Meta-Level Abduction

Katsumi Inoue
National Institute of Informatics, Japan
April, 2015

Meta-level abduction—Contents—
1. Meta-Reasoning and Abduction (Introduction)
2. Patterns of Abduction (Summary)
3. Simple Logic of Causality (ILP 2009)
4. General Logic of Causality (New)
5. Causality Logic with Inhibition (MLJ 2013)
6. Law/Theoretical-Model Meta-level Abduction and
Representation Meta-level Abduction (New)
7. Applications to SBGN-AF networks (FMMB 2014)
The items 1—6 will appear in: Inoue, K., “Meta-level abduction”, IfCoLog Journal of
Logics and their Applications, Special Issue of "Frontiers of Abduction“ (2015).

Meta-reasoning• Pereira, Bundy, Russel, et al.
– The meta-level control of the reasoning process itself.
– The introspective monitoring of the reasoning process at theobject level.

• Kowalski initiated meta-programming in logic
programming, e.g., [Bowen & Kowalski, 1982]

“Vanilla” meta-interpreter (Hill & Lloyd)
solve(true).
solve(A & B) ← solve(A) & solve(B).
solve(￢A) ← ￢solve(A).
solve(A) ← rule(A←B) & solve(B).
• In general, any inference rule can be expressed in such a metarule, e.g.,solve(A←B) ← solve(A←C) & solve(C←B).
solve(￢A) ← solve(B←A) & solve(￢B).
• All constructs with predicates solve, clause, demo are atoms.
• Those meta-level axioms are used for deduction only.

Abductive inference• Abduction augments sufficient conditions missing in
the premises (background knowledge) to enable aderivation (proof) of the given observation (goal).
• This inference fills the gap in a proof of theobservation from the premises.
• Inferred sufficient conditions are called hypotheses
or explanations. Often, they are called explanans.
The observation is called the explanundum.
• A hypothesis can be any formula, e.g., a set(conjunction) of atoms/literals/rules.

Three modes of inference(C.S. Peirce)
deduction

AC
A
C

A
induction

abductionCACEveryday LifeAC

BusinessC

ScienceAMeta-level abduction• Abduction is performed on meta-level axioms.
• For example, given a meta-theory:solve(A) ← rule(A←B) ∧ solve(B),solve(B),and a meta-goal:
solve(A),we can abduce
rule(A←B).
• In this example, we can realize rule abduction.
• Yet, this is an ordinary abduction since it abduce atoms.

Meta-level abduction (Inoue et al., 2009-)
 A method to abduce rules, enabling to infer hypotheses• explaining empirical rules by means of hidden rules,• representing multiple missing links / causal relations,• simultaneously accounting for multiple observations,• containing unknown nodes as new predicates.
• A new way of induction based on full-clausal abduction.
 Combination of rule abduction and fact abduction is possible by
way of conditional query answering.
 First motivating examples were from cognitive modeling for
improving the skill of music playing (ILP ‘09, LNAI 5989)
 The method has been applied to scientific discovery fromnetwork data, e.g., genetic/biochemical pathways (ILP ‘10, LNAI
6489; Machine Learning 91, 2013; FMMB ‘14, LNBI 8738).

