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1 leqcia III-IV. interpretacia. Sesrulebadoba da WeSmaritoba. modelebi. formulebs gaacnia azri mxolod masin, roca matsi Semaval simboloebs gaacnia interpretacia. rogorc vicit, simboloebi warmoadgenen Teoriis enis nawils. Teoriis ena arvnisnot L-iT. interpretaciis qves Cven gvesmis funqcia :L D, romelic asaxavs enas algebrul sistemasi (D,{g i j },{R i j }), sadac (a i ) D, (f i j )=g i j, (P i j )=R i j, da g i j :D j D da R i j D j aris Sesabamisad j adgiliani operacia da mimarteba D-Si. D-s ewodeba interpretaciis are. mocemul interpretaciasi cvladebi moisazreba rogorc cvladebi D ze, xolo kavsirebs, da kvantorebs eniweba mati Cveulebrivi azri. mocemuli interpretaciisatvis nebismieri formula Tavisufali cvladebis garese (an sxvanairad, Caketili formula) warmoadgens gamonatqvams, romelic WeSmaritia an mcdaria, xolo nebismieri formula Tavisufali cvladebit gamoxatavs romelirac mimartebas interpretaciis aresi; es mimarteba SeiZleba iyos Sesrulebadi (WeSmariti) cvladebis romelirac mnisvnelobebisatvis interpretaciis aredan da ar iyos Sesrulebadi (mcdari) sxva mnisvnelobebisatvis. m a g a l i T e b i: (i) P 2 (x, x 2 ) ; (ii) x 2 P 2 (x, x 2 ) ; (iii) x 2 x P 2 (x 2, x ). Tu Cven virebt interpretaciis ared dadebit mtel ricxvta simravles da P 2 (y,z)-is interpretacia aris y z, masin (i) warmoadgens mimartebas y z, romelic Sesrulebadia yvela iset dadebit mtel ricxvta (a,b) dalagebul wyvilebistvis, sadac a naklebia an tolia b-ze; (ii) warmoadgens Tvisebas (e. i. unarul (ertadgilian) mimartebas) yoveli dadebiti mteli y ricxvistvis, y z, romelic sruldeba mxolod ricxvistvis ; dabolos, (iii) aris WeSmariti gamonatqvami, romelic amtkicebs umciresi dadebiti mteli ricxvis arsebobas. Cven rom interpretaciis ared agvero yvela mteli ricxvebis simravle, masin (iii)aarmocndeboda mcdari. Sesrulebadobis da WeSmaritobis cneba intuiciurad gasagebia, magram moviyvanot ufro zusti formulireba. davusvat, gvaqvs romelime interpretacia Tavisi interpretaciis areti D da iyos yvela Tvlad mimdevrobata simravle, romlis elementebi ekutvnis D-s. ganvsazrvrot, ras nisnavs, rom formula Sesrulebadia s = (b, b 2,. ) mimdevrobaze - dan mocemuli interpretaciisatvis. winaswar Cven ganvsazrvravt ertadgilian funqcias s*, romlis domeinia yvela termta simravle, xolo kodomeini aris D, e. i. s*: T D. () Tu termi t aris sagnobrivi cvladi x i, masin s*(t) = b i.
