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1、On the Logic of Vector Space ModelsThe basic idea:When one argues for/against B,ones premises A1,A2,.are meantto rationally“push”the receiver“in the direction of”/“away from”B,and they are doing so with a certain“strength”:content is a vector.A1A2BIf the premises manage to do so perfectly,this can b
2、e viewed as adeductive inference(A1,A2,.B)in the logic of vector space models.If they do so approximately,this can be viewed as an inductive inference.AAAAAAAAAAAAAAAAA2A2AAAA1.5AAAAAAAAAAAAA2A2AAAA1.5AAAAAAAAAAAAA2A2AAAA1.5AAAAAAAAAAAAA2A2AAA0.5A1.5AAAAAAAAAAAAA2A2AAA0.5A1.5A1.5AAAB(BA)CC1.7(AB)(BA
3、)C0.7(AB)C1.7(AB)(BA)C0.7A0.7BC1.7(AB)(BA)C0.35A0.35B0.5CAB(BA)CC1.7(AB)(BA)C0.7(AB)C1.7(AB)(BA)C0.7A0.7BC1.7(AB)(BA)C0.35A0.35B0.5CAB(BA)CC1.7(AB)(BA)C0.7(AB)C1.7(AB)(BA)C0.7A0.7BC1.7(AB)(BA)C0.35A0.35B0.5CAB(BA)CC1.7(AB)(BA)C0.7(AB)C1.7(AB)(BA)C0.7A0.7BC1.7(AB)(BA)C0.35A0.35B0.5C1.7(AB)(BA)C0.35A0
4、.35B0.5C1.7(AB)(BA)CD(AB)0.35D0.5CA(BC)(AB)CAABB1.7(AB)(BA)C0.35A0.35B0.5C1.7(AB)(BA)CD(AB)0.35D0.5CA(BC)(AB)CAABB1.7(AB)(BA)C0.35A0.35B0.5C1.7(AB)(BA)CD(AB)0.35D0.5CA(BC)(AB)CAABB1.7(AB)(BA)C0.35A0.35B0.5C1.7(AB)(BA)CD(AB)0.35D0.5CA(BC)(AB)CAABBEBF0.707Now let us make all of that more precise by de
5、veloping the logic and semanticsof vector space models.Plan:1The Basic Language of Vector Space Models2The Basic Logic of Vector Space Models3The Basic Semantics of Vector Space Models4Basic Belief Revision in Vector Space Models5Extensions:Similarity,Induction,Disjunction,Conditionals,.6Interpretat
6、ions:Similarity,Probability,Machine Learning7Application:The Linda Example8Application:The Queen Example9Conclusions and ProspectsThe Basic Language of Vector Space ModelsDefinition of terms:For every real number r,the numeral r of r is a term.Definition of(factual)formulas:Every propositional varia
7、ble piis a formula.If A and B are formulas,their conjunction(AB)is a formula.If A is a formula,its negationA is a formula.The logical verumis a formula.If A is a formula andis a term,thenA is a formula.Definition of equivalence formulas:If A and B are(factual)formulas,then AB is an equivalence formu
8、la.Metalinguistic abbreviations:AB=df(AB)(“Material”implication)=dfThe Basic Logic of Vector Space ModelsThe derivability relation has the following format:hF,Ei CwhereFis a finite multi-set of factual formulas,Eis a set of equivalence formulas,C is a factual formula or an equivalence formula.Struct
9、ural rules:is reflexive,monotonic,and transitive w.r.t.equivalence formulas.satisfies the following restricted form of transitivity w.r.t.factual formulas:F11,.,Fk1,A1B1,.,AmBmF2,F2,A01B01,.,A0nB0nF3F11,.,Fk1,A1B1,.,AmBm,A01B01,.,A0nB0nF3Logical axioms and rules:(when the conclusion is factual,all f