Patterns of Abduction (Schurz, Synthese, 2008)
Kind of abduction

Explanandum

Explanans

Factual abduction

Single facts

New facts— Observable-fact abduction

Single facts

Factual reasons— 1st-order existential abduction

Single facts

Facts with new unknown
individuals— Unobservable-fact abduction

Single facts

Unobservable facts

Law abduction

Empirical laws

New laws

Theoretical-model abduction

General empiricalphenomena

New theoretical models

2nd-order existential
abduction

General empiricalphenomena

New laws with newconcepts— Micro-part abduction

General empiricalphenomena

Microscopic compositions— Analogical abduction

General empiricalphenomena

New laws with analogicalconcepts— Hypothetical cause abduction

General empiricalphenomena

Hidden (unobservable)causesFactual Meta-Level Abduction
Kind of abduction

Explanandum

Explanans

Factual abduction

Single facts

New facts— Observable-fact abduction

Single facts

Factual reasons— 1st-order existential abduction

Single facts

Facts with new unknown
individuals— Unobservable-fact abduction

Single facts

Unobservable facts

Law abduction

Empirical laws

New laws

Theoretical-model abduction

General empiricalphenomena

New theoretical models

2nd-order existential
abduction

General empiricalphenomena

New laws with newconcepts— Micro-part abduction

General empiricalphenomena

Microscopic compositions— Analogical abduction

General empiricalphenomena

New laws with analogicalconcepts— Hypothetical cause abduction

General empiricalphenomena

Hidden (unobservable)causesA simple logic of causality

To express relations between events, we use causal chains.
Causality can be represented in first-order predicate logic.
Two meta-predicates:
1. linked(X,Y): X is directly caused by Y.
2. caused(X,Y): There is a causal chain from Y to X.

 Basic axioms:

caused(X,Y) ← linked(X,Y).
caused(X,Y) ← linked(X,Z) ∧ caused(Z,Y).

Representing logical connectivesgslinked(g, s).

gs￢linked(g, s).

g
s

OR

linked(g, s) ∨ linked(h, s)hsglinked(g, s) ∨ linked(g, t)ANDt

Object and meta level representation
 Object domain (object level)A ← B.
B ← C ∧ D.
 Each rule in the object level is represented
as a fact in the meta level.
 Each literal in the object level is represented
as a term in the meta level.

A

B

AND
D

C

 Causal relations (meta level)Alinked(A, B).

linkedBlinked(B, C) ∨ linked(B, D).
 Rule abduction in the object level is realized
by abducing literals of the form linked(_,_)
at the meta level.

linkedORClinkedDFormalizing rule abduction
 g: a goal, s: an input, r: a (hidden) node
B: linked(g, r).
￢linked(g, s).

rgs

That is, g is directly caused by r, but g is notdirectly caused by s.
 g is not directly caused by s, but we knowthat there is a causal chain to g from s.
This is given by an observation:
G: caused(g, s).
 SOLAR computes a hypothesis
H: linked(r, s),given the abducibles {linked(_,_)}.

rgs

Patterns of Abduction (Schurz, Synthese, 2008)
Kind of abduction

Explanandum

Explanans

Factual abduction

Single facts

New facts— Observable-fact abduction

Single facts

Factual reasons— 1st-order existential abduction

Single facts

Facts with new unknown
individuals— Unobservable-fact abduction

Single facts

Unobservable facts

Law abduction

Empirical laws

New laws

Theoretical-model abduction

General empiricalphenomena

New theoretical models

2nd-order existential
abduction

General empiricalphenomena

New laws with newconcepts— Micro-part abduction

General empiricalphenomena

Microscopic compositions— Analogical abduction

General empiricalphenomena

New laws with analogicalconcepts— Hypothetical cause abduction

General empiricalphenomena

Hidden (unobservable)causesNode introduction = Predicate invention
 g, h : goal nodes， s: an input node.
B: ￢ linked(g, s).
￢ linked(h, s).
That is, there are no direct causal relations from s
to g and from h to s, but there are causal chainsas the observations:gsh

G: caused(g, s) ∧ caused(h, s).

 Given the abducibles {linked(_,_)},SOLAR generates a hypothesis H:∃X. ( linked(g, X) ∧ linked(h, X) ∧ linked(X, s) ).


g
X
h

Variable X represents a newly introduced node, whichcorresponds to predicate invention or object invention.

sRepresenting different structures
B: ￢ linked(g, s).
￢ linked(h, s).
G: caused(g, s).
caused(h, s).

gshAbducibles: {linked(_,_)}.

H with 2 intermediate nodes:gXs∃X∃Y. ( linked(g, X) ∧ linked(h, Y)∧ linked(X, s) ∧ linked(Y, s) ).

h∃X∃Y. ( linked(g, X) ∧ linked(h, Y)∧ linked(X, Y) ∧ linked(Y, s) ).

g

h

Y
X
s
Y

Correctness of meta-level abduction
Proposition: Let λ(B) be the theory obtained by replacing every
linked(g, s) appearing in B with the formula (g ← s).
B ∪ Meta-Axioms ╞═ (caused(g, s1) ∨ ･･･ ∨ caused(g, sn))iff λ(B) ╞═ (g ← s1 ∧ ･･･∧ sn).