2 (2) Tu termi t aris sagnobrivi konstanta a i, masin s*( a i ) emtxveva mis interpretacias (a i ) D. (3) Tu f n i funqcionaluri asoa, romlis interpretacia ( f n n i ) = g i aris n adgiliani operacia D ze da t,, t n termebia, masin s*(f n i (t,, t n )) = g n i (s*(t ),, s*(t n )). amrigad s* aris funqcia, romelic ganisazrvreba s mimdevrobit da asaxavs termta simravles D-Si. Tu vilaparakebt araformalurad, masin nebismieri s = (b, b 2,. ) mimdevrobistvis da nebismieri t termistvis s*(t) aris D simravlis elementi, romelic miireba x i cvladis Canacvlebis Sedegad b i elementit yoveli i Tvis term t-si da Semdeg interpretaciis yvela operaciis SesrulebiT, romlebic Seesabamebian term t-si Semaval funqcionalur asoebs. magalitad, Tu t aris f 2 2 (x 3,f 2 (x, a )), interpretaciis 2 are aris mtel ricxvta simravle, f 2 da f 2 interpretirdeba Sesabamisad rogorc Cveulebrivi gamravleba da jami, xolo a rogorc 2; masin nebismieri s=(b,b 2,.) mimdevrobistvis s*(t) warmoadgens mtel ricxvs b 3 (b + 2). axla Cven SevudgeT ZiriTad gansazrvrebas, romelsac ganvmartavt formulis induqciur gansazrvraze dayrdnobit. (i) Tu aris elementaruli formula P n n j (t,, t n ) da R j aris misi Sesabamisi mimarteba (P n j ) mocemul interpretaciasi, masin formula Sesrulebulia s mimdevrobaze masin da mxolod masin, roca R n j (s*(t ),,s*(t n )), e. i. n elementiani mimdevroba n (s*(t ),,s*(t n )) ekutvnis mimarteba R j, e. i. (s*(t ),,s*(t n )) R n j. (ii) formula Sesrulebulia s-ze masin da mxolod masin, roca formula ar aris Sesrulebuli s-ze. (iii) formula Sesrulebulia s-ze masin da mxolod masin, roca formula ar aris Sesrulebuli mimdevroba s-ze an roca formula Sesrulebulia s-ze. (iv) formula x i Sesrulebulia s-ze masin da mxolod masin, roca formula Sesrulebulia nebismier mimdevrobaze -dan, romelic gansxvavebulia s-gan araumetes Tavisi i-uri komponentit. sxvagvarad rom vtqvat, formula Sesrulebadia s=(b,b 2,.) mimdevrobaze masin da mxolod masin, roca formulis Tavisufali x i cvladis nebismieri b D simboloti Canacvleba gvazlevs WeSmarit winadadebas mocemul interpretaciasi. formula -s ewodeba WeSmariti (mocemul interpretaciasi) masin da mxolod masin, roca is sruldeba nebismier mimdevrobaze dan. formula -s ewodeba mcdari (mocemul interpretaciasi), Tu is ar sruldeba arcert mimdevrobaze dan. mocemul interpretacias ewodeba modeli formulata mocemuli simravlisatvis, Tu yoveli formula -dan WeSmaritia mocemul interpretaciasi. 2
3 SeamowmeT Semdegi debulebebi, romlebic gamomdinareoben gansazrvrebidan. (I) mcdaria mocemul interpretaciasi masin da mxolod masin, roca WeSmaritia igive interpretaciasi, da WeSmaritia masin da mxolod masin, roca mcdaria. (II) arcerti formula ar SeiZleba iyos ertdroulad WeSmariti da mcdari ertdaigive interpretaciasi. (III) Tu mocemul interpretaciasi WeSmaritia da, masin WeSmaritia. (IV) mcdaria mocemul interpretaciasi masin da mxolod masin, roca WeSmaritia amave interpretaciasi, xolo mcdaria. (V) (i) Sesrulebulia s-ze masin da mxolod masin, roca formula Sesrulebulia s ze da Sesrulebulia s ze. Sesrulebulia s-ze masin da mxolod masin, roca formula Sesrulebulia s ze an Sesrulebulia s ze. Sesrulebulia s-ze masin da mxolod masin, roca an formula Sesrulebulia s ze da Sesrulebulia s ze, an ar aris Sesrulebuli s ze da ar aris Sesrulebulia s ze. (ii) x i Sesrulebulia s ze masin da mxolod masin, roca Sesrulebulia ert mainc s mimdevrobaze, romelic gansxvavdeba s gan araumetes mxolod i uri komponentit. (VI) WeSmaritia mocmul interpretaciasi masin da mxolod masin, roca amave interpretaciasi WeSmaritia x i. mocemuli