Using Rules for the Visualization of Tableaux Proof Techniques for Propositional Logic

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1 Using Rules for the Visuliztion of Tbleux Proof Techniques for Propositionl Logic Nd Shrf, Slim Abdenndher Computer Science nd Engineering Deprtment The Germn University in Ciro Ciro, Egypt {nd.hmed, Thom Frühwirth Institute of Softwre Engineering nd Compiler Construction University of Ulm Ulm, Germny Abstrct This pper discusses rule-bsed pproch for visulizing proofs using tbleux techniques. Semntic Tbleu is usully used to prove by refuttion. In ddition, tbleux techniques re commonly studied in different courses. Visuliztion is n effective teching methodology. The vilbility of visuliztion methods tht could be used by instructors is thus importnt. Index Terms Tbleu Proof, Visuliztion, Constrint Hndling Rules I. INTRODUCTION Semntic tbleux is procedure used for checking stisfibility [1] in propositionl logic. Tbleux bsiclly tries to serch for model. If such stisfying interprettion ws found then the expression is mrked s stisfible. Semntic Tbleux could be pplied for first-order logic formuls s well s propositionl logic. In the pper, we focus on propositionl logic for the proof of the concept. In this cse, formuls could be checked for stisfibility. In cse the formul is stisfible, the tbleux would be ble to find model for it. For exmple formul b ( b) is unstisfible. In this cse, ll the brnches of the corresponding tbleu will be closed brnches indicting the lck of model. This methodology hs been used in different contexts nd logics [2] [3]. Due to the importnce of this proof technique, it is lso studied in different curricul. The existence of visuliztion methodologies hve proven to be effective in teching [4]. Throughout different studies, groups using lgorithm visulistion techniques hd better results thn groups with clssicl methodologies [5]. Such visuliztion pltforms could dditionlly be used by instructors in clsses nd office hours. The im of the presented work is to provide tool tht students nd instructors would benefit from in the context of tbleux proof techniques. With the new tool, the student could evlute ny expression. The tool would thus be helpful in lerning s students get to see in step-by-step mnner how the computtion is done. In other words, the tool incorportes the strength of visuliztion into studying tbleux proof techniques. Instructors cn lso use visuliztion to provide esy-to-follow exmples in the clssrooms. The tool uses rules to represent the tbleux proof technique. In ddition, the visuiztion detils re represented using the so-clled nnottion rules [6] [8]. This mkes the engine of the tool declrtive nd extensible. The pper is orgnized s follows: Section II introduces semntic tbleux in propositionl logic using exmples. Section III introduces Constrint Hndling Rules (CHR). Section IV shows how the Tbleux technique ws implemented using rules through the use of CHR. Section V shows how the tbleu proof technique ws visulized. The section gives some exmples s well. The pper finlly concludes with directions to future work. II. SEMANTIC TABLEAUX Semntic Tbleux tries to build bck the truth tble. Semntic tbleux is in mny cses used to prove by refuttion. In other words, in cse the sttement p is to be checked, we would try to prove tht p is flse. Tht wy, p must be true [1], [9]. With propositionl logic however, we cn use semntic tbleu to prove the stisfibility of n expression. Without loss of generlity, hving rules for, nd would produce sound nd complete results. The rules re given below: b b b An expression b mens tht nd b hve to be true. On the other hnd, the expression b mens tht if or if b is true, the expression holds. Tht is why brnching occurs since it could be tht one of the brnches fils but the expression still holds. Finlly, mens tht hs to be true. These rules re ble to hndle different types of expressions. A brnch tht contins contrdiction is closed brnch nd cn never led to solution. For exmple the expression: c (( d) (b c)) produces the tbleu: b

