=Paper=
{{Paper
|id=Vol-2678/paper8
|storemode=property
|title=Actual Causality in Contextual Abduction
|pdfUrl=https://ceur-ws.org/Vol-2678/paper8.pdf
|volume=Vol-2678
|authors=Degoldie Sonny,Ari Saptawijaya,Raja Oktovin Parhasian Damanik
|dblpUrl=https://dblp.org/rec/conf/iclp/SonnySD20
}}
==Actual Causality in Contextual Abduction==
https://ceur-ws.org/Vol-2678/paper8.pdf
Actual Causality in Contextual Abduction?
Degoldie Sonny1 , Ari Saptawijaya2 , and Raja Oktovin Parhasian Damanik2
1
Shopee, Singapore
degoldiesonny98@gmail.com
2
Faculty of Computer Science, Universitas Indonesia
{saptawijaya, rajaoktovin}@cs.ui.ac.id
1 Introduction
In daily life, causality is often employed to understand of why particular things
happen. Different from general causality that captures the general laws that
describe the cause-and-effect relationships, actual causality aims at explaining
why a certain event happened using observations of what have happened prior
to the event.
One of the well-known approaches to deal with actual causality has been
developed by Halpern and Pearl (HP) [7]. HP causal model is based on structural
models [14], where structural equations are introduced to capture the causal
influence of some variables to another. The variables are distinguished according
to how their values are determined: those whose values are determined by factors
outside of the model are called exogenous variables, whereas endogenous variables
have their values determined by other variables within the model. A signature
S is a triple (U, V, R), where U is the set of exogenous variabels, V is the set of
endogenous variables, and R is a function such that for each X ∈ U ∪ V, R(X) is
the set of possible values that X can take. In this extended abstract, we assume
that the variable is binary or boolean. A causal model is then a tuple M = (S, F)
where S = (U, V, R) is a signature and F = {FX : X ∈ V} is a set of causal
functions FX , called structural equations, one for each endogenous variables X.
Bochman and Lifschitz (BL) shows that structural equations can be repre-
sented in causal calculus [4]. In boolean causal model, function FX corresponds
to boolean structural equations of the form X = F , where X is an endogenous
variable and F is a propositional formula in U ∪ (V \ {X}). An interpretation of
propositions satisfying biconditional A ⇔ F for every boolean structural equa-
tion A = F in a boolean causal model M gives a causal world for M . Given a
boolean causal model M , a (propositional) causal theory ∆M can be defined as
the set of the rules F ⇒ A and ¬F ⇒ ¬A for each structural equation A = F
in M and all rules A ⇒ A and ¬A ⇒ ¬A for every exogenous atoms A in M .
As actual causality amounts to explaining a specific event, abduction seems
to be a natural approach in finding such explanations. In abduction, one chooses
from available hypotheses those that would best explain the observed evidence
?
Copyright c 2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
2 Sonny et al.
[11]. Abduction has already been well studied in logic programming with vari-
ous applications [5]. Abductive logic programming is realized by extending logic
programs with hypotheses, called abducibles. An abducible is an atom Ab or its
negation not Ab (syntactically an atom, but denoting not Ab) whose truth value
are initially not assumed. Typically, abduction may be restricted by integrity
constraints, but they are not needed for the purpose of this extended abstract
and therefore are omitted.
The goal of this ongoing research is therefore to further examine the notion
of actual causation in abductive logic programming. In particular, we look into
contextual abduction with its implementation Tabdual [17, 16] to model actual
causality. Contextual abductions allows an observation to be explained with
respect to a given context. This context can be viewed as a constraint in finding
plausible explanations. In the first contribution, we propose a causal abductive
framework to represent causal model of Halpern and Pearl [7] by translating their
structural equations into abductive logic programs. In the second contribution,
we provide a practical procedure, on top of Tabdual, to enact actual causality
directly from an abductive logic program by contextual abduction. Therein, the
given actual causal world is treated as the context of abduction and actual causes
are then computed by finding consistent abductive solutions with respect to that
given context.
2 Technical Means
The rules in BL causal theory that correspond to structural equations in HP
boolean causal model cannot immediately be represented as a logic program.
This is because rules with a negative head, ¬F ⇒ ¬A, do not qualify as logic pro-
gram’s rules by definition. Nevertheless, abduction in logic programming lends
itself to capturing boolean structural equations. Recall that exogenous variables
differ from endogenous ones, in that the values of the former are determined by
factors outside of the model. That is, in contrast to endogenous variables, they
are not causally explained by variables in the model. In abductive logic program-
ming, this notion of exogenous variables can suitably be mapped into abducibles,
thus BL’s rule A ⇒ A for an exogenous variable A needs no corresponding logic
program’s rule.
Definition 1 (Abducibles wrt. Boolean Causal Model). Let M be a boolean
causal model. The set of abducibles with respect to model M , denoted by AM , is
the set of exogenous atoms U in M .
Definition 2 (Logic Program wrt. Boolean Causal Model). Let M be a
boolean causal model. The logic program with respect to model M , denoted by
PM , is the set of rules, A ← F , obtained from the structural equation A = F in
model M for each endogenous atom A.
