=Paper=
{{Paper
|id=Vol-2261/paper1
|storemode=property
|title=Ethical Decision Making under the Weak Completion Semantics
|pdfUrl=https://ceur-ws.org/Vol-2261/paper1.pdf
|volume=Vol-2261
|authors=Steffen Hölldobler
|dblpUrl=https://dblp.org/rec/conf/ijcai/Holldobler18
}}
==Ethical Decision Making under the Weak Completion Semantics==
https://ceur-ws.org/Vol-2261/paper1.pdf
Ethical Decision Making under the Weak Completion Semantics
Steffen Hölldobler
International Center for Computational Logic, Technische Universität Dresden, Germany
and North-Caucasus Federal University, Stavropol, Russian Federation
sh@iccl.tu-dresden.de
Abstract all 12 cognitive theories discussed in [Khemlani and Johnson-
Laird2012]. As all human reasoning tasks are solved within
The weak completion semantics is a novel com- one framework, the WCS is an integrated and computational
putational theory based on logic programs. It is cognitive theory. We are unaware of any other theory of this
extended to deal with equalities, which is a pre- kind and with such a wide variety of applications.
requisite to represent and reason about actions and Recently, ethical decision making has received much at-
causality as in the fluent calculus. This is discussed tention as autonomous agents become part of our daily life.
in the context of ethical decision making. In or- In particular, we were inspired by [Pereira and Saptawi-
der to decide questions about the moral permissi- jaya2016], who studied computational models of machine
bility of actions, counterfactuals need to be consid- ethics. Various ethical problems are implemented as logic
ered. Somewhat surprisingly, this can be straight- programs and these programs can be queried for moral per-
forwardly done in the extended approach. missibility. Unfortunately, their approach does not provide
a general method to account for ethical dilemmas and is not
integrated into a cognitive theory about human reasoning.
1 Introduction The problems studied in [Pereira and Saptawijaya2016]
The weak completion semantics (WCS) is a novel cognitive were trolley problems or variants thereof like the bystander
theory. Its original idea is based on [Stenning and van Lam- case. In these problems, actions with direct and indirect ef-
balgen2008] who proposed to model human reasoning tasks fects must be considered. Hence, in order to model and rea-
by, firstly, reasoning towards a normal logic program to rep- son about these problems within the WCS, the WCS must be
resent the reasoning task and, secondly, by reasoning with re- extended to deal with actions and causality. We have chosen
spect to the least model of the normal logic program. Unfor- the fluent calculus [Hölldobler and Schneeberger1990] for
tunately, Stenning and van Lambalgen’s approach contained modeling actions and causality because it treats fluents as re-
a technical bug which was corrected in [Hölldobler and Ken- sources which can be consumed and produced. This property
cana Ramli2009]. is shared with Petri networks [Hölldobler and Jovan2014], the
The WCS is based on many techniques and methods from latter of which have already been used in computational mod-
logic programming and computational logic. However, these els for human reasoning [Barrett2010].
techniques and methods are usually tweaked a little bit in or- In the fluent calculus [Hölldobler and Schneeberger1990]
der to model human reasoning tasks adequately. For exam- states are represented as multisets of fluent. Multisets are rep-
ple, programs are not completed in the sense of [Clark1978], resented with the help of a binary function symbol ◦ written
but only weakly completed. Instead of the semantic operator infix and a constant 1 such that ◦ is commutative, associative,
introduced in [Fitting1985], a modified operator introduced and 1 is its unit element. For example, the multisets {˙ }˙ and
in [Stenning and van Lambalgen2008] is used. Instead of the ˙ b, b}˙ are represented by the fluent terms 1 and a ◦ b ◦ b,
{a,
three-valued Kripke-Kleene logic used in [Fitting1985, Sten- respectively. In order to deal with function symbols like ◦ in
ning and van Lambalgen2008], the three-valued Łukasiewicz the WCS, we need to extend WCS to handle equality. Luck-
logic [Łukasiewicz1920] is used. Because of the latter, nor- ily, as shown in [Dietz Saldanha et al.2018] the key properties
mal logic programs admit a least model and reasoning is per- of the WCS, viz. the existence of a least model and the fact
formed with respect to this model (see [Hölldobler and Ken- that this model can be computed as the least fixed point of an
cana Ramli2009]). appropriate semantic operator, hold also for logic programs
The approach has been applied to various human rea- with equality.