10、actual premises are stated completely)AA(REFL)A(BC)(AB)C(ASS)ABBA(COMM)A A(VER)AA(BOT)(A)A(COMP)1AA(ID)(AB)(A)(B)(DIS1)(+)A(A)(A)(DIS2)A1,.,An,B1C1,.,BmCmA1.An(AGG)A,B1C1,.,BmCm A(for 0)(WEAK)A,BA B/A(SUB)(In particular:A,(BA)B)And finally the only axiom that is perhaps surprising at first glance:(E
11、QUIV)(and“push”the receiver in the same waythat is,not at all.)(AB)(A)(B)(DIS1)(+)A(A)(A)(DIS2)A1,.,An,B1C1,.,BmCmA1.An(AGG)A,B1C1,.,BmCm A(for 0)(WEAK)A,BA B/A(SUB)(In particular:A,(BA)B)And finally the only axiom that is perhaps surprising at first glance:(EQUIV)(and“push”the receiver in the same
12、waythat is,not at all.)(AB)(A)(B)(DIS1)(+)A(A)(A)(DIS2)A1,.,An,B1C1,.,BmCmA1.An(AGG)A,B1C1,.,BmCm A(for 0)(WEAK)A,BA B/A(SUB)(In particular:A,(BA)B)And finally the only axiom that is perhaps surprising at first glance:(EQUIV)(and“push”the receiver in the same waythat is,not at all.)Derivable rules:A
13、1,.,AnBA1,.,An B(for 0)AB,BAABABBA(SYMM)AB,BCAC(TRANS)AAAA,AAA AAAAB,BCACA,(AB)BWhileA,AAAis derivable,this is not derivable:AAA(A single A may not be“strong enough”!),is substructural.There are also surprising inferences,such as:(AB)ABThis is(somewhat)justified in view of:AB(AB)ABABBoth(AB)andAB sh
14、ould reverse the“push”exerted by AB.Lemma0A Proof:1.0A(0+0)A(REFL)(Note that 0=(0+0)!)2.(0+0)A(0A0A)(DIS2)3.0A(0A0A)1.,2.(TRANS)4.(0A0A)(0A)(0A0A)(0A)(REFL)5.(0A0A)(0A)(0A(0A)4.,3.(SUBST0A0A0A)6.(0A(0A(0A)(0A0A)(0A)(ASS)7.(0A(0A(0A)(0A(0A)6.,5.(TRANS)8.(0A(0A)(BOT).9.(0A(0A)8.(SYMM).10.(0A)7.,9.(SUB
15、ST0A(0A)11.(EQUIV)12.(0A)10.,11.(SUBST)13.(0A)0A(VER)14.0A(0A)13.(SYMM)15.0A 14.,12.(TRANS)Lemma A(1)AProof:1.1AA(ID)2.(AA)(BOT)3.0A(LEMMA)4.0A 3.(SYMM)5.(AA)0A 2.,4.(TRANS)6.(1AA)0A 5.,1.(SUBST1AA)7.0A(1AA)6.(SYMM)8.(1)A(1AA)(1)A(1AA)(REFL)9.(1)A(1AA)(1)A0A)8.,7.(SUBST0A1AA)10.(1)A(1AA)(1)A1A)A(ASS
16、)11.(1)A1A)A(1)A(1AA)10.(SYMM)12.(1)A1A)A(1)A0A)11.,9.(TRANS)13.(1+1)A(1)A1A)(DIS2)14.0A(1+1)A(REFL)15.0A(1)A1A)14.,13.(TRANS)16.(0AA)(1)A0A)12.,15.(SUBST0A(1)A1A)17.(A)(1)A)16.,4.(SUBST0A)18.(EQUIV)19.(A)(1)A)17.,18.(SUBST)20.(A)A(VER)21.(A)(A)(COMM)22.(A)A 21.,20.(TRANS)23.A(A)22.(SYMM)24.A(1)A)19
17、.,23.(SUBSTA(1)A)25.(1)A)(1)A(VER)26.A(1)A 24.,25.(TRANS)Derivations like that can be tedious.But ultimately we will see that one can simply infer by vector calculation!Lemma(AB)B(1)AProof:1.(AB)(AB)(REFL)2.(AB)(AB)1.(ABBREV)3.B(1)B(LEMMA)4.(1)B B 3.(SYMM)5.(AB)(A(1)B)2.,4.(SUBST(1)BB)6.(A(1)B)(1)(A