2 c ( d) (b c) d b c d b c In this cse, the first brnch is closed one due to the existence of nd in it. The lst brnch is lso closed becuse it contins c nd c. The other two open brnches represent two possible models to the expression. The first model is hving, c nd d s true sttements. The second model is to hve, c nd b s true sttements. III. CONSTRAINT HANDLING RULES This section introduces Constrint Hndling Rules (CHR) [10] [12]. CHR is declrtive rule-bsed lnguge. CHR ws introduced for writing constrint solvers. However, due to its declrtivity nd ese-of-use, it hs developed into generl purpose lnguge. The rules of CHR progrm ct on constrints in the constrint store. There re two types of constrints: CHR or user-defined constrints nd builtin constrints hndled by the host lnguge. A CHR rule consists of hed (H), body (B) nd n optionl gurd (G). The gurd is condition for pplying the rule. A rule could lso hve nme. A hed contins CHR constrints only. A gurd could only hve built-in constrints. The body, on the other side, could contin CHR or built-in constrints. A CHR progrm consists of set of simpgtion rules. A CHR rule is pplicble if the store contins constrints mtching the constrints in the hed of the rule nd if the gurd of the rule is stisfied [11]. A simpgtion rule hs the form: H K \ H R G B. A simpgtion rule hs two types of hed constrints: H K nd H R. H K re the kept hed constrints while H R re the removed ones. On pplying simpgtion rule, constrints mtching H R re removed from the constrint store while the ones mtching H K re kept in the store. There re two specil types of simpgtion rules: simplifiction nd propgtion rules. A simplifiction rule is simpgtion rule with empty H K while propgtion rule hs n empty H R. A simplifiction rule hs the form: H R G B. Simplifiction rules replce the hed constrints (H R ) with the body constrints (B). A propgtion rule hs the form: H K G B. If propgtion rule is pplied, it dds the body constrints (B) to the constrint store. No constrints re thus removed from the store [11]. A wide rnge of lgorithms were implemented through CHR. The following single-ruled progrm is ble to find the smllest number mong set of numbers. The numbers re represented by the constrint min/1. Every time the rule find_min is pplied, two numbers (A nd B) re compred ginst ech other. The gurd mkes sure tht A is less thn B. The smller number is kept in the constrint store nd the lrger number is removed from the store. On successive ppliction of this rule, only the smllest number survives. f i n d _ m i min (A) \ min (B) <=> A<B t rue. The following exmple shows the constrint store fter ech ppliction of the rule for the query min(6), min(5), min(4). The underlined constrints represent the ones tht mtched the hed of the rule. The first ppliction of the rule removes 6 nd keeps 5. The second ppliction removes 5 nd keeps only the constrint holding 4 in the constrint store. 4 is indeed the minimum number. min(6), min(5), min(4) min(5), min(4) min(4) IV. TABLEAUX WITH CONSTRAINT HANDLING RULES CHR could be used to implement Boolen constrint systems [13], [14]. To implement the Tbleux proof technique with CHR, different constrints were used. Ech brnch is lbeled with n identifier. Once brnch B is dded, it is ssumed to be successful brnch by utomticlly dding the constrint success(b). Ech component in brnch is represented by the constrint brnch/3. For exmple, the constrint, brnch(bnum,l,e), represents the existence of expression E in brnch BNum t level L. In ddition, the totl number of brnches is represented by the constrint brnchest(num). The constrint brnch/2 represents the number of nodes in brnch. brnch(2,1) mens tht the brnch identified by 2 hs one node only. A brnch fils once it contins n expression nd its negtion. This is represented by the CHR rule: b r n c h (BNum, Level1,A), b r n c h (BNum, Level2, not (A) ) ==> b F i l (BNum). In cse brnch hs disjunction, brnching hppens. The rule oring, hndles this cse. The rule is fired once brnch BNum hs disjunction t level (Level). In this cse, two new brnches re dded. The first brnch is identified by the number NumB. The second brnch is identified with NumB1. Using the constrint bances/2, the old brnch is mrked s n ncestor for the two new brnches. The rule is only executed if the disjunction occurs t so-fr successful brnch. Otherwise there is no need to continue processing the brnch. o r i n s u c c e s s (BNum), b r n c h (BNum, Mx ) \ b r nch (BNum, Level, or (A, B) ), brnchest (NumB) <=> NumB1 i s NumB + 1, NumB2 i s NumB + 2 b r n c h (NumB, 1 ), s u c c e s s (NumB), b r n c h (NumB, 0,A) bances (NumB,BNum)