Theorem: Observation: O = (caused(g, s1) ∨ ･･･ ∨ caused(g, sn)).
H is an abductive explanation of O with respect to B and ΓM ={linked(_,_)} iff λ(H) is a hypothesis such that
• λ(B) ∪ λ(H) ╞═ (g ← s1 ∧ ･･･∧ sn), and• λ(B) ∪ λ(H) is consistent.

Implementation• Although the idea of meta-level abduction is simple,
its implementation requires an abductive procedurefor first-order full clausal theories.
• Currently, SOLAR (Nabeshima et al., 2010) is only
such a state-of-the-art procedure.
• There will be a possibility to use logic programming
and answer set programming.

Application to “knack” discovery
B: linked(inc_sound, bow_close_to_the_bridge).
linked(bow_close_to_the_bridge, stable_bow_movement) ∨linked(bow_close_to_the_bridge, smooth_bow_direction_change).
linked(smooth_bow_direction_change, flexible_wrist).
← linked(inc_sound, keep_arm_close).
← linked(stable_bow_movement, keep_arm_close).
← linked(smooth_bow_direction_change, keep_arm_close).
G: caused(inc_sound, keep_arm_close).

 SOLAR generates 52 hypotheses (maximum search depth: 15,
maximum length of produced clauses: 5). One of them is:∃X. (linked(stable_bow_movement, X) ∧ linked(flexible_wrist, X)∧ linked(X, keep_arm_close) ).

The obtained hypothesis
increase_sound_volume①

H: ∃X. (linked(stable_bow_movement, X)②∧ linked(flexible_wrist, X)∧ linked(X, keep_arm_close) ).
③stable_bow_
movement

flexible_
wrist

①

②

bow_close_
to_the_bridgesmooth_bow_
direction_changestable_bow_
movement

flexible_wrist

increase_
upper_arm_
impedance③

increase_
upper_arm_
impedance

keep_one’s_
arm_close
keep_one’s_
arm_close

Abducing facts
New axioms:
caused(X, X) ← abd(X). % for abducibles
caused(X, Y) ← linked(X, Y).
caused(X, Y) ← linked(X, Z) ∧ caused(Z, Y).
The top clause:← caused(g, X) ∧ abd(X).
Note: abd plays the role of an answer predicate.
An integrity constraint that p and q cannot hold simultaneously:← caused(p, X) ∧ caused(q, Y) ∧ abd(X) ∧ abd(Y ).

Abducing facts and rules• Abducing facts is nothing but answer extraction.

• Abducing facts and rules is then conditional queryanswering.

Patterns of Abduction (Schurz, Synthese, 2008)
Kind of abduction

Explanandum

Explanans

Factual abduction

Single facts

New facts— Observable-fact abduction

Single facts

Factual reasons— 1st-order existential abduction

Single facts

Facts with new unknown
individuals— Unobservable-fact abduction

Single facts

Unobservable facts

Law abduction

Empirical laws

New laws

Theoretical-model abduction

General empiricalphenomena

New theoretical models

2nd-order existential
abduction

General empiricalphenomena

New laws with newconcepts— Micro-part abduction

General empiricalphenomena

Microscopic compositions— Analogical abduction

General empiricalphenomena

New laws with analogicalconcepts— Hypothetical cause abduction

General empiricalphenomena

Hidden (unobservable)causesGeneral Logic of Causality (GLC)• Now we do not consider any particular interpretation of themeta-predicate linked so that its semantics is left open.
• This is desirable, since caused(g, s) cannot be interpreted as
the material implication (g ← s) in general. Instead, themeaning of causality at the object level is now abstracted away.
• In particular, the correspondence such that (caused(g, s1) ∨･･･
∨ caused(g, sn))  (g ← s1 ∧･･･∧ sn) is abandoned.
• In GLC, a causal network is given as a set of atoms of the form
linked(F,G), where F and G are any Boolean formulas. Similarly,
atoms of the form caused(F,G) are allowed with any F and G.
• This logic, called GLC, is defined by structural inference rules.
• Here, we denote the meta-atoms linked(g, s) and caused(g, s)
as the meta-relations (s  g) and (s ⊨ g), respectively.

