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Abductive Reasoning
CS621 – Artificial Intelligence
Aarif Jindani(113050050)
Alex Poovathingal(113050035)
Ashok Rawat(113050018)
Department of CSE
IIT, Bombay
5th Nov, 2011
Logical Reasoning
Precondition ==Rule==> Conclusion
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Deduction – determine the conclusion
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Induction – determine the rule
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Abduction – determine the precondition
”Abduction is the source of all human knowledge”
History of Abductive Reasoning
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Charles Sanders Pierce (1839-1914)
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”Abduction is no more nor less than guessing”
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Rule: All beans from this bag are white.
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Result: The beans are white.
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Case: Therefore these beans are from this bag.
(Add more)
Is Abductive Inference Correct?
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Rule: If an eclipse occurs, sky suddenly turns dark.
Case: The sky is dark.
Result: Eclipse occured.
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Rule: If it rained last night, the lawn will be wet.
Case: The lawn is wet.
Result: It rained last night.
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”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.”
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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
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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
•
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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:
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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.
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Monotonicity
If Φ1 is a subset of Φ2, then e(Φ1) is a subset of e(Φ2).
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Accountability
α(φ) is the set of observations that cannot be explained without hypothesis φ.
Set Cover Based Approach
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Abductive algorithm of Allemang,4 parts:
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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) ‫ ﬤ‬flies(x))
(2) ∀x(ufo(x) ‫ ﬤ‬flies(x))
(3) ∀x(penguin(x) V ostrich(x) ‫ ﬤ‬ab(x))
(4) ∀x(songbird(x) ‫ ﬤ‬bird(x))
(5) ∀x(songbird(x) ‫ ﬤ‬eats insects(x))
(6) ∀x(frog(x) ‫ ﬤ‬eats_insects(x))
(7) ∀x(frog(x) ‫ ﬤ‬green(x) Ʌ croaks(x))
(8) ∀x(frog(x) ‫ ﬤ‬ab(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),croax(F)}
Result of Collection Phase:
~flies(F) => φ={ penguin(F) ,ostrich(F), no_bird(F),frog(F)}
croax(F) => φ={ frog(F)}
Hypotheses Set, HYP = { penguin(F),frog(F)}
modify HYP to HYP = {no_bird(F),frog(F)}
---- incompatible
Result of Parsimony Phase:
e({frog(F)})‫~{ ﬤ‬flies(F),croax(F)} => HYP = {frog(F)}
Result of Critique Phase:
HYP = {frog(F)}
Set Cover Based Approach
Limitations:
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Basically the 4 assumptions
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The computability of mapping ‘e’ of the subsets of hypotheses
set to those of the observations in order to initiate the
process.(Computational)
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The independent assumption is quite strong, so restricting it to
“easy to manage” domains only.
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Changes in the theory can lead to extensive respecification of
the mapping ‘e’.
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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.
Casual Logic Theory
Konolige analyses:
(C,E,T) is a simple causal theory defined over the firstorder 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
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”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 nonverbal because it is rooted in visual and kinesthetic
perception.
Applications of Abduction in Real-Life
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
”Early men hypothesized the existence of God inorder to
explain the design and existance of the world.”
Applications in Computer and AI
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Fault Diagnosis
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Automated Planning
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Medical Reasoning
Conclusion
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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
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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).
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http://commonsenseatheism.com/?p=3703
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http://en.wikipedia.org/wiki/Abductive_reasoning