18、(1)B)(LEMMA)7.(1)(A(1)B)(A(1)B)6.(SYMM)8.(AB)(1)(A(1)B)5.,7.(SUBST(1)(A(1)B)(A(1)B)9.(1)(A(1)B)(1)A(1)(1)B)(DIS1)10.(1)A(1)(1)B)(1)(A(1)B)9.(SYMM)11.(AB)(1)A(1)(1)B)8.,10.(SUBST(1)A(1)(1)B)(1)(A(1)B)12.(1)(1)B)(1)(1)B(COMP)13.(1)(1)B1B(REFL)14.(1)(1)B)1B 12.,13.(TRANS)15.1BB(ID)16.(1)(1)B)B 14.,15.(
19、TRANS)17.B(1)(1)B)16.(SYMM)18.(AB)(1)AB 11.,17.(SUBSTB(1)(1)B)19.(1)ABB(1)A(COMM)20.(AB)B(1)A 18.,19.(TRANS)A(1)A(AB)B(1)AppqqABAABroyal:x is royal.king:x is a king.man:x is male.(royalking),(king(royalman)man1.royalking(P1)2.king(royalman)(P2)3.(royalking)(king(1)royal)(LEMMA)4.(king(1)royal)(royal
20、king)3.(SYMM)5.king(1)royal 1.,4.(SUBSTking(1)royalroyalking)6.(1)royalking)(king(1)royal)(COM)7.(1)royalking 5.,6.(SUBST(1)royalkingking(1)royal)8.(royalman)king 2.(SYMM)9.(1)royal(royalman)7.,8.(SUBSTroyalmanking)10.(1)royal(royalman)(1)royalroyal)man(ASS)11.royal(1)royal(LEMMA)12.(1)royal(royalma
21、n)(royalroyal)man 10.,11.(SUBSTroyal(1)royal)13.(royalroyal)(royalroyal)(COM)14.(royalroyal)(BOT)15.(royalroyal)13.,14.(TRANS)16.(EQUIV)17.16.(SYMM)18.(royalroyal)15.,17.(TRANS)19.(royalroyal)18.(SYMM)20.(1)royal(royalman)(man)12.,19.(SUBSTroyalroyal)21.(man)(man)(COM)22.(1)royal(royalman)(man)20.,2
22、1.(TRANS)23.(man)man(VER)24.(1)royal(royalman)man 22.,23.(TRANS)25.man(1)royal(royalman)24.(SYMM)26.man 9.,25.(SUBSTman(1)royal(royalman)The Basic Semantics of Vector Space ModelsDefinitionhV,Viis a vector space model iff(i)V=hV,+,iis a vector space overR,and(ii)Vis a valuation over V.A valuationVov
23、er V is a function that is defined on terms,factual formulas,and equivalence formulas,such that the following is the case:V(r)is the real-valued scalar r.V(pi)is a vector in V.V(AB)=V(A)+V(B).V(A)=V(A).V()is the zero vector0 of V.V(A)=V()V(A).V(AB)=t iffV(A)=V(B)(and f otherwise),Definition(Logical
24、Consequence)A1,.,Am,B1B01,.,BnB0n,.|=C ifffor all vector space modelshV,Vi:ifV(B1B01)=.=V(BnB0n)=.=t,thenif C is an equivalence formula,thenV(C)=t;if C is a factual formula,then there is awith 0 1,such that(V(A1)+.+V(Am)=V(C).E.g.,(royalking),(king(royalman)|=man:ifV(king)=V(royal)+V(man),then 1(V(k
25、ing)V(royal)=V(man).pp6|=q.p|=p but p,q6|=p.(p,q may take one in a different direction than p.),|=is paraconsistent and nonmonotonic with respect to factual formulas.TheoremThe basic logic of vector space models is sound and complete with respect tothe basic semantics of vector space models:hF,Ei C
26、iffhF,Ei|=C.Related:Dynamic semantics(e.g.Heim 1983,Groenendijk and Stokhof 1991,van Benthem 1995).Quantum logic(Birkhoff&von Neumann 1936,Baltag and Smets 2011).Quantum models of cognition(Busemeyer and Bruza 2012).Conceptual spaces(G ardenfors 2000,2014).Compositional distributional semantics(e.g.
