CS344: Introduction to Artificial Intelligence

Transcription

1 CS344: Introduction to Artiicil Intelligence Lecture: Herbrnd s Theorem roving stisibilit o logic ormule using semntic trees rom Smbolic logic nd mechnicl theorem proving B Runk ilni Under the guidnce o ro.. Bhttchr

2 Bsic Deinitions Interprettion: Assignment o mening to the smbols o lnguge Interprettions o redicte logic requires deining: Domin O Discourse D which is set o individuls tht the quntiiers will rnge over Mppings or ever cons stnt n-r unction nd n-r predicte to elements n-r unctions D n D nd n- r reltions on D respe ectivel

3 Bsic Deinitions c contd. Stisibilit Consistenc A ormul G is stisibl e i there eists n interprettion I such tht G is evluted to T True in I I is then clled model o G nd is sid to stis G Unstisibilit Inconsiste enc G is inconsistent i theree eists no interprettion tht stisies ii G

4 Need or the theo orem roving stisibilit o ormul is better chieved b proving the unstisibilit o its negtion roving unstisibilit over lrge set o interprettions is resource intensive Herbrnds Theorem reduces the number o interprettions tht need to be checked ls undmentl role in Automted Theorem roving

5 Skolem Stndrd Form Logic ormule need to irst be converted to the Skolem Stndrd Form m which leves the ormul in the orm o set o cluses This is done in three ste eps Convert to rene Form Convert to CNF Conjunctive Norml Form Eliminte eistentil unitiiers using Skolem unctions

6 Step 1: Converting to rene Form Involves bringing ll quntiiers to the beginning o the ormul i i M i= n Where - i is either V Universl untiier or Ǝ Eistentil ti uniti ier nd is clled the prei - M contins no untiiers nd is clled the mtri ti

7 Emple Emple z z z z z z z z z z z z z z

8 Step 2: Convertin g to CNF Remove nd Appl De Morgn s lws Appl Distributive lws Appl Commuttive s well s Associtive lws

9 Emple Emple z z z R z R z R z z R z z R z z R z z R

10 Step 3: Skolemiz tion Consider the ormul 1 1 n n M I n eistentil quntiier r is not preceded b n universl quntiier then r in M cn be replced b n constnt c nd r cn be removed Otherwise i there re m universl quntiiers beore r then An m-plce unction p 1 p 2 p m cn replce r where p 1 p 2 p m re the vribles tht hve been universll quntiied Here c is skolem vrible while is skolem unction

11 Emple z R z z R z We eliminte i nd z using sko olem unctions becomes nd z becomes g s is the onl preceding universl quntiier R g g R g

12 Herbrnd Univers se It is inesible to consider ll interprettions over ll domins in order to prove unstisibilit Insted we tr to i specil domin clled Herbrnd universe such tht the ormul S is unstisible i it is lse under ll the interprettions over this domin

13 Herbrnd Univers se contd. H 0 is the set o ll constnts in set o cluses S I there re no constnts in S then H 0 will hve single constnt s H 0 = } For i=123 let H i+1 be the union o H i nd set o ll terms o the orm n t1 t n or ll n-plce unctions in S where t j where j=1 n re members o the set H H is clled the Herbrnd universe o S

14 Herbrnd Univers se contd. Atom Set: Set o the ground toms o the orm n t 1 t n or ll n-plce predictes n occuring in S where t 1 t n re elements o the Herbrnd Universe o S Also clled the Herbrnd Bse A ground instnce o cluse C o set o cluses is cluse obtined b replcing vribles in C b members o the Herbrn nd Universe o S

15 Emple Emple } H 0 S = = } } } 2 1 H H = = } 2 H = M C Let H = = : Atom Set nd Here A = } } } K...} C re both ground instnces o }

16 H-Interprettions For set o cluses S with its Herbrnd universe H we deine I s n H-Interprettion i: I mps ll constnts in S to themselves An n-plce unction is ssigned unction tht mps h 1 h n n element in H n to h 1 h n n element in H where h 1 h n re elements in H Or simpl stted s I=m 1 m 2 m n } where m j = A j or ~A j i.e e. A j is set to true or lse nd A = A 1 A 2 A n }

