qualitative and semi-quantitative inference and revision ... · 1 introduction: why conditionals 2...

Department ofComputer Science

Qualitative and Semi-quantitativeInference and Revision

with ConditionalsWorkshop on Human Reasoning and Computational Logic

Dresden

Christian Eichhorn, Gabriele Kern-IsbernerFebruary 9th, 2017

Eichhorn, Kern-Isberner Inference and Revision with Conditionals February 9th, 2017 1/33


Introduction: Why Conditionals Conditional Structures Getting Semi-quantitative by Adding Numbers: OCF Conclusion

1 Introduction: Why Conditionals

2 Conditional StructuresNo Numbers NeededInference

3 Getting Semi-quantitative by Adding Numbers: OCFOCF and C-representationsC-inferenceC-revision

4 Conclusion





2 Conditional Structures

3 Getting Semi-quantitative by Adding Numbers: OCF

4 Conclusion




Defeasible Rules and Conditionals

Defeasible rules “If A then (usually, probably, plausibly, . . . ) B”

Uncertain, defeasible connections between antecedent A andconsequent B.

Not similar to classical (material) implications“If A then (definitely) B” (written A⇒ B) but substantiallydifferent. (Leading to well known “fallacies”1).

Can be (formally) implemented by conditionals (B|A).

1E.g. Linda being a feminist bank teller, or Lisa visiting a library, or Janetaking drugs, . . .




Conditionals

Conditionals (B|A)Formally implement defeasible rules.

Encode semantical relationships between antecedence A andconsequent B.Encode nonmonotonic inference via

(B|A) is accepted iff A|∼ B holdsA|∼ B holds iff the verification (A and B) of the encoded ruleis more plausible2 than its falsification.

2or probable, or possible, . . .Eichhorn, Kern-Isberner Inference and Revision with Conditionals February 9th, 2017 5/33



Why Are Conditionals so Special?

Conditionals (B|A)Focus on cases where the premise A is fulfilled andDo not say anything about cases where A does not hold.Go beyond classical logic, as they are three valued entities.

A conditional is evaluated in a world ω to [de Finetti 1937]

J(B|A)Kω =

true iff ω |= AB (ω satisfies A and B) (verification)false iff ω |= AB (ω satisfies A and ¬B) (falsification)u iff ω |= A (ω satisfies ¬A) (neutrality)



Introduction: Why Conditionals Conditional Structures Getting Semi-quantitative by Adding Numbers: OCF ConclusionNo Numbers Needed Inference


2 Conditional StructuresNo Numbers NeededInference

3 Getting Semi-quantitative by Adding Numbers: OCF

4 Conclusion




Inference by ConditionalsIf ≺ is a (well-behaved) preference relation on possible worlds, thennonmonotonic inference can be defined by verification /falsification of conditionals:

Definition (Preferential entailment [Makinson1994])Let A,B be propositional formulas.We nonmonotonically infer B from A in ≺ if and only if for everyworld ω′ that falsifies the conditional (B|A) there is a world ω thatverifies the conditional which is ≺-preferred to ω′.

A|∼≺ B iff ∀ ω′|= AB ∃ ω|= AB such that ω ≺ ω′

But where does ≺ come from?




Where to Get the Preferential Relation ≺ From?

Preference between worlds should be based on backgroundknowledge R.Usually, ≺ can be obtained from < (or >) on the plausibility3

of worlds.

Caveat: for some worlds, this relation may be an artefact ofthe underlying formalism.

Why not to rely on R directly?

3or probability, or possibility, or. . .Eichhorn, Kern-Isberner Inference and Revision with Conditionals February 9th, 2017 10/33



Conditional StructureLet impacts GR = a+

1 ,a−1 , . . . ,a+

n ,a−n ni=1 be generator of freeabelian group FR = (GR, ·, 1).

Definition (Conditional structure [Kern-Isberner 2001])Let R = (B1|A1), . . . , (Bn|An) be a knowledge base. TheConditional Structure of a world ω is defined by the functionσR : Ω 7→ FR such that σR(ω) =

∏ni=1 σR,i(ω).

Example (Tweety knowledge base RPBF = (f |b), (f |p), (b|p))

ω p b f p b f p b f p b f p b f p b f p b f p b f

verifies δ1,δ3 δ2,δ3 — δ2 δ1 — — —

falsifies δ2 δ1 δ2,δ3 δ3 — δ1 — —

σRPBF (ω) a+1 a−2 a+

3 a−1 a+2 a+

3 a−2 a−3 a+2 a−3 a+

1 a−1 1 1




Structural PreferenceDefine the preference relation ≺σR⊆ Ω× Ω such that ω ≺σR ω′ iffω′ falsifies, c.p., more conditionals than ω:

Definition (Structural preference [Kern-Isberner 2001])ω is structurally preferred to ω′ if and only if∀ 1 ≤ i ≤ n σR,i(ω) = a−i implies σR,i(ω′) = a−i and∃ 1 ≤ j ≤ n σR,j(ω) ∈ a+

i , 1 and σR,i(ω′) = a−i .




3 a−1 a+2 a+

3 a−2 a−3 a+2 a−3 a+

1 a−1 1 1

pbf ≺σRPBFpbf




Structural InferenceWe structurally infer B from A if and only if the verification or theconditional (B|A) is structurally preferred to its falsification:

Definition (Structural inference cf. [Kern-Isberner, Eichhorn ’14])

A|∼σR B iff ∀ ω′ |= AB ∃ ω |= AB s.t. ω ≺σR ω′.


pbf ≺σRPBFpbf

thereforepf |∼σRPBF

b

p b f

p b f

p b f

p b f

p b f

p b f

p b f

p b f




Why System P?

System P is well established in nonmonotonic reasoning.

Experimental research indicated that humans. . .. . . use System P when making inferences or . . .

[Da Silva Neves et al. 2002; Pfeifer & Kleiter 2005]. . . at least use rules from System P for their inferences.

[Kuhnmunch & Ragni 2014].

Hence satisfaction of System P is an important qualitycriterion both formally and cognitively.




Summing up Structural Approach

Structural information in the knowledge base provided. . .. . . by the qualitative information of conditionals and . . .. . . the trivalent evaluation of the conditionals

. . . allows for high qualitative inference (caveat: DirectInference is not fulfilled).

Next stepNumbers to impacts → semi-quantitative inference and revision.



Introduction: Why Conditionals Conditional Structures Getting Semi-quantitative by Adding Numbers: OCF ConclusionOCF and C-representations C-inference C-revision




4 Conclusion




Ranking Functions

An Ordinal Conditional Function (OCF) or ranking function κ is afunction that assigns a degree of disbelief to each world ω ∈ Ω .

Definition (OCF [Spohn 1988,2012])κ := Ω→ N∞0 such that:

κ−1(0) 6= ∅κ(φ) = minκ(ω)|ω |= φ

κ(ψ|φ) = κ(φψ)− κ(φ)κ |= (ψ|φ) iff κ(φψ) < κ(φψ)

Example (Tweety OCF)

κ(ω) = 4 p b fκ(ω) = 2 p b f , p b fκ(ω) = 1 p b f , p b fκ(ω) = 0 p b f , p b f , p b f




C-representations generate admissible OCFs for a knowledge base

Let R = (B1|A1), . . . , (Bn|An) be a knowledge base.Let κ−i ∈ N0 be integer impacts associated to each a−i .4

Definition (Kern-Isberner 2001,2004)A c-representation is an OCF

κcR(ω) =

∑i:ω|=AiBi

κ−i

where the impacts are chosen s.t. κcR |= R which is the case iff

κ−i > minω|=φiψi

∑j:ω|=φjψj

i 6=j

κ−j

− minω|=φiψi

∑j:ω|=φjψj

i 6=j

κ−j

∀ 1 ≤ i ≤ n

4σR,i(ω) = a−i iff J(Bi|Ai)Kω = false iff ω |= AiBi




Example with Tweety knowledge base RPBF = (f |b), (f |p), (b|p)


verifies δ1,δ3 δ2,δ3 — δ2 δ1 — — —

falsifies δ2 δ1 δ2,δ3 δ3 — δ1 — —

κcRPBF2 1 4 2 0 1 0 0

κ−1 > minκ−2 , 0 −min0, 0 ⇒ κ−1 > 0→ κ−1 = 1

κ−2 > minκ−1 , κ−3 −min0, κ−3 ⇒ κ−2 > minκ−1 , κ

−3

→ κ−2 = 2

κ−3 > minκ−1 , κ−2 −minκ−2 , 0 ⇒ κ−3 > minκ−1 , κ

−2

→ κ−2 = 2




Inference with C-representations

Definition (c-inference [Kern-Isberner 2001; Spohn 1988])We infer B from A with a c-representation κcR iff κcR accepts theconditional (B|A), formally

A|∼κcRB iff κcR |= (B|A) iff κcR(AB) < κcR(AB)

Properties (excerpt) [Kern-Isberner 2001; Kern-Isberner & Eichhorn 2014]

C-representation for R exists iff R is consistent.Not a single, unique OCF, but schema for R-admissible OCF.For positive impacts κ−i , the preference relation induced byκcR and < on Ω embeds ≺σR .C-inference satisfies System P, System R and Direct Inference.




Example with Tweety knowledge base RPBF = (f |b), (f |p), (b|p)


κcRPBF2 1 4 2 0 1 0 0


3 a−1 a+2 a+

3 a−2 a−3 a+2 a−3 a+

1 a−1 1 1

κcRPBF(ω) = 4 p b f p b f

κcRPBF(ω) = 2 p b f p b f , p b f p b f

κcRPBF(ω) = 1 p b f p b f , p b f

κcRPBF(ω) = 0 p b f , p b f , p b f

p b f

p b f

p b f

p b f

p b f

p b f

p b f

p b f

p b f

p b f

p b fp b f

p|∼κcRPBF

f because κ(pf) = 1 < 2 = κ(pf).

p|σRPBFf because no ≺σRPBF

preferred ω |= pf to pbf .




Summing up C-representations

Generate ranking function on top of conditional structuresNot a single OCF (like, e.g., System Z) but schema forR-admissible OCFInduced ordering embeds structural preference

(for positive impacts)Induced inference stronger than structural inference:

System P, System RDirect Inference

What’s next?Use c-representations for revision




Understanding Human Inferences with c-representations

C-representations applied to the Suppression Task1 Formalize background knowledge as conditional knowledge

bases R. (e.g. (library|essay), (library|open), (essay|>))2 Construct R for κcR to match inferences drawn.

If (library|essay) ∈ R, no suppression in c-inferenceor any other System P inference.

[Ragni et al. 2016a, 2016b]

Thus if suppression occurs, either the conditional was not believed,or suppression does not occur because of inference, but of revision.







4 Conclusion




Advanced Belief Revision for Ranking Functions

Belief revision task for OCFGiven a prior OCF κ and some new information consisting of a setof conditionals ∆ = (B1|A1), . . . , (Bn|An), find a posterior OCFκ∗ = κ ∗∆ such that κ∗ |= ∆ and the revision complies with thecore ideas of AGM.

This task involves:Iterated revision, since an epistemic state κ is changed;Conditional revision, since the prior is revised by conditionalinformation;Multiple revision, since ∆ can be a set of plausiblepropositions by setting A ≡ (A|>).




A Principle of Conditional Preservation for ranking functions

OCF principle of conditional preservationLet Ω = ω1, . . . , ωm and Ω′ = ω′1, . . . , ω′m be two sets ofpossible worlds (not necessarily different).If for each conditional (Bi|Ai) in ∆, Ω and Ω′ behave the same,i.e., they show the same number of verifications resp. falsifications,then prior κ and posterior κ∗ are balanced by

(κ(ω1) + . . .+ κ(ωm))− (κ(ω′1) + . . .+ κ(ω′m))= (κ∗(ω1) + . . .+ κ∗(ωm))− (κ∗(ω′1) + . . .+ κ∗(ω′m))




A Simple Principle of Conditional Preservation

The general principle of conditional preservation yields a simple,straightforward consequence:

Simple PCP [Kern-Isberner & Huvermann 2013]

(SCondPres) If two possible worlds ω1, ω2 ∈ Ω verify resp. falsifyexactly the same conditionals in ∆, thenκ∗(ω1)− κ(ω1) = κ∗(ω2)− κ(ω2).

(SCondPres) claims that the amount of change between prior andposterior epistemic state depends only on the conditionals in thenew information set, more precisely, on the so-called conditionalstructure of the respective world.




C-revisions revisions that satisfy the principle of conditional preservation

New information ∆ = (B1|A1), . . . , (Bn|An)

OCF C-revision

κ∗ = κ ∗∆ : κ∗(ω) = κ0 + κ(ω) +∑

1≤i≤nω|=AiBi

κ−i ,

κ−i ’s have to be chosen appropriately to ensure κ∗ |= R (Success).

(Success) is satisfied iff for all i, 1 ≤ i ≤ n,

κ−i > minω|=AiBi

κ(ω) +

∑j 6=i

ω|=AjBj

κ−j

− minω|=AiBi

κ(ω) +

∑j 6=i

ω|=AjBj

κ−j

.

55C-representation = c-revision with uniform κu




Summing up Thank you for your attention.

Conditional StructuresUse the qualitative information in a knowledge base in the form ofabstract impacts for a preferential relation over the worldsgenerating a System P satisfying inference relation therewith.

C-representationsAdd numeric weights to these impacts and create an OCF bysumming up the numeric impacts, generating a System R satisfyinginference therewith.

C-revisionsIncorporate the principle of conditional preservation for a multiple,iterated, conditional, AGM satisfying revision.




Bibliography I

Da Silva Neves, R., Bonnefon, J.-F., and Raufaste, E. (2002). An Empirical Test of Patterns forNonmonotonic Inference. Annals of Mathematics and Artificial Intelligence, 34(1-3):107–130.

de Finetti, B. (1974). Theory of Probability, volume 1,2. John Wiley and Sons, New York, NY, USA.

Kern-Isberner, G. (2001). Conditionals in Nonmonotonic Reasoning and Belief Revision – ConsideringConditionals as Agents. Number 2087 in Lecture Notes in Computer Science. Springer Science+BusinessMedia, Berlin, DE.

Kern-Isberner, G. (2004). A thorough axiomatization of a principle of conditional preservation in beliefrevision. Annals of Mathematics and Artificial Intelligence, 40:127–164.

Kern-Isberner, G. and Eichhorn, C. (2014). Structural Inference from Conditional Knowledge Bases. InUnterhuber, M. and Schurz, G., editors, Logic and Probability: Reasoning in Uncertain Environments,number 102 (4) in Studia Logica, pages 751–769. Springer Science+Business Media, Dordrecht, NL.

Kern-Isberner, G. and Huvermann, D. (2016). What kind of independence do we need for multiple anditerated revision? International Journal of Applied Logic, Special Issue on Uncertain Reasoning. (acceptedfor publication).

Kuhnmunch, G. and Ragni, M. (2014). Can Formal Non-monotonic Systems Properly Describe HumanReasoning? In Proceedings of the Cognitive Science Conference (COGSCI2014), pages 1806–1811.

Makinson, D. (1994). General Patterns in Nonmonotonic Reasoning. In Gabbay, D. M., Hogger, C. J., andRobinson, J. A., editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume 3,pages 35–110. Oxford University Press, New York, NY, USA.




Bibliography II

Pfeifer, N. and Kleiter, G. D. (2005). Coherence and Nonmonotonicity in Human Reasoning. Synthese,146(1–2):93–109.

Ragni, M., Eichhorn, C., Bock, T., Kern-Isberner, G., and Tse, A. P. P. (2016a). Formal NonmonotonicTheories and Properties of Human Defeasible Reasoning. Minds and Machines – Journal for ArtificialIntelligence, Philosophy and Cognitive Science. accepted.

Ragni, M., Eichhorn, C., and Kern-Isberner, G. (2016b). Simulating human inferences in the light of newinformation: A formal analysis. In Kambhampati, S., editor, Proceedings of the Twenty-Fifth InternationalJoint Conference on Artificial Intelligence (IJCAI’16), pages 2604–2610, Palo Alto, CA, USA. AAAI Press.

Spohn, W. (1988). Ordinal Conditional Functions: A Dynamic Theory of Epistemic States. In Causation inDecision, Belief Change and Statistics: Proceedings of the Irvine Conference on Probability and Causation,volume 42 of The Western Ontario Series in Philosophy of Science, pages 105–134, Dordrecht, NL. SpringerScience+Business Media.

Spohn, W. (2012). The Laws of Belief: Ranking Theory and Its Philosophical Applications. OxfordUniversity Press, Oxford, UK.


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