27、Coecke et al.2010).Arrow logic(Venema 1997).Modal logics of space(Van Benthem and Bezhanishvili 2007).Vector logic systems(e.g.Westphal and Hardy 2005).Basic Belief Revision in Vector Space ModelsVector space models have a plausible doxastic interpretation:V(B):doxastic“change vector”corresponding t
28、othe rational change effected by receiving evidence B(0:the agents present doxastic state)But given a“natural doxastic origin”,one may also interpret one of the vectorsas summarizing what the agent has learned so far:V(A):doxastic“place vector”corresponding tothe agents present state A(0:the“natural
29、 doxastic origin”)Either way,the logic of vector space models may also be used as a logic ofbelief revision in the spirit of AGM(1985).Basic Belief Revision in Vector Space ModelsVector space models have a plausible doxastic interpretation:V(B):doxastic“change vector”corresponding tothe rational cha
30、nge effected by receiving evidence B(0:the agents present doxastic state)But given a“natural doxastic origin”,one may also interpret one of the vectorsas summarizing what the agent has learned so far:V(A):doxastic“place vector”corresponding tothe agents present state A(0:the“natural doxastic origin”
31、)Either way,the logic of vector space models may also be used as a logic ofbelief revision in the spirit of AGM(1985).The agents present state:A(=1.5p0.9q)ppqqANew evidence comes along:B(=0.8p1.2q)ppqqABThe agent revises her beliefs:AB(=AB)ppqqABBABThe agent contracts her beliefs:(AB)B(=(AB)B=B(AB)p
32、pqqABBABThe agent expands her beliefs:A+(r1)A(=A(r1)A=rK)ppqqArAExtensions:Similarity,Induction,Disjunction,Conditionals,.Next,we are going to introduce some extensions to the basic system.They can be introducedinto the metalanguage,orinto the object language by metalinguistic abbreviation,orinto th
33、e object language as new primitives.In what follows,I will merely sketch some options.Introduce a norm|.|into V:ppqqBB“Pure”Content of B(rel.toV):the oriented half-line ofV(B)Subject matter of B(rel.toV):the non-oriented line ofV(B)Strength of B(rel.toV,|.|):the norm|V(B)|ofV(B).(,|=0;|AB|A|+|B|;|AA
34、|=|A|+|A|;|A|=|A|.)Consider Euclidean V with dot product:ppqqABBCSimilarity of A and B(rel.toV):V(A)V(B)0.Alternatively:V(A)V(B)|V(A)|2|V(B)|20(where|V(A)|2=pV(A)V(A)B is neutral to A(inV)iffV(A)V(B)=0.Consider Euclidean V with dot product:ppqqABBCHere:V(A)V(B)=2.280.V(A)V(B)|V(A)|2|V(B)|2=0.905.=co
35、s 25.177.V(B)V(B)|V(B)|2|V(B)|2=1=cos 0.V(B)V(B)|V(B)|2|V(B)|2=1=cos 180.C is neutral to B(inV)and hence also neutral toB.Consider Euclidean V with dot product:ppqqABBCA1,.,Am|=V,rB iffV(A1.Am)V(B)|V(A1.Am)|2|V(B)|2=r.(Inductive Inference)Here:A|=V,0.905B.If A1,.,AmB,then for allV(withV(Ai),V(B),0):
36、A1,.,Am|=V,1B.Introduce disjunction:ppqqABABV(AB)=|V(A)|V(A)|+|V(B)|V(A)+|V(B)|V(A)|+|V(B)|V(B)(if|V(A)|+|V(B)|0;otherwise letV(AB)=0).AAAAAABBA(But there are also alternative treatments of disjunction/negated-conjunction!)Introduce Ramsey test conditionals:ppqqABABAcc(AB)=V(A)V(B)|V(A)|2:the extent
37、 to which change by A conforms to change by B.Introduce Ramsey test conditionals:ppqqKABABKAAcc(AB)=V(A)V(B)|V(A)|2:the extent to which change by A conforms to change by B.AccK(AB)=V(KA)V(B)|V(KA)|2:the extent to which change by A(in K)conforms to change by B.Introduce Ramsey test conditionals:ppqqA
38、BABAcc(AB)=V(A)V(B)|V(A)|2:the extent to which change by A conforms to change by B.A(BC)(AB)(AC)Introduce extended belief revision/DEL:ppqqABCDE.g.:V(?=?2p12qz|2p12qA)=?20012?V(A).Introduce extended belief revision/DEL:ppqq?A?B?C?DE.g.:V(?=?2p12qz|2p12qA)=?20012?V(A).?(AB)?A?B0?(AB)?A?B?(A)?AInterpr