formulis Caketva ewodeba formulas, romelic miireba zogadobis kvantorebis minawerit -s win, romlebic Seicaven indeqsebis klebadobis mixedvit -s yvela Tavisufal cvladebs. formulis Caketvas, romelic ar Seicavs Tavisufal cvladebs, vuwodebt TviT am formula -s. (magalitad, Tu aris P 2 (x 2, x 5 ) x 2 P 3 (x, x 2, x 3 ), masin -s Caketva iqneba x 5 x 3 x 2 x.) (VII) tavtologiis yoveli kerzo SemTxveva WeSmaritia nebismier interpretaciasi. (mocemuli propoziciuli formis kerzo SemTxvevas Cven vuwodebt nebismier formulas, romelic miireba am formulasi propoziciuli asoebis nacvlad I rigis Teoriis (predikatta arricxvis) formulebis CanacvlebiT im pirobit, rom ertdaigive propoziciuli asoebis Semavlobis adgilas ertdaigive formula Cainacvleba.) (VIII) davusvat, rom formula -s yvela Tavisufali cvladi imyofeba x,..., cvladebs Soris. masin Tu s da s i x in mimdevrobebis komponentebi i,,i n nomrit emtxveva ertmanets, masin formula Sesrulebulia s-ze masin da mxolod masin, roca is Sesrulebulia s -ze. (m i T i T e b a. induqcia kavsirebis da kvantorebis raodenobis mixedvit formula 3
4 Si. jer damtkicdes, rom Tu term t-s cvladebi imyofeba x,..., Soris, xolo s da s mimdevrobebis wevrebi i,, i n i x in nomrit emtxveva ertmanets, masin s*(t) = (s )*(t). kerzod, Tu t ar Seicavs cvladebs, masin s *(t) = s 2 *(t) nebismieri s da s 2 mimdevrobebisatvis. yvela iseti n-elementiani b,..., ) mimdevrobebis ( i b ik simravles, sadac misi elementebi ekutvnis interpretaciis ares D, rom formula Sesrulebulia yovel mimdevroba s ze, romlis i,, i n komponentebi emtxveva Sesabamisad b,...,, i b in ewodeba interpretaciis mimarteba (an Tviseba), romelic Seesabameba formula s. davusvat, magalitad, rom D aris yvela adamianta simravle, P 2 (x,y) da P 2 2 (x,y) interpretaciebi Sesabamisad arian x aris y-is Zma da x aris y-is msobeli ; masin binaruli mimarteba D_Si, romelic Seesabameba formulas x 3 (P 2 (x,x 3 ) & P 2 2 (x 3,x 2 )), warmoadgens mimartebas natesaur kavsirs, romelic akavsirebs ZmisSvils da bizas. Tu interpretaciis ared avirebt mtel dadebit ricxvta simravles, xolo P 2, f 2 da a interpretirebulia Sesabamisad rogorc =, gamravleba da, masin formulas P 2 (x, a ) & x 2 ( x 3 P 2 (x,f 2 (x 2, x 3 )) P 2 (x 2, x ) P 2 (x 2, a )) (IX) (X) Seesabameba mocemuli azrit ricxvis Tviseba iyos martivi. Tu formula Caketilia, masin nebismier interpretaciasi WeSmaritia an, an (e. i. mcdaria). aracaketili, e. i. romelic Seicavs Tavisufal cvladebs, formula SeiZleba zogiert interpretaciasi arc WeSmariti da arc mcdari. davusvat, magalitad, aris P 2 (x,x 2 ). ganvixilot interpretacia, romlis are aris mtel ricxvta simravle da sadac P 2 (x,x 2 ) interpretacia aris x < y. am interpretaciasi Sesrulebadia mxolod mimdevroba s = (b, b 2, ) ze, romelic akmayofilebs pirobas b < b 2. masasadame, am interpretciasi ganxiluli formula arc WeSmariti da arc mcdari. lema. davusvat, rom t da v termebia, s mimdevroba -dan, t miireba t-dan yvela x i Semavlobebis CanacvlebiT v termit da s miireba s-dan massi i-uri komponentis SecvliT s*(v) Ti; masin s*(t ) = (s )*(t). damtkiceba. davusvat, rom (x i ) formulaa, t termia, romelic Tavisufalia x i -Tvis (x i )-Si, da (t) formulaa, romelic mirebulia (x i )-is yvela Tavisufali x i cvladis SenacvlebiT t-ti. masin formula (t) Sesrulebadia mimdevroba s = (b, b 2, )-ze masin da mxolod masin, roca is Sesrulebadia s, romelic mirebulia s-dan element b i -is SecvliT s*(t)-t. 4