Note that the logic program PM corresponds only to a subset of rules in the
causal theory ∆M for the reason explained in the beginning of this section.
Actual Causality in Contextual Abduction 3
Next, we state the correspondence between an abductive framework and HP
boolean causal model.
Definition 3 (Causal Abductive Framework). Let M be a boolean causal
model. A causal abductive framework FM = hPM , AM i is an abductive framework
that corresponds to M , where PM and AM are the logic program and the set of
abducibles wrt. model M , respectively.
We adopt NESS (Necessary Element of a Sufficient Set) test [19] for defining
actual causality as post factum attribution of causal responsibility for actual
outcome: “a particular condition c was a cause of event e”. That is, in addition
to a causal abductive framework F , the definition of actual causality also assumes
that an actual world (an exact causal world) α of F is observed.
NESS test has also been used recently by Bochman [3] to define actual causal-
ity albeit using BL causal theory as its logical setting. We now refer to our causal
abductive framework and define actual cause by formalizing the following NESS
test: “a particular condition was a cause of (condition contributing to) a specific
consequence if and only if it was a necessary element of a set of antecedent actual
conditions that was sufficient for the occurrence of the consequence”.
Definition 4 (Actual Cause via NESS Test). Let F be a causal abductive
framework and α be an actual world wrt. F . A literal L ∈ α is an actual cause
of a literal G ∈ α wrt. F if S is a minimal abductive solution to query G wrt.
F , S is consistent wrt. α, and L ∈ S.
The consistency requirement for a minimal abductive solution wrt. an actual
world can be achieved in contextual abduction by imposing the given actual
world as the abductive context of the query.
Definition 5 (Abductive solution wrt. actual world). Let FM be a causal
abductive framework corresponds to boolean causal model M . Given a query G
wrt. FM , the abductive solution Sα wrt. an actual world α is obtained by having
the abductive context I for G, where I is formed from literals in α whose atoms
are exogenous variables in M .
We now set up a procedure for determining, in a particular actual world α,
actual causes of an event l ∈ α wrt. a causal abductive framework F . The proce-
dure relies on Tabdual to carry out contextual abduction and to automatically
compute the set of dual rules dual(P ) of the abductive logic program P [17].
In Tabdual, a set of dual rules are introduced, by means of the dual program
transformation [1], to deal with abduction under negative goals. The idea of the
dual transformation is to define, for each atom A and its set of rules R in a logic
program P , a set of dual rules whose head not A is true if and only if A is false
by R in the employed semantics of P . Note that, instead of having a negative
goal not A as the rules’ head, its corresponding ‘positive’ literal not A is used,
thus conforming the syntax of rules in a logic program.
In the following procedure, complA (A) denotes the negation complement
of an abducible A, where the complement of a positive abducible Ab and its
4 Sonny et al.
negation not Ab is defined as complA (Ab) = not Ab and complA (not Ab) = Ab,
respectively.
Algorithm 1: Finding actual causes via contextual abduction in
Tabdual
Input : F = hP, Ai, actual world α, literal l ∈ α
Output: actual cause of l wrt. F and causal world α
1. Compute dual(P ). Let P + = P ∪ dual(P ).
2. Compute minimal abductive solution S of query l with empty context
under P + . Let W be the set of all such minimal abductive solutions.
3. Compute the abductive solution Sα of query l w.r.t. the given actual world
α under P + .
4. Construct set Wα that consists of only abductive solution S ∈ W satisfying
for every t ∈ Sα , complA (t) ∈
/ S.
5. Every literal c ∈ S where S ∈ Wα is an actual cause of l in α.
The experiment in modeling examples from the literature (including Loader
[10], Window [3], Backup [9], Bottle [3], Bogus Prevention [8], Push [13], In-
evitable Shock [13], Purple Flame [6]) shows that the returned actual causes
mostly agree with those delivered by Bochman’s approach [3], albeit restricted
to abducibles only.
3 Conclusions
In this extended abstract, we have tried to examine the notion of actual causation
in a causal abductive framework, i.e., by translating the structural equations
in the causal model of Halpern and Pearl [7] into abductive logic programs.
Employing this framework, a procedure has been proposed that makes use of
the tabled abduction system Tabdual, to enact actual causality directly from
an abductive logic program by contextual abduction.
Our approach is similar to the causal theory of Bochman and Lifschitz (BL)
[4] in that situations are modeled using a set of rules. Different from Bochman’s
approach in computing actual causes [3], we focus on abduction in defining
and computing actual causality. Moreover, our framework consists of normal
logic programs, whereas BL causal theory allows rules with a negative head.
But thanks to the dual program transformation of Tabdual, such rules with
negative head (dual rules) are automatically computed.
This research is an ongoing work. An important next step will be to evalu-
ate further the obtained results and conduct a comparative analysis with other
approaches that define actual causality in logic programming, e.g., [12, 18, 2]. It
is also part of future work to explore the application of the present approach to
machine ethics, extending our previous counterfactual approach [15]—also built
from abduction in logic programming—for distinguishing causes and side effects
in justifying morally permissible actions.
Actual Causality in Contextual Abduction 5
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