Actual Causality in Contextual Abduction?

1 Introduction

In daily life, causality is often employed to understand of why particular things happen. Di erent from general causality that captures the general laws that describe the cause-and-e ect 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 in uence 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 2 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 = fFX : X 2 Vg 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 represented 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 n fXg). An interpretation of propositions satisfying biconditional A , F for every boolean structural equation 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 de ned 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 speci c event, abduction seems to be a natural approach in nding 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 Commons License Attribution 4.0 International (CC BY 4.0). [ 11 ]. Abduction has already been well studied in logic programming with various 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 nding plausible explanations. In the rst 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 nding 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 program's rules by de nition. Nevertheless, abduction in logic programming lends itself to capturing boolean structural equations. Recall that exogenous variables di er 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 programming, 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.

De nition 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 .

De nition 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. Next, we state the correspondence between an abductive framework and HP boolean causal model.

De nition 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 Su cient Set) test [ 19 ] for de ning 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 de nition of actual causality also assumes that an actual world (an exact causal world) of F is observed.

NESS test has also been used recently by Bochman [ 3 ] to de ne actual causality albeit using BL causal theory as its logical setting. We now refer to our causal abductive framework and de ne actual cause by formalizing the following NESS test: \a particular condition was a cause of (condition contributing to) a speci c consequence if and only if it was a necessary element of a set of antecedent actual conditions that was su cient for the occurrence of the consequence". De nition 4 (Actual Cause via NESS Test). Let F be a causal abductive framework and be an actual world wrt. F . A literal L 2 is an actual cause of a literal G 2 wrt. F if S is a minimal abductive solution to query G wrt. F , S is consistent wrt. , and L 2 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.

De nition 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 S wrt. an actual world is obtained by having the abductive context I for G, where I is formed from literals in whose atoms are exogenous variables in M .

We now set up a procedure for determining, in a particular actual world , actual causes of an event l 2 wrt. a causal abductive framework F . The procedure 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 de ne, 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 negation not Ab is de ned as complA(Ab) = not Ab and complA(not Ab) = Ab, respectively.

Algorithm Tabdual

Input : F = hP; Ai, actual world , literal l 2 Output: actual cause of l wrt. F and causal world

1: Finding actual causes via contextual abduction in 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 S of query l w.r.t. the given actual world under P +. 4. Construct set W that consists of only abductive solution S 2 W satisfying for every t 2 S , complA(t) 2= S. 5. Every literal c 2 S where S 2 W is an actual cause of l in .

The experiment in modeling examples from the literature (including Loader [ 10 ], Window [ 3 ], Backup [ 9 ], Bottle [ 3 ], Bogus Prevention [ 8 ], Push [ 13 ], Inevitable 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. Di erent from Bochman's approach in computing actual causes [ 3 ], we focus on abduction in de ning 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 evaluate further the obtained results and conduct a comparative analysis with other approaches that de ne 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 e ects in justifying morally permissible actions.

References

1. Alferes , J.J. , Pereira , L.M. , Swift , T. : Abduction in well-founded semantics and generalized stable models via tabled dual programs . Theory and Practice of Logic Programming 4 ( 4 ), 383 { 428 ( 2004 )

2. Beckers , S. , Vennekens , J.: A general framework for de ning and extending actual causation using CP-logic . International Journal of Approximate Reasoning 77 , 105 { 126 ( 2016 )

3. Bochman , A. : Actual causality in a logical setting . In: IJCAI . pp. 1730 { 1736 ( 2018 )

4. Bochman , A. , Lifschitz , V. : Pearl's causality in a logical setting . In: AAAI ( 2015 )

5. Denecker , M. , Kakas , A.C. : Abduction in logic programming . In: Computational Logic: Logic Programming and Beyond, Essays in Honour of Robert A . Kowalski , Part I. p. 402 { 436 . Springer-Verlag, Berlin, Heidelberg ( 2002 )

6. Ehring , D. : Causal relata . Synthese pp. 319 { 328 ( 1987 )

7. Halpern , J.Y. , Pearl , J. : Causes and explanations: A structural-model approach. Part I: Causes . British Journal for the Philosophy of Science 56 ( 4 ), 843 { 887 ( 2005 )

8. Hiddleston , E.: Causal powers . British Journal for the Philosophy of Science 56 ( 1 ), 27 { 59 ( 2005 )

9. Hitchcock , C. : Prevention, preemption, and the principle of su cient reason . The Philosophical Review 116 ( 4 ), 495 { 532 ( 2007 )

10. Hopkins , M. , Pearl , J. : Clarifying the usage of structural models for commonsense causal reasoning . In: Proc. AAAI Spring Symposium on Logical Formalizations of Commonsense Reasoning . pp. 83 { 89 ( 2003 )

11. Josephson , J.R. , Josephson , S.G. : Abductive Inference: Computation, Philosophy, Technology. Cambridge U. P. ( 1995 )

12. LeBlanc , E., Balduccini , M. , Vennekens , J.: Explaining actual causation via reasoning about actions and change . In: JELIA . Springer ( 2019 )

13. McDermott , M. : Redundant causation . British Journal for the Philosophy of Science pp. 523 { 544 ( 1995 )

14. Pearl , J. : Causality: Models, Reasoning and Inference . Cambridge University Press ( 2000 )

15. Pereira , L.M. , Saptawijaya , A. : Counterfactuals, logic programming and agent morality . In: Urbaniak, R. , Payette , G . (eds.) Applications of Formal Philosophy, vol. 14 . Springer ( 2017 )

16. Perkasa , S. , Saptawijaya , A. , Pereira , L. : Tabling in contextual abduction with answer subsumption . In: ICACSIS . pp. 459 { 464 . IEEE ( 2017 )

17. Saptawijaya , A. , Pereira , L.M.: Tabdual: a tabled abduction system for logic programs . IfCoLog Journal of Logics and their Applications 2 ( 1 ), 69 { 124 ( 2015 )

18. Vennekens , J.: Actual causation in CP-logic . Theory and Practice of Logic Programming 11 ( 4-5 ), 647 { 662 ( 2011 )

19. Wright , R.: Causation in tort law . California Law Review 73 , 1735 ( 1985 )