3 , b r n c h (NumB1, 1 ), s u c c e s s (NumB1), b r n ch (NumB1, 0, B), bances (NumB1,BNum), brnchest (NumB2). The rule filsucc sets brnch to fil if one of its ncestors is filed brnch. f i l S u c b F i l (BNum), bances ( NewBrnch,BNum) ==> b F i l ( NewBrnch ). The rules brnchf2, nd brnchf3, hndle the cses where brnch nd its ncestor hve contrdicting nodes in this cse the child brnch fils. b r n c h (BNum1, Level1,A), b r n c h (BNum2, Level2, not (A) ), bances (BNum, BNum1) ==> b F i l (BNum). b r n c h (BNum1, Level1, not (A) ), b r n c h (BNum2, Level2,A), bances (BNum, BNum1) ==> b F i l (BNum). The rule newanc hndles the trnsitivity of the ncestor reltion. It sttes tht whenever B is n ncestor of A nd C is n ncestor of B then C is n ncestor for A. bances (A, B), bances (B, C) ==> bances (A, C). In cse the unstisfibility is to be checked for, the following rule is dded: f i n i s h \ s u c c e s s ( _ ) <=> f i l. The rule clenup mkes sure the progrm fils if there is brnch tht succeeds. An exmple of querying the progrm is s t r t, e x p r e s s i o n ( nd ( [, not ( ), or ( b, or ( c, d ) ) ] ) ), f i n i s h. In this cse the expression to be checked is () (b (c d)). As seen from the query the expression is embedded within the constrint expression. Two uxiliry constrints re lso dded: strt nd finish. The constrint strt/0 initilizes the procedure. It triggers the rule: r u l e S t r s t r t <=> brnchest ( 1 ), b r n c h ( 0, 0 ), s u c c e s s ( 0 ). The rule rulestrt dds the initil brnch identified with 0. The totl number of brnches re thus one. The totl number of nodes in brnch 0 is set to be zero. The rule expr hndles the initil expression entered through the query. The rule expr dds the conjuncts of the expression to the min brnch (brnch 0). e b r n c h ( 0,Num), e x p r e s s i o n ( nd ( [A B ] ) ) <=> Num1 i s Num + 1,N1 i s Num 1 b r n c h ( 0,Num1), b r n c h ( 0,Num,A), e x p r e s s i o n ( nd (B) ). V. VISUALIZATION OF TABLEAUX PROOFS The ide is to use the grph visuliztion methodology introduced in [15] in ddition to the nimtion procedure introduced in [6] to nimte the CHR implementtion of the proof technique. Fig. 1. Annotting brnch A. Annottion Rules for Animting CHR In [6], nnottion rules were used in CHRAnim to visulize the execution of CHR progrms. The ide ws to ssocite n occurrence of textsfchr constrint with visul object. Thus, whenever the constrint ws dded/removed, the corresponding visul objects gets dded/removed. By time, this would produce n nimtion of the lgorithm implemented by the progrm. The nnotted constrints in this cse re the interesting constrints whose occurrence ffect the dt structure mnipulted by the progrm. Users re provided wit n interfce through which they cn ssocite constrint with grphicl object s shown in Figure 1. Once the user choose the grphicl object, the node is populted with the corresponding prmeters. CHRAnim could be used with ny visuliztion pltform. The engine is ble to red the vilble objects file. On otherwords, CHRAnim outsources the ctul visuliztion to existing pltform to move the intelligence of the system to the nnottions insted. Ech prmeter cn hve different type of vlue including: 1) A constnt vlue e.g. green, ) The result of the function vlueof(arg) such tht Arg is one of the rguments of the nnotted constrint. The vlue could lso be n expression ht contins combintions of constnt vlues nd outputs of the vlueof function. The nnottion could lso hve precondition for ppliction. In ddition, rule could be nnotted to hve visul effect [6]. Due to spce constrints the detils of the rule nnottion re not given. In generl the ide, is tht the rule is nnotted with constrint tht is then given norml grphicl nnottions. This is lso done through the interfce. In this cse, once the rule is executed, visul effect hppen