Postulates for GLCαψ• (Reflection)
α⊨ψ
α⊨β
β≽ψ
• (Weakening)α⊨ψα ≽β β ⊨ ψ• (Strengthening)
α⊨ψ
α⊨φ
α⊨ψ
• (And)α ⊨ φ∧ψ• (Cut)• (Or)α⊨β

α∧β ⊨ ψα⊨ψα⊨ψ β⊨ψα∨β ⊨ ψNote: ≽ is a classical Tarski consequence operator; X ≽Y iff X entails Y.

GLC meta-program
caused(X, Y) ← linked(X, Y).
caused(X, Y) ← (X ⪯ Z) ∧ caused(Z, Y).
caused(X, Y) ← caused(X, Z) ∧ (Z ⪯ Y).
caused(X∧Y, Z) ← caused(X, Z) ∧ caused(Y, Z).
caused(X, Y) ← caused(X, Y∧Z) ∧ caused(Z, Y).
caused(X, Y∨Z) ← caused(X, Y) ∧ caused(X, Z).
Note: ≼ is a classical consequence operator;
X ≼ Y iff Y ≽ X iff Y entails X, e.g., s ≼ s∧t

GLC is more general than SLC• Proposition. Suppose a theory T for SLC, and let T′ be a theory for GLC
obtained from T by replacing every disjunctive fact of the form
(linked(p, q1) ∨···∨ linked(p, qm)) with a fact of the form linked(p ,q1∧···∧qm). If G = (caused(g, s1) ∨···∨ caused(g, sn)) is derived fromT in SLC, then G′ = caused(g, s1∧···∧sn) is derived from T′ in GLC.
• Proof (sketch): Reflection in GLC corresponds to the first meta-axiom in SLC.
The second meta-axiom in SLC is obtained by Transitivity (which is aconsequence of Strengthening and Cut) and Reflection.
The case inference in SLC can be simulated in GLC as follows. Suppose(linked(q, p1) ∨ linked(q, p2)) in SLC. From caused(p1, s1) and caused(p2, s2),
(caused(q, s1) ∨ caused(q, s2)) is derived in SLC. The corresponding
inference can be obtained in GLC as:s1 ⊨ p 1
s2 ⊨ p 2
s1 ∧ s2 ⊨ p 1
s1 ∧ s2 ⊨ p 2
s1 ∧ s2 ⊨ p1 ∧ p2
s1 ∧ s2 ⊨ q

p1 ∧ p2  q
p1 ∧ p2 ⊨ q

Transitivity in GLC• Strengthening and Cut imply the transitivity:• (Transitivity)

α⊨β
β⊨ψ
α⊨ψ

• There is a controversy on the validness of transitivity, e.g.,
Hitchcock, J. Philos., 98 (2001). Also, McDermott (1995), Pearl(2000, 2009), Halpern & Pearl (2000), Woodward (2004), etc.
• But, we can interpret the causality here simply as an influence.
Moreover, many anomaly cases can be avoided if we carefullymodel the causal theory.

Monotonicity in GLC• (Left) Strengthening is also called Monotonicity (Kraus,
Lehmann & Magidor, 1990):(Strengthening)α≽β β⊨ψα⊨ψ• Monotonicity apparently fails to hold in nonmonotonic logics.
• Example. “Striking a match causes it to light” obviously does
not imply “Putting a match in water and then striking it causes
it to light.” (Bochman & Gabbay, 2012)

Networks with positive and negative causal
links (Inoue, Doncescu & Nabeshima, 2010-2013)
 Consider networks with both positive and negative causal effects.
 In biology, such networks appear in gene regulatory/transcription
systems, signaling networks, and metabolic pathways.
 Two types of direct causal relations: triggered and inhibited.
 triggered(g, t) : a positive direct cause (t is a trigger of g)gt
in a causal graph, whose meaning is (g ⇐ t) in the object level,
where ⇐ means that the causation appears if it is not prevented.
 inhibited(g, s) : a negative direct cause (s is an inhibitor of g)gsin a causal graph, whose meaning is (￢g ⇐ s) in the object level.