soning tasks like the suppression task [Byrne1989, Dietz et In this paper, we will focus on the representation of the
al.2012], the selection task [Wason1968, Dietz et al.2013], bystander case. We will show how to represent this prob-
and human syllogistic reasoning [Khemlani and Johnson- lem in the extended approach. In particular, we formalize
Laird2012, Oliviera da Costa et al.2017]. In fact, WCS per- a purely utilitarian view [Bentham2009] and the doctrine of
formed better on the human syllogistic reasoning tasks than double effect [Aquinas1988]. In order to decide which action
is morally permissible in the bystander case we need to rea- → m ••
(initial state)
son about a counterfactual [Nickerson2015]. It turns out, that 0 1
this can be straightforwardly done in the extended approach. s •
2
2 The Weak Completion Semantics with m → ••
(trolley moving to track 1)
Equality 0 1
s •
We assume the reader to be familiar with the WCS as pre-
2
sented in [Hölldobler2015, Dietz Saldanha et al.2017]. In
the weak completion semantics with equality (WCSE) a logic m d→•
(trolley killing first human)
program P is considered together with a set E of equations. 0 1
As shown in [Jaffar et al.1984], E defines a finest congru- s •
ence relation on the set of ground terms. Let [t] denote the 2
congruence class defined by the ground term t. For example, m dd ↓
[a ◦ b ◦ b] = [b ◦ a ◦ b] = [b ◦ b ◦ a ◦ 1]. Furthermore, (trolley killing second human)
0 1
let [p(t1 , . . . , tn )] be an abbreviation for p([t1 ], . . . , [tn ]), s •
where p is an n-ary relation symbol and all ti , 1 ≤ i ≤ n, 2
are ground terms. [p(t1 , . . . , tn )] = [q(s1 , . . . , qm )] if and
only if p = q, n = m, and [ti ] = [si ] for all 1 ≤ i ≤ n.
For example, [p(a ◦ b ◦ b, 1)] = [p(b ◦ a ◦ b, 1 ◦ 1]. We con- Figure 1: The bystander case (initial state) and its ramifications if
sider E-interpretations and E-models as usual (see e.g. [Jaffar Hank decides to do nothing, where ↓ denotes that no further action
et al.1984]). is applicable.
As shown in [Dietz Saldanha et al.2018], a logic program
P together with a set E of equation has a least E-model un-
der the three-valued Łukasiewicz logic [Łukasiewicz1920]. the side track. Hank can change the switch, killing him. Or
This model is the least fixed point of the following semantic he can refrain from doing so, letting the two die. Is it morally
E
operator: Let I be an E-interpretation. We define ΦP (I) = permissible for Hank to change the switch?
> ⊥
hJ , J i where The case is illustrated in Figure 1 (initial state). The tracks
J >
= {[A] | there exists A ← Body ∈ gP are divided into segments 0, 1, and 2, the arrow represents that
and I(Body) = >}, the trolley t is moving forward and that the track is clear (c),
J⊥ = {[A] | there exists A ← Body ∈ gP the switch is in position m (main) but can be changed into po-
and for all A0 ← Body ∈ gP sition s (side), and a bullet above a track segment represents a
with [A] = [A0 ] human (h) on this track. t, c, and h may be indexed to denote
we find I(Body) = ⊥}, the track to which they apply. In addition, we need a fluent d
denoting a dead human.