39、etation:Similarity,Probability,Machine LearningppqqpqpqpqpqABCSentence vectors may be interpreted by operationalizing similarity:similarity between sentences,andsimilarity between sentences and(vector representations of)worlds.Or one interprets sentence vectors(subjectively)probabilistically:Let us
40、assume thatV(p1),.,V(pn)is a basis of V.Let vectors over that basis represent probability measures on formulas,where measures are restricted to those in which propositional variablesare mutually conditionally independent(P(p1p2)=P(p1)P(p2),etc.).Probabilistically,evidence E is given by multiplicativ
41、e odds-changefor each pi:Pnew(pi)Pnew(pi)=P(pi)P(pi)ei(0ei)In V,E is given by adding the vector of logarithms of odds-changes:V(E)=ln(e1).ln(en)(ln(ei)Thus,the origin0 represents the agents present probability measure P.Probabilistically,we have for each i,lnPnew(pi)Pnew(pi)P(pi)P(pi)=lnPnew(pi)Pnew
42、(pi)P(pi)P(pi)which justifies in V:V(E)=V(E)Furthermore,let P0(pi)=dfUpdate(P,ei)be the probability of pithatresults from updating the P-odds of pimultiplicatively by ei.Then one can show for each i,P00(pi)=dfUpdate(Update(P,ei),e0i)=Update(P,eie0i)andlnP00(pi)P00(pi)=ln(P(pi)P(pi)eie0i)=lnP(pi)P(pi
43、)+ln(ei)+ln(e0i)which justifies in V:V(EE0)=V(E)+V(E0)The agents present state:ppqq0P(pq)=0.25P(pq)=0.25P(pq)=0.25P(pq)=0.25P(p)=P(q)=12.New evidence comes along:A(=1.5p0.9q)ppqqAP0(pq)=0.58P0(pq)=0.24P0(pq)=0.13P0(pq)=0.05P0(p)P0(p)=P(p)P(p)e1.5=4.48P0(q)P0(q)=P(q)P(q)e0.9=2.4596.Thus,P0(p)=0.81757
44、,P0(q)=0.710949.Further evidence comes along:B(=0.8p1.2q)ppqqABABP00(pq)=0.81P00(pq)=0.10P00(pq)=0.08P00(pq)=0.01P00(p)P00(p)=P0(p)P0(p)e0.8=9.973879P00(q)P00(q)=P0(q)P0(q)e1.2=8.16616.Thus,P00(p)=0.90887,P00(q)=0.8909.Or in one fell swoop:AB(=1.5p0.9q0.8p1.2q)ppqqABABP00(pq)=0.81P00(pq)=0.10P00(pq)
45、=0.08P00(pq)=0.01P00(p)P00(p)=P0(p)P0(p)e0.8=9.973879P00(q)P00(q)=P0(q)P0(q)e1.2=8.16616.P00(p)P00(p)=P(p)P(p)e1.5e0.8=9.973879P00(q)P00(q)=P(q)P(q)e0.9e1.2=8.16616This probabilistic interpretation of vector space models is closely related tologistic regression models/simple artificial neural networ
46、kswith(i)log-odds-changes as inputs,(ii)linear combinations thereof,and(iii)outputs resulting from applications of the logistic function.From that point of view,for each A of interest,learning algorithms aim todetermine valuationsVand suitable coefficientsA1,A2,.,such thatV(A)V(A1p A2q.)where p,q,.a
47、re the inputs.Application:The Linda Example(Tversky&Kahneman 1983)B:bank tellerB:bank tellerF:feministF:feministEBFE:Linda is 31 years old,single,outspoken,and very bright.She majoredin philosophy.As a student,she was deeply concerned with issues ofdiscrimination and social justice,and also particip
48、ated in anti-nucleardemonstrations.Which is more probable:B or BF?Application:The Linda Example(Tversky&Kahneman 1983)B:bank tellerB:bank tellerF:feministF:feministEBFV(E)V(B)|V(E)|2|V(B)|2=cos 00.707cos=V(E)V(BF)|V(E)|2|V(BF)|2Which is more“probable”given E:B or BF?Answer:BF.Application:The Linda E
49、xample(Tversky&Kahneman 1983)B:bank tellerB:bank tellerF:feministF:feministEBFV(E)V(B)|V(E)|2|V(B)|2=cos 0 (EQUIV)20.(1man(1)man)19.,18.(TRANS)21.(manking)woman)(royal)woman)12.,20.(SUBST1man(1)man)22.(royal)royal(VER)23.royal(royal)22.(SYMM)24.(manking)woman)(royalwoman)21.,23.(SUBSTroyalroyal)25.(
50、manking)woman)|z kingman+womanqueen 24.,2.(SUBSTqueenroyalwoman)Conclusions and ProspectsThere is a reasonably well-behaved logic of vector space models.It can be used to formally reconstruct belief revision,subject matter,strength,similarity,neutrality,induction,conditionals,.It has a probabilistic