17 H-Interprettions contd. Not ll interprettions re H-Interprettions Given n interprettion I over domin D n H- Interprettion I* corresponding to I is n H- Interprettion tht: Hs ech element rom the Herbrnd Universe mpped to some elemen nt o D Truth vlue o h 1 h n in I* must be sme s tht o d 1 d n in I

18 Emple Emple Herbrnd Universe } H H R S Let = given b is Atom Set Herbrnd Universe A A H H = = Some Herbrnd Interprettion A = 2 1 I I = = 3 I = } } } } K R = re ns } K K K R } } K K K K K K R R } K K K R

19 Use o H-Interpre ettions I n interprettion I stisies set o cluses S over some domin D then n one o the H- Interprettions I* corresponding to I will lso stis H A set o cluses S is unstisible i S is lse under ll H-Interprettions o S

20 Semntic Trees Finding proo or set o cluses is equivlent to generting semntic tree A semntic tree is tree where ech link is ttched with inite set o toms or their negtions such tht: Ech node hs onl in nite set o immedite links For ech node N the union o sets connected to links o the brnch down to N do oes not contin complementr pir I N is n inner node the en its outgoing links re mrked with complementr literls

21 Semntic Trees C Contd. Ever pth to node N does not contin complementr literls in IN where IN is the set o literls long the edges o the pth A Complete Semntic Tree is one in which ever pth contins ever literl in Herbrnd bse either +ve or ve but not both A ilure node N is one which lsiies I N but not I N where N is predecessor o N A semntic tree is closed i ever pth contins ilure node

22 Emple Imge courtes: S is stisible becuse it hs t lest one brnch without ilure node

23 Emple Imge courtes: S is unstisible s the tree is closed

24 Herbrnd s Theor rem Ver. 1 Theorem: A set S o cluses is unstisible i corresponding to ever complete semntic tree o S there is inite closed semntic tree roo: rt 1: Assume S is unstisible ii - Let T be the complete semntic tree or S - For ever brnch B o T we let I B be the set o ll literls ttched to the links in B

25 Version 1 roo c contd. -I B is n interprettion o S b deinition - As S is unstisible I B must lsi ground instnce o cluse C in S let s cll it C - T is complete so C must be inite nd there must eist ilure node N B inite distnce rom root on brnch B - Ever brnch o T hs ilure node so we ind closed semntic tree T or S - T hs inite no. o nodes Konig s Lemm Hence irst hl o thm. is proved

26 Version 1 roo c contd. rt 2: I there is inite closed semntic tree or ever complete semntic tree o S - Then ever brnch contins ilure node - i.e. ever interprettion n lsiies S - Hence S is unstisible Thus both hlves o the theorem re proved

27 Herbrnd s Theor rem Ver. 2 Theorem: A set S o cluses is unstisible i there is inite unstisible set S o ground instnces o cluses o S roo: rt 1: Assume S is unstisible ii - Let T be completee semntic tree o S - B ver. 1 o Herbrnd Thm. there is inite closed semntic tree T corresponding to T

28 Version 2 roo c contd. - Let S be set o ll the ground instnces o cluses tht re lsiied t ll ilure nodes o T - S is inite since T contins inite no. o ilure nodes - Since S is lse in ever interprettion o S S is lso unstisible Hence irst hl o thm. is proved

29 Version 2 roo c contd. rt 2: Suppose S is inite unstisible set o gr. instnces o cluses in S - Ever interprettion I o S contins n interprettion I o S - So i I lsiies S then I must lso lsi S - Since S is lsiied b ever interprettion i I it must lso be lsiied b ever interprettion I o S - i.e. S is lsiied b ever interprettion o S -Hence S is unstisible Thus both hlves o the thm. re proved

30 Emple Let S = } This set is unstisible Hence One b Herbrnd's Th heorem there is inite unstisible set S' o ground instnces o cluses o S o these sets cn be S ' = }

31 Reerences Chng Chin-Ling nd Lee Richrd Chr-Tung Smbolic Logic nd Mechnicl Theorem roving Acdemic ress New York NY 1973