Alternating axioms for causality


Causal chains have two kinds too:
1. promoted(X,Y): X is positively caused by Y.
2. suppressed(X,Y): X is negatively caused by Y.
caused(X,Y) ← linked(X,Y).
caused(X,Y) ← linked(X,Z)∧caused(Z,Y).

promoted(X, Y) ← triggered(X, Y).
promoted(X, Y) ← triggered(X,Z) ∧ promoted(Z, Y).
promoted(X, Y) ← inhibited(X,Z) ∧ suppressed(Z, Y).
suppressed(X, Y) ← inhibited(X, Y).
suppressed(X, Y) ← inhibited(X,Z) ∧ promoted(Z, Y ).
suppressed(X, Y) ← triggered(X,Z) ∧ suppressed(Z, Y).
← promoted(X, Y) ∧ suppressed(X,Y).

Monotonic property• Meta-level abduction is defined for an observation
promoted(g, s) or suppressed(g, s)with the abducibles
Γ = { triggered(_,_), inhibited(_,_) }.
• Given positive and negative observations, both positive andnegative direct causes are abduced and new nodes areproduced when necessary.
• Proposition: For any suppression (resp. promotion) for g from s,
there is a causal chain P from s to g such that there exist an odd
number of (resp. an even number of) direct inhibitions in P.

Antagonistic factorstgs

triggered(g, t)

inhibited(g, s)

Suppose that both the trigger t and the suppressor s areactivated. Is g promoted or suppressed? Intuitions are:(1) If t works and s does not work, then g is promoted by t;(2) If s works and t does not work, then g is suppressed by s;(3) If both t and s work, then g is suppressed by s.
Namely, an inhibitor is preferred to a trigger.

Axioms with defaults


Causal chains should have nonmonotonic effects.

promoted(X, Y) ← triggered(X, Y) ∧ no_inhibitor(X).
promoted(X, Y) ← triggered(X,Z) ∧ no_inhibitor(X) ∧ promoted(Z, Y).
promoted(X, Y) ← inhibited(X,Z) ∧ suppressed(Z, Y).
suppressed(X, Y) ← inhibited(X, Y).
suppressed(X, Y) ← inhibited(X,Z) ∧ promoted(Z, Y).
suppressed(X, Y) ← triggered(X,Z) ∧ no_inhibitor(X) ∧ suppressed(Z, Y).
← promoted(X, Y) ∧ suppressed(X,Y).

no_inhibitor(_) : treated as a default, which can be assumed
during inference unless contradiction occurs.

Abduction with defaults• For default assumptions of the form no_inhibitor(_), their
negations are skipped in SOLAR by putting them in the
production field.
• Membership of a clause C in an extension of a default theoryis guaranteed for each obtained consequence of the formC ← no_inhibitor(t1) ∧ no_inhibitor(t2) ∧ ・・・[Inoue et al., 2004, 2006].

Correspondence between object-level inference
and meta-level consequence finding
object-level
inference

top clause in SOLAR *production fieldproving rules￢caused(g, s)none

abducing facts￢caused(g,X)∨ans(X)

ans(_)

predicting facts￢caused(X, s) ∨ ans(X)

ans(_)

predicting rulesnone

promoted(_,_), suppressed(_,_)

abducing rules￢caused(g, s)￢triggered(_,_),￢inhibited(_,_)

abducing rules and facts￢caused(g,X)∨￢abd(X)￢triggered(_,_),￢inhibited(_,_),
ans(_)

predicting conditionalfacts￢caused(X, s) ∨ ans(X)￢triggered(_,_),￢inhibited(_,_),
ans(_)

predicting conditional
rules

none

￢triggered(_,_),￢inhibited(_,_),
promoted(_,_), suppressed(_,_)*￢caused(X,Y) is instantiated by either ￢promoted(X,Y) or ￢suppressed(X,Y).