5 Sedegi. Tu mimdevroba s-ze Sesrulebadia formula x i (x i ), masin Sesrulebadia agretve formula (t). masasadame, formula x i (x i ) (t) WeSmaritia nebismier interpretaciasi. (XI) Tu formula ar Seicavs x i rogorc Tavisufal cvlads, masin formula x i ( ) ( x i ) WeSmaritia nebismier interpretaciasi. savarjiso daamtkicet (I) (XI). magalitis saxit Cven davamtkicot (XI). dausvat, rom (XI) arasworia. es nisnavs, rom romelirac da formulebistvis formula x i ( ) ( x i ) ar aris WeSmariti romelirac interpretaciasi. gansazrvris me -(iii) punqtis Tanaxmad unda arsebobdes mimdevroba s, romelzedac x i ( ) Sesrulebadia, xolo ( x i ) ar aris Sesrulebadi. masin, isev imave me -(iii) punqtis Tanaxmad am mimdevroba s-ze Sesrulebadia da ar aris Sesrulebadi x i.. e. i., igive gansazrvrebis me- (iv) punqtis Tnaxmad, arsebobs mimdevroba s, romelic SeiZleba gansxvavdebodes s gan mxolod erti i-uri komponentit, romelzedac ar aris Sesrulebadi. vinaidan x i ar aris Tvisufali arc x i ( ) Si da arc -Si, xolo TviT es formulebi Sesrulebadia s-ze, masin, Tanaxmad (VIII), iseni Sesrulebadia agretve s -ze, (iv) Tanaxmad gamomdinareobs, rom agretve Sesrulebadia s -ze. amgvarad, da Sesrulebadia s -ze, saidanac, punqti (iii) mixedvit, gamomdinareobs, rom -c Sesrulebadia s -ze. miviret winaamrdegoba, rom ar aris Sesrulebadi s -ze. amit (XI) damtkicebulia. formula -s ewodeba logikurad zogadmartebuli (predikatta arricxvasi), Tu is WeSmaritia nebismier interpretaciasi. formula -s ewodeba Sesrulebadi (predikatta arricxvasi), Tu arsebobs interpretacia, romelsic Sesrulebadia ertze mainc mimdevrobaze -dan. cxadia, rom formula logikurad zogadmartebulia masin da mxolod masin, roca formula ar aris Sesrulebadi, da Sesrulebadia masin da mxolod masin, roca formula ar aris logikurad zogadmartebuli. rogorc vicit, nebismier interpretaciasi Caketili formula an WeSmaritia an mcdari, e. i. Sesrulebadia an nebismier mimdevrobaze, an arcertze. amgvarad, nebismieri Caketili formula Sesrulebadia masin da mxolod masin, roca is WeSmaritia romelirac interpretaciasi. formula vuwodebt winaamrdegobrivs (predikatta arricxvasi), Tu formula logikurad zogadmartebulia, an, rac igivea, Tu formula mcdaria nebismier interpretaciasi. 5
6 vityvit, rom formula logikurad gamomdinareobs formula - dan (predikatta arricxvasi), Tu nebismier interpretaciasi formula Sesrulebadia nebismier mimdevrobaze, romelzedac Sesrulebadia formula. (ufro zogadad, formula logikurad gamomdinareobs (predikatta arricxvasi) formulata simravlidan, Tu nebismier interpretaciasi formula Sesrulebadia nebismier mimdevrobaze, romelzedac Sesrulebadia, romelzedac Sesrulebadia yoveli formula -dan.) formulebs vuwodebt logikurad eqvivalenturebs (predikatta arricxvasi), Tu yoveli matgani logikurad gamomdinareobs meoredan. am gansazrvrebebidan usualod gamomdinareobs Semdegi mtkicebebi: (a) formula logikurad gamomdinareobs formula -dan masin da mxolod masin, roca formula logikurad zogadmartebulia. (b) da formulebi logikurad eqvivalenturia masin da mxolod masin, roca formula logikurad zogadmartebulia. (c) Tu formula logikurad gamomdinareobs formula -dan da WeSmaritia mocemul interpretaciasi, masin amave interpretaciasi WeSmaritia. (d) Tu formula logikurad gamomdinareobs formulebis simravlidan, romlebic WeSmaritia mocemul interpretaciasi, masin amave interpretaciasi WeSmaritia. yovel winadadebas romelime formalur an bunebriv enasi ewodeba logikurad WeSmariti (predikatta arricxvasi), Tu is aris romelirac logikurad zogadmartebuli formulis kerzo SemTxveva, da ewodeba logikurad mcdari (predikatta arricxvasi), Tu is aris kerzo SemTxveva romelirac winaamrdegobrivi formulisa. 6
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