4 independent of whether the constrints dded/removed re interesting nnotted constrints. B. Producing Animtions In our cse, the min im is to nnotte every brnch/3 constrint with grphicl node object. Ech brnch(bnum,level,e) is nnotted with node hving green bckground. The nme of the node is the result of conctenting the brnch number nd the level. The text inside the node is the expression E. The grphicl interfce through which the user cn do the nnottions is given in Figure 1. In ddition the rule oring is nnotted with two edge grphicl objects between the node contining the ored expression nd the two new nodes. In this cse the rule oring is nnottd with constrint e(oldnode,newnode) where OldNode nd newnode contin the two identifiers. The constrint e/2 is then nnotted with n edge object. An exmple of the visuliztion is given in Figure 2(). In Figure 2(c), different nnottion ws given. In this nnottion the constrint expression produces node with the remining conjuncts s long s there re more thn one. Since the brnch fils due to the existence of the two expressions nd (), the rest of the tree is not shown. Another nnottion exmple would ffect the result of the visuliztion. In this cse, the rule brnchf could be nnotted with the ction updtenode introduced in [15] to chnge the color of the two nodes tht cuse the brnch to fil to the color pink. In this cse, the new visulized tree is shown in Figure 2(b). Another exmple is given in Figure 3, in ddition to the bsic nnottions, the rules brnchf2 nd brnchf3 were similrly nnotted to highlight the contrdicting nodes in the sub-brnch only. As seen from the previous exmples, throughout chnging the nnottions, different visuliztions were produced. This could thus llow n instructor to djust the visuliztion in the wy they see suitble nd most fitting to the tught mteril. VI. CONCLUSIONS & FUTURE WORK The tool presented enbles students to see how expressions get evlution with the Tbleux techniques visully. The engine behind the pltform uses nnottions rules to bstrct the detils of the visuliztion nd the proof production. The nnottions could thus be pre-set for one time nd used by ll the students. Annottions could be lso provided through n interfce to eliminte the need of knowing ny of the detils. In the future, the pltform should be extended with First Order logic constructs. Vribles should be lso hndled nd different types of expressions re to be explored. [2] R. Hähnle nd B. Beckert, Proof confluent tbleu clculi, in Automted Resoning with Anlytic Tbleux nd Relted Methods, Interntionl Conference, TABLEAUX 99, Srtog Springs, NY, USA, June 7-11, 1999, Proceedings, ser. Lecture Notes in Computer Science, N. V. Murry, Ed., vol Springer, 1999, pp [Online]. Avilble: [3] B. Beckert nd R. 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Bnntyne, F. Bouli, R. Burkhrd, J. Counsell, U. Cvek, M. J. Eppler, G. G. Grinstein, W. Hung, S. Kernbch, C. Lin, F. Lin, F. T. Mrchese, C. M. Pun, M. Srfrz, M. Trutschl, A. Ursyn, G. Venturini, T. G. Wyeld, nd J. J. Zhng, Eds. IEEE Computer Society, 2016, pp [Online]. Avilble: [8], Using rules to nimte prolog progrms, in Proceedings of the Doctorl Consortium, Chllenge, Industry Trck, Tutorils nd RuleML+RR 2017 hosted by Interntionl Joint Conference on Rules nd Resoning 2017 (RuleML+RR 2017), London, UK, July 11-15, 2017., ser. CEUR Workshop Proceedings, N. Bssilides, A. Bikkis, S. Costntini, E. Frnconi, A. Giurc, R. Kontchkov, T. Ptkos, F. Sdri, nd W. V. Woensel, Eds., vol CEUR-WS.org, [Online]. Avilble: [9] M. D Agostino, D. M. Gbby, R. Hähnle, nd J. Posegg, Hndbook of tbleu methods. Springer Science & Business Medi, [10] T. W. Frühwirth, Theory nd prctice of constrint hndling rules, Journl of Logic Progrmming. Specil Issue on Constrint Logic Progrmming, vol. 37, no. 1âĂŞ-3, pp , [Online]. Avilble: [11] T. Frühwirth, Constrint Hndling Rules. Cmbridge University Press, August [Online]. Avilble: org [12] T. W. Frühwirth, Constrint Hndling Rules - Wht Else? in Rule Technologies: Foundtions, Tools, nd Applictions - 9th Interntionl Symposium, RuleML 2015, Proceedings, ser. Lecture Notes in Computer Science, N. Bssilides, G. Gottlob, F. Sdri, A. Pschke, nd D. Romn, Eds., vol Springer, 2015, pp [Online]. Avilble: [13] T. W. Frühwirth nd S. Abdenndher, Essentils of constrint progrmming, chpter 9, ser. Cognitive Technologies. Springer, [Online]. Avilble: [14], Principles of Constrint Systems nd Constrint Solvers, Archives of Control Sciences, vol. 16, no. 2, pp , [15] N. Shrf, S. Abdenndher, nd T. W. 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5 () Output (b) Output with different nnottion Fig. 2. Exmple (c) Output with nother different nnottion () Exmple with Brnching (b) Filed sub-brnch (c) Another Filed sub-brnch Fig. 3. Another Exmple