Consistency of meta-level abduction
B0 = { inhibited(p, p) }.

p

B0 U Meta-Axioms is inconsistent.
B0 represents a negative feedback loop.
Theorem (Inoue et al., 2013): B is consistent if and only ifthere are NO nodes g, s in N such that there are both a proof
Π+ of promoted(g, s) and a proof Π− of suppressed(g, s)satisfying (i) neither Π+ nor Π− have a trigger; (ii) there arean even number of occurrences of inhibitors in Π+; and (iii)
there are an odd number of occurrences of inhibitors in Π−.
prq

B1 : This is an inconsistent network, sinceboth suppressed(r, p) and promoted(r, p)
are derived.

p53 signal network (Tran & Baral, 2009)
UV

p53
Mdm2
A
B

cancerX

Meta-level representation for
p53 signal network
triggered(cancer, uv),
triggered(p53, uv),
inhibited(cancer, a),triggered(a, p53),
inhibited(a, b),
triggered(b, p53 ∧ mdm2),
triggered(X, Y ∧ Z) ≡ (triggered(X, Y ) ∨ triggered(X,Z)).

Goal and abducibles for p53 signal network• Consider a tumor suppressor gene X such that mutants of X are
highly susceptible to cancer. Suppose exposure of the cell to highlevel UV does not lead to cancer, given that the initial concentrationof Mdm2 is high. These initial conditions are represented assource(uv) ∧ source(mdm2),
i.e., both UV and Mdm2 can be abduced whenever necessary.
• Objective: hypothesize about the possible influences of X on thep53 pathway, explaining how the cell can avoid cancer.
• Goal:∃S (suppressed(cancer, S) ∧ source(S))• Abducibles: Γ = {triggered(_,_), inhibited(_,_)}• Top clause: (￢suppressed(cancer, S)∨￢source(S) ∨ans(S))• Production field: {￢L | L∈Γ} ∪ { ans(_) , ￢no_inhibitor(_) }• SOLAR produces 26 minimal hypotheses in 3 seconds.

Hypothesis I
UV

p53
Mdm2
A

X

B

cancerC

triggered(x, uv) ∧ ∃Y (triggered(Y, mdm2 ∧ x)∧inhibited(b, Y))Hypothesis II
UV

p53
Mdm2
A

X

B

cancerC

triggered(x, uv) ∧ ∃Y (triggered(Y, p53 ∧ x)∧inhibited(b, Y))

Cyclin-dependent kinases (Schneider et al., 2002)
DNA
damage

(5)

p53
(4)

(6)

cyclin H
p21/WAF1

cdk7

CAK

cdk2

(3)

cyclin E

DNA
synthesis

cyclin E/
cdk2
Rb
(1)

(2)

Cut-off experiment of CDK network••••

The original CDK network is cut off at (1)—(6).
Whether the same links are restored by meta-level abduction?
Interestingly, the network has a redundancy (→robustness).
Some links are recovered by being generalized. E.g., {(1g),(2g)}becomes ∃X(inhibited(dna_synthesis, X) ∧ inhibited(X, cyclin e/cdk2))• New pathways are also created. (7) corresponds to analogy.
Removed links#HP21-mediatedCdk2 active

cyclin E inhibit

(1)

4

(1)

—

—

(1),(2)9(1g),(2g)——(1),(3)17(1),(3)(1)(7)(2),(4)24(2),(4)(2)(7)(4),(6)

22

(4)

(6)

(7)

(1),(2),(3),(4),(5)392(1g),(2g),(3g),(4),(5)(1g),(2g),(5)(1g),(2g),(5),(7)DNAdamagepromoted

(5)

p53

(4)

(6)
(7)

cyclin H

p21/WAF1

cdk7

CAK
cdk2

suppressed(3)cyclin E

suppressedDNAsynthesis

cyclin E/
cdk2

promoted
(1)