and gP denotes the set of all ground instances of clauses oc-
curring in P. We choose to represent a state by a pair of multisets con-
One should observe that the set E of equations is built sisting of the casualties in its second element and all other
into the computation of the ΦP E
-operator: In the computation fluents in its first element. Multisets are represented by so-
> called fluent terms in the fluent calculus, i.e., the initial state
of J , if a ground atom A is mapped to true because it is the
of the bystander case is the pair
head of a rule whose body is true, then all members of the
congruence class containing A are mapped to true. Likewise,
in the computation of J ⊥ we do not only have to consider all (t0 ◦ c0 ◦ m ◦ h1 ◦ h1 ◦ h2 , 1) (1)
rules with head A, but all rules whose head A0 is in the same
congruence class as A, and if A is mapped to false, then all of fluent terms. The casualties are represented in the second
members of the congruence class containing A are mapped to element of (1) by the constant 1 encoding the empty multi-
false. set. Initially, there are no casualties, but casualties will play
a special role when preferring one action over another as will
3 The Bystander Case be discussed later in this section. The first element of (1) en-
A trolley, whose conductor has fainted, is headed towards two codes the multiset {t˙ 0 , c0 , m, h1 , h1 , h2 }.
˙
people walking on the main track.1 The banks of the track are
There are two kinds of actions, the ones which can be per-
so steep that these two people will not be able to get off the
formed by Hank (the direct actions donothing and change),
track in time. Hank is standing next to a switch, which can
and the actions which are performed by the trolley (the indi-
turn the trolley onto a side track, thereby preventing it from
rect actions downhill and kill ). We will represent the actions
killing the two people. However, there is a man standing on
by the trolley explicitly with the help of a five-place relation
1
Note that in the original trolley problem, five people are on the symbol action specifying the preconditions, the name, and
main track. For the sake of simplicity, we assume that only two the immediate effects of an action. As a state is represented
people are on the main track. by two multisets, the preconditions anf the immediate effects
have also two parts: → m ••
(initial state)
0 1
action(t0 ◦ c0 ◦ m, 1, downhill , t1 ◦ c0 ◦ m, 1) ← > s •
action(t0 ◦ c0 ◦ s, 1, downhill , t2 ◦ c0 ◦ s, 1) ← > 2
action(t1 ◦ h1 , 1, kill , t1 , d) ← > m ••
(trolley moving to track 2)
action(t2 ◦ h2 , 1, kill , t2 , d) ← > 0 1
s →•
If the trolley is on track 0, this track is clear, and the switch is 2
in position m, then it will run downhill onto track 1 whereas
track 0 remains clear and the switch will remain in posi- m ••
(trolley killing human
tion m; if, however, the switch is in position s, the trolley will 0
d ↓ on track 2)
s
run downhill onto track 2. If the trolley is on either track 1
or 2 and there is a human on this track, it will kill the human 2
leading to a casualty.
The possible actions of Hank are the base cases in the def- Figure 2: The bystander case (initial state) and its ramifications if
inition of causality:2 Hank decides to change the switch. One should observe that now
causes(donothing, t0 ◦ c0 ◦ m ◦ h1 ◦ h1 ◦ h2 , 1) ← > the switch points to the side track.
