pblishing Company, USA, 986. [3] Negnevitsky, M., Artificial Intelligence: A Gie to Intelligent Systems, Aison-Wesley Pblishing Company, Great Britain, 22. [4] Chen, Zh., Comptational Intelligence for Decision Spport, CRC Press LLC, USA, 2. [7] Cormen, T.H., Ch. E. Leiserson, R.L. Rivest, C. Stein: Introction to Algorithms, McGraw-Hill Pblishing Company, USA, 2. [8] Uyemra, J.P., A First Corse in Digital Systems Design: An Integrate Approach, Brooks-Cole Pblishing Company, USA, 2. [9] J.P. Roth, Diagnosis of Atomata Failres: a Calclation an a Metho, IBM Jornal of Research an Development, USA, Jly 966, Vol., pp , [] Abramovici, M., M.A. Breer, an A.D. Frieman, Digital Systems Testing an Testable Design, Compter Science Press, USA, 99. [] J. Sziray, Test Calclation for Logic Networks by Composite Jstification, Digital Processes, Great Britain, 979, Vol. 5, No. -2, pp [2] J. Sziray, Test Calclation for Logic an Timing Falts, IEEE International Workshop on RTL an High Level Testing, (WRTLT-24), Osaka, Japan, November -2, 24. Proceeings, pp [5] Lewis, H.R., Ch. H. Papaimitrio, Elements of the Theory of Comptation, Prentice-Hall, Inc., USA, 998. [6] Hopcroft, J.E., R. Motwani, J.D. Ullman, Introction to Atomata Theory, Langages, an Comptation, Secon Eition, Aison-Wesley Pblishing Company, USA, 2.