Rb

(2)

suppressedLaw Meta-Level Abduction
Kind of abduction

Explanandum

Explanans

Factual abduction

Single facts

New facts— Observable-fact abduction

Single facts

Factual reasons— 1st-order existential abduction

Single facts

Facts with new unknown
individuals— Unobservable-fact abduction

Single facts

Unobservable facts

Law abduction

Empirical laws

New laws

Theoretical-model abduction

General empiricalphenomena

New theoretical models

2nd-order existential
abduction

General empiricalphenomena

New laws with newconcepts— Micro-part abduction

General empiricalphenomena

Microscopic compositions— Analogical abduction

General empiricalphenomena

New laws with analogicalconcepts— Hypothetical cause abduction

General empiricalphenomena

Hidden (unobservable)causesLaw meta-level abduction• Could be called “meta-level induction”• Theoretically, any ILP system that is capable of law
abduction can be used for law MLA.
• By law MLA, we can devise a part of deductive or
abductive system.
• For example, let T be
(caused(X,Y) ← linked(X,Z) ∧ caused(Z,Y)) ∧linked(a, b) ∧ linked(b, c), and G = caused(a, c).
Then, CF-induction (Inoue, 2014) can induce ameta-axiom: (caused(X,Y) ← linked(X,Y)). This is
abduction of a meta-axiom for SLC.

Representational Meta-Level Abduction
Kind of abduction

Explanandum

Explanans

Factual abduction

Single facts

New facts— Observable-fact abduction

Single facts

Factual reasons— 1st-order existential abduction

Single facts

Facts with new unknown
individuals— Unobservable-fact abduction

Single facts

Unobservable facts

Law abduction

Empirical laws

New laws

Theoretical-model abduction

General empiricalphenomena

New theoretical models

2nd-order existential
abduction

General empiricalphenomena

New laws with newconcepts— Micro-part abduction

General empiricalphenomena

Microscopic compositions— Analogical abduction

General empiricalphenomena

New laws with analogicalconcepts— Hypothetical cause abduction

General empiricalphenomena

Hidden (unobservable)causesRepresentational meta-level abduction(thought experiment)• A representation change often appears in scientific discovery
as a paradigm shift.
• For example, non-Euclidean geometry changed geometricalpostulates of Euclidean geometry. In the course of such a
discovery, a new axiom set is introduced into the newframework. Hence abducing an essential change in a domain
is representation abduction at the object level.
• For such invented non-Euclidean geometries, Hilbert studied
axiomatic systems of both Euclidian and non-Euclidian
geometries in his “Foundations of Geometry”. As Hilbert later
invoked the term metamathematics, his work on axiomatic
systems exactly describes a meta-theory of those variants ofgeometries. This is exactly a case of representation MLA.

Can machines learn logics?
• Extraction of a hidden logic from a domain or inventing a newlogic from observations in a new situation invokes
representation MLA.
• Sakama and Inoue (2015) consider the question:– There are two machines A and B. The machine A is capable of deductive
reasoning with an underlying logic L. Given a set S of formulas as an
input, the machine A produces (a subset of) the logical consequences
Th(S) as an output. The machine B has no axiomatic system fordeduction. Given input-output pairs (S1, Th(S1)), . . . , (Si, Th(Si)), . . . ofA as an input to B, the problem is whether one can develop an algorithm
C which successfully produces the axiomatic system L for deduction.

• Meta-level one-step deduction rules including MP are learned.
• The scenario can be applied to learning abduction and othernon-standard logics.

Problem solving with meta-level abduction• Consists of:
1. design of meta-level axioms,
2. representation of domain knowledge at the meta level,
3. restriction of the search space to treat large knowledge.
• The task (2) is rather tractable.
• Law abduction can contribute to the task (1) is important. Butother axiomatizations are considerable, e.g., introduction of time,modality, majority logic.
• The task (3) can be realized by introducing more constraints.
Automation of constraint generation is future work.
• Hypothesis evaluation/ranking is also important, c.f. (Inoue et al.,IJCAI-09), (Gat-Viks & Shamir, 2002).
• Analogy is useful to get a particular form of hypotheses.

Conclusion/Perspectives• The method of meta-level abduction has been analyzed from theviewpoint of patterns of abduction.
• Fact MLA is strong enough to realize 2nd-order existential
abduction at the meta-level. Two logics of causality, SLC and GLC,
have been focused in the form of causal networks.
• New nodes can also be abduced and predicate invention isrealized as existentially quantified hypotheses.
• Law abduction is another form of induction—SOLAR as an
inductive inference engine.
• Meta-level abduction has been extended to allow abduction of
positive and negative links and abduction with defaults.
• Law MLA can produce meta-axioms in logics. Representation
MLA can abduce logics themselves.