causes(change, t0 ◦ c0 ◦ s ◦ h1 ◦ h1 ◦ h2 , 1) ← > (2)
The recursive case of the definition of causality is given as Let P be the program consisting of the clauses mentioned
in this section so far and E be the set of equations specifying
causes(A, E1 ◦ Z1 , E2 ◦ Z2 ) ← that ◦ is associative, commutative, and 1 being its unit ele-
action(P1 , P2 , A0 , E1 , E2 ) ∧ ment. Hank has the choice to do nothing or to change the
causes(A, P1 ◦ Z1 , P2 ◦ Z2 ) ∧ (3) switch. Depending on his decision, the trolley will execute
¬ab(A0 ). its actions which are computed as ramifications in the fluent
It checks whether in a given state (P1 ◦ Z1 , P2 ◦ Z2 ) an calculus [Thielscher2003]. If Hank is doing nothing, then the
action A0 is applicable, which is the case if the precondi- least E-model of P – which is equal to the least fixed point
E E
tions (P1 , P2 ) are contained in the given state. If this holds, of ΦP – is computed by iterating ΦP starting with the empty
then the action is executed leading to the successor state interpretation h∅, ∅i. The following equivalence classes will
(E1 ◦Z1 , E2 ◦Z2 ), where (E1 , E2 ) are the direct effects of the be mapped to true in subsequent iterations:4
action A0 . In other words, if an action is applied, then its pre- [causes(donothing, t0 ◦ c0 ◦ m ◦ h1 ◦ h1 ◦ h2 , 1)]
conditions are consumed and its direct effects are produced. [causes(donothing, t1 ◦ c0 ◦ m ◦ h1 ◦ h1 ◦ h2 , 1)]
Such an action application is considered to be a ramifica- [causes(donothing, t1 ◦ c0 ◦ m ◦ h1 ◦ h2 , d)]
tion [Thielscher2003] with respect to the initial, direct action [causes(donothing, t1 ◦ c0 ◦ m ◦ h2 , d ◦ d)]
performed by Hank. Hence, the first argument A of causes is
not changed. The execution of an action is also conditioned They correspond precisely to the four states shown in Fig-
by ¬ab(A0 ), where ab is an abnormality predicate. Such ab- ure 1. No further action is applicable to the elements of the
normalities were introduced in [Stenning and van Lambal- final congruence class. The two people on the main track will
gen2008] to represent conditionals as licenses for inference. be killed.
In this example, there is nothing abnormal known with re- On the other hand, if Hank is changing the switch, then the
E
spect to the actions downhill and kill and, consequently, the least fixed point of ΦP contains
assumptions [causes(change, t2 ◦ c0 ◦ s ◦ h1 ◦ h1 , d)].
ab(downhill ) ← ⊥
The two people on the main track will be saved but the person
ab(kill ) ← ⊥
on the side track will be killed. This case is illustrated in
are added to the program. But we can imagine situations, Figure 2.
where the trolley will only cross the switch if the switch is The two cases can be compared by means of a prefer
not broken.3 clause:
2
In the original version of the fluent calculus, causes is a ternary prefer (A1 , A2 ) ←
predicate stating that the execution of a plan transfers an initial into causes(A1 , Z1 , D1 ) ∧
a goal state. Its base case is of the form causes(X, [ ], X), i.e., causes(A2 , Z2 , D1 ◦ d ◦ D2 ) ∧
the empty plans transforms arbitrary states X into X. Generating ¬abprefer (A1 )
models bottom up using a semantic operator one has to consider all abprefer (change) ← ⊥
ground instances of this atom, which is usually too large to consider
abprefer (donothing) ← ⊥
as a base case for human reasoning episodes. The solution presented
in this paper overcomes this problem in that we only have a small Comparing D1 and D1 ◦ d ◦ D2 , action A2 leads to at least
number of base cases depending on the number of options an agent one more dead person than action A1 . Hence, A1 is preferred
like Hank may consider. over A2 if nothing abnormal is known about A1 .
3
If the switch is broken, the trolley may derail. Such a scenario
4 E
can be modeled in WCSE as well, but it is beyond the scope of this The first two iterations of ΦP are shown in detail in the Ap-
paper to discuss it in detail. pendix.
→ m •• moral machine project (moralmachine.mit.edu) can
(initial state)
0 1 be automatically treated under WCSE. We also would like to
s
generalize the reasoning such that if an action does something
2 good and nothing abnormal is known, then it is permissible.
m •• This, however, requires a formalization of ‘something good’
(trolley moving to track 2) and very likely a formalization of ‘something bad’. And,
0
s ↓ we should have a closer look at counterfactuals and minimal
2 change.
Acknowledgements I’d like to thank Dominic Deck-
Figure 3: The bystander case (initial state) and its ramifications if
Hank is considering the counterfactual.
ert, Emmanuelle-Anna Dietz Saldