M APPING P ARAMETRIZED D IFFERENCE R EVISION O PERATORS TO B ELIEF C ONTRACTION Maria Andrikopoulou Theofanis Aravanis
S T URE O F P RESENT IO N RUC T AT Be lie f C ha ng e T he AG M Pa ra d ig m Re visio n a nd C o ntra c tio n Inte rd e fina b ility o f O p e ra tio ns Pa ra me trize d Diffe re nc e Re visio n O p e ra to rs (PDR O p e ra to rs) PDR O p e ra to rs in the Re a lm o f C o ntra c tio n C o nc lusio ns
B EL IEF C HANG E (R EVISIO N ) [1/ 2] Ma ry ha s just d isc o ve re d tha t G e o rg e a nd Dim itra a re no t he r • true p a re nts. She w a s a d o p te d w he n she w a s 6 m o nths o ld fro m a n • o rp ha na g e in Sa o Pa o lo . T he ne w s re a lly sho o k Ma ry. • Muc h o f w ha t she use d to b e lie ve a b o ut he rse lf a nd he r • fa m ily w a s w ro ng . She m ust, no w , p ut he r tho ug hts b a c k in o rd e r. • M. Andrikopoulou and T. Aravanis
B EL IEF C HANG E (R EVISIO N ) [2/ 2] A typ ic a l (a ltho ug h ra the r d ra m a tic ) insta nc e o f a • b e lie f-re visio n sc e na rio . A ra tio na l a g e nt re c e ive s ne w (c o ntra d ic ting ) • info rm a tio n, tha t m a ke s he r c ha ng e he r b e lie fs. Withd ra w so m e o f the o ld b e lie fs, b e fo re she c a n • (c o nsiste ntly) a c c o m m o d a te the ne w info rm a tio n. Ac c e p t the c o nse q ue nc e s tha t mig ht re sult fro m the • inte ra c tio n o f the ne w info rm a tio n w ith the o ld b e lie fs. M. Andrikopoulou and T. Aravanis
T HE AG M P ARADIG M [1/ 2] T he stud y o f Be lie f C ha ng e c a n b e tra c e d b a c k to the • e a rly ’ 80s, w ith the se m ina l wo rk o f Alc ho urró n, G ä rd e nfo rs, a nd Ma kinso n. T he y e sta b lishe d the AG M p a ra d ig m ; to this d a te , the • d o m ina nt fra m e w o rk in Be lie f C ha ng e . ( φ , ψ ) α Be lie fs a re m o d e le d a s se nte nc e s o f • p ro p o sitio na l la ng ua g e . Be lie f se ts ( Κ ) a re mo d e le d a s se ts o f se nte nc e s c lo se d • und e r lo g ic a l im p lic a tio n (the o rie s). T he p ro c e ss o f b e lie f c ha ng e is e nc o d e d a s a func tio n • tha t m a p s a the o ry a nd a se nte nc e to a ne w the o ry. M. Andrikopoulou and T. Aravanis
T HE AG M P ARADIG M [2/ 2] T he AG M p a ra d ig m stud ie s b o th b e lie f re visio n a nd b e lie f • c o ntra c tio n . Be lie f Re visio n: Inc o rp o ra tio n o f a se nte nc e tha t is • inc o nsiste nt w ith a b e lie f se t. Be lie f C o ntra c tio n: Withd ra w a l o f a se nte nc e fro m a • b e lie f se t. T he AG M p a ra d ig m c ha ra c te rize s ra tio na l b e lie f re visio n • a nd b e lie f c o ntra c tio n func tio ns, b y m e a ns o f a se t o f ra tio na lity p o stula te s fo r e a c h c a se . Princ ip le o f Minim a l C ha ng e : T he ne w b e lie f se t d iffe rs a s • little a s p o ssib le fro m the o ld b e lie f se t. M. Andrikopoulou and T. Aravanis
T HE AG M P O ST ES FO R R EVISIO N UL AT M. Andrikopoulou and T. Aravanis
T HE AG M P O ST ES FO R C O NT IO N UL AT RAC T M. Andrikopoulou and T. Aravanis
I NT Y O F O PERAT O RS ERDEFINABIL IT • Re visio n a nd c o ntra c tio n c a n b e d e fine d in te rm s o f e a c h o the r thro ug h L e vi a nd Ha rp e r Id e ntitie s . M. Andrikopoulou and T. Aravanis
C O NC RET E R EVISIO N O PERAT O RS • Se ve ra l c o nc re te “ o ff the se lf” re visio n func tio ns (o p e ra to rs), imp le me nting the p ro c e ss o f b e lie f re visio n, ha ve b e e n p ro p o se d . • Amo ng the we ll-kno wn p ro p o sa ls, o nly Da la l’ s re visio n o p e ra to r sa tisfie s the full se t o f AG M p o stula te s fo r re visio n (Κ* 1) – (Κ* 8). • Simp le a nd intuitive c o nstruc tio n. M. Andrikopoulou and T. Aravanis
P ARAMET RIZED D IFFERENC E R EVISIO N O PERAT O RS • Intro d uc e d b y Pe p p a s a nd Willia ms (2016). • Sa tisfy the AG M p o stula te s fo r re visio n. • Na tura l g e ne ra liza tio ns o f Da la l’ s re visio n o p e ra to r, with a muc h g re a te r ra ng e o f a p p lic a b ility . • Lo w re p re se nta tio na l a nd c o mp uta tio na l c o st, tha t ma ke s the m id e a l fo r re a l-wo rld a p p lic a tio ns. • Ha ve b e e n a xio ma tic a lly d e fine d , ve ry re c e ntly, in the re a lm o f re visio n. • We p ro vid e a n a xio ma tic c ha ra c te riza tio n in the re a lm o f c o ntra c tio n . M. Andrikopoulou and T. Aravanis
PDR O PERAT O RS IN T HE R EAL M O F R EVISIO N [1/ 2] • PDR o p e ra to rs ha ve b e e n a xio ma tic a lly d e fine d in the re a lm o f re visio n b y Pe p p a s a nd Willia ms (2018). M. Andrikopoulou and T. Aravanis
PDR O PERAT O RS IN T HE R EAL M O F R EVISIO N [2/ 2] • PDR o p e ra to rs a re a p ro p e r sub c la ss o f the who le c la ss o f AG M re visio n func tio ns. M. Andrikopoulou and T. Aravanis
PDR O PERAT O RS IN T HE R EAL M O F C O NT IO N RAC T • We p ro vid e d the a xio ma tic c ha ra c te riza tio n o f PDR o p e ra to rs in the re a lm o f b e lie f c o ntra c tio n . • Le vi a nd Ha rp e r Id e ntitie s we re utilize d . M. Andrikopoulou and T. Aravanis
O NE - T O - O NE C O RRESPO NDENC E M. Andrikopoulou and T. Aravanis
C O NC L USIO NS • PDR o p e ra to rs b ring us a ste p c lo se r to the d e ve lo p me nt o f a suc c e ssful AG M b e lie f-c ha ng e syste m, d ue to the ir fa vo ra b le p ro p e rtie s. • We ha ve ma p p e d PDR o p e ra to rs in the re a lm o f b e lie f c o ntra c tio n, b y me a ns o f the Le vi a nd Ha rp e r Id e ntitie s, c ha ra c te rizing the c la ss o f PDC o p e ra to rs. • T he a xio ma tic c ha ra c te riza tio n o f this ne w c la ss o f o p e ra to rs ha s b e e n c o mp le te d fo r the two p ro c e sse s o f b e lie f c ha ng e (re visio n a nd c o ntra c tio n).
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