Abductive Reasoning

CS621 – Artificial Intelligence

Aarif Jindani(113050050)

Alex Poovathingal(113050035)

Ashok Rawat(113050018)

Department of CSE

IIT, Bombay

5th Nov, 2011

Precondition ==Rule==> Conclusion

Deduction – determine the conclusion

Induction – determine the rule

Abduction – determine the precondition

”Abduction is the source of all human knowledge”

Logical Reasoning

History of Abductive Reasoning

Charles Sanders Pierce (1839-1914) ”Abduction is no more nor less than guessing”

Rule: All beans from this bag are white. Result: The beans are white. Case: Therefore these beans are from this bag.

(Add more)

Is Abductive Inference Correct?

Rule: If an eclipse occurs, sky suddenly turns dark.

Case: The sky is dark.

Result: Eclipse occured. Rule: If it rained last night, the lawn will be wet.

Case: The lawn is wet.

Result: It rained last night. ”Induction and Abduction are fallible forms of reasoning.

Their conclusions are susceptable to retraction.”

”Success of our guesses far exceed that of random luck and seems born of attunement to nature by insticts.”

Why is abductive inference useful?

”Abduction works often enough and is the only source for new ideas.”

When Newton saw the apple falling down, he must have done an abductive inference and came up with the theory of gravity.

A possible Thought Process

Apple fell down. If earth pulled everything towards it, then ofcourse,

apple too would fall down. So earth is pulling everything towards it.

Stages in Mental Process of Abduction

”Just as abduction originates with an emotional reaction, it ends with one.”

Abductive Reasoning Process Model

• Set-Cover Based

• Defining a theory from a set of hypotheses based on the current observations.

• Logic Based

• Defining a logical theory based on a set of sentences (explanations) that describe the observations.

Set Cover Based Approach

• A triplet (Φ,Ω,e) defines a domain of hypothesis assembly.

• Φ – Set of Hypotheses

• Ω – Set of Observations

• E – Mapping from subsets of Φ to subsets of Ω.

• Assumptions:

• Computational

For every subset Φ’ of Φ, e(Φ’) is computable.

• Independence

e(Φ1 U Φ2) = e(Φ1) U e(Φ2); for all Φ1, Φ2 that are subsets of Φ.

• Monotonicity

If Φ1 is a subset of Φ2, then e(Φ1) is a subset of e(Φ2).

• Accountability

α(φ) is the set of observations that cannot be explained without hypothesis φ.

Set Cover Based Approach

• Abductive algorithm of Allemang,4 parts:

• Screening: Acceptability of all the possible hypotheses is decided and allocated in a hierarchical classification system.

• Collection: Collection of hypotheses accounting for the observations. A set of hypotheses is made by adding every hypothesis that explains the observations.

• Parsimony: Narrows down the collection to its most applicable subset. If a subset of the set of collected hypotheses is able to explain the observations, that is the new (narrowed down) hypothesis set.

• Critique: Marks the most essential hypotheses among the available ones. Individually every hypothesis is excluded from the set and then the set is tested against the observations. If they cannot be proved, then the excluded hypothesis is marked essential.

Set Cover Based Approach

Example:

Consider a theory T consisting of the following propositions:

(1) ∀x(bird(x) ∧ ~ ab(x) כ flies(x))

(2) ∀x(ufo(x) כ flies(x))

(3) ∀x(penguin(x) V ostrich(x) כ ab(x))

(4) ∀x(songbird(x) כ bird(x))

(5) ∀x(songbird(x) כ eats insects(x))

(6) ∀x(frog(x) כ eats_insects(x))

(7) ∀x(frog(x) כ green(x) Ʌ croaks(x))

(8) ∀x(frog(x) כ ab(x))

Set Cover Based Approach

For the above theories we have the following domain of hypothesis assembly:

Φ = {frog(x), songbird(x), bird(x), ufo(x), no_bird(x)}

Ω = {flies(x), green(x), croaks(x),~flies(x), eats_insects(x)}

e({ frog(x)}) = { eats_insects (x), ~ flies(x), green(x), croaks(x)}

e({ songbird(x)}) = { eats_insects(x), flies(x)}

e({ ufo(x),bird(x)}) = { flies(x)}

e({ penguin(x)} ) = { ~flies(x)}

e({ ostrich(x)}) = { ~flies(x)}

e({ no_bird(x)}) = { ~flies(x)}

Set Cover Based Approach

For a given set of observations:

Ω’ = {~flies(F),croaks(F)}

Result of Collection Phase:

~flies(F) => φ={ penguin(F) ,ostrich(F), no_bird(F),frog(F)}

croaks(F) => φ={ frog(F)}

Hypotheses Set, HYP = { penguin(F),frog(F)} ---- incompatiblemodify HYP to HYP = {no_bird(F),frog(F)}

Result of Parsimony Phase:

e({frog(F)}) כ{~flies(F),croax(F)} => HYP = {frog(F)}

Result of Critique Phase:

HYP = {frog(F)}

Set Cover Based Approach

Limitations:

• Basically the 4 assumptions

• The computability of mapping ‘e’ of the subsets of hypotheses set to those of the observations in order to initiate the process.(Computational)

• The independent assumption is quite strong, so restricting it to “easy to manage” domains only.

• Changes in the theory can lead to extensive respecification of the mapping ‘e’.

• Restricted to diagnostic tasks or repair problems only as the mapping ‘e’ and causal relationship is known for them.

Logic Based Approach

An abduction system consists of

• a logical theory ‘T’ defined over the language ‘L’, and

• a set of sentences A of ‘L’ that are called abducible.

If a sentence φ is found as the result of an abductive process in searching for an explanation of ω, it must satisfy the following conditions:

• T U φ is consistent,

• T U φ ⊢ ω,

• φ is abducible, i.e., φ ε A.

Causal Logic Theory

Konolige analyses:

(C,E,T) is a simple causal theory defined over the first-order language ‘L’, where C is a set of causes, E a set of effects and T is a logical theory defined over L.

An explanation of a set of observations Ω subset of E is a finite set of sentences Φ such that:

• Φ is consistent with T

• T U Φ ⊢ Ω, where Ω is the conjunction of all ω ∈ Ω.

• Φ is a subset-minimal.

Example Representation

Example:

The simple causal theory (C, E, T) is defined as follows:

T is same as our example specified earlier.

C = {frog(x), songbird(x), bird(x), ostrich(x), penguin(x}

E = {flies(x), green(x), croaks(x), eats insects(x)}

If we have the set of observations Ω = {~flies(F), croaks(F)}, then Φ = {frog(F)} is an explanation because

• frog(F) is consistent with T,

• T U frog(F) ⊢ ~flies(F) and croaks(F), and

• frog(F) is subset-minimal.

Abductive Logic Programming

• Extension to logic programming with abduction

• Separates theory in two parts

– Normal Logic Program to identify Φ (Backward Reasoning)

– Integrity Constraints to filter set of possible candidates

Example Abductive Logic Programming

• Logic Program

– Grass is wet if it rained.Grass is wet if the sprinkler was on.The sun was shining.

• IC

– False if it rained and the sun was shining.

Power of Abduction

”Abduction is not a feeble-minded cousin of deductive principles like modus ponens. Its actually, a far richer and more powerful form of thinking”.

People's understanding of causality is inherently non-verbal because it is rooted in visual and kinesthetic perception.

Applications of Abduction in Real-Life

”Early men hypothesized the existence of God inorder to explain the design and existance of the world.”

Domains Targets to be Explanained

Explanatory hypothesis

Science Experimental Results

Theories

Medicine Symtoms Diseases

Crime Evidence Culprits and Motives

Machines Operations, Breakdowns

Parts, Interactions and Flaws

Social Behaviour Mental States and Traits

Applications in Computer and AI

Fault Diagnosis Automated Planning Medical Reasoning

Conclusion

Abduction is the qaulitative,everyday reasoning. Deduction shows that something must be,

induction shows that something exists, and abduction shows that something mabye.

Abduction is that starting point of all research.

References

Thagard, P. (2007). Abductive inference: From philosophical analysis to neural mechanisms. Cambridge: Cambridge University Press.

Gabriele Paul (1993). Approaches to abductive reasoning: An overview. German Research Center for Artificial Intelligence.

Erkki Patokorpi: Role of abductive reasoning in digital interaction, 2006

Drew V. McDermott and Jon Doyle, Non-monotonic logic I:MIT AI Lab Memo 468 (1978).

http://commonsenseatheism.com/?p=3703

http://en.wikipedia.org/wiki/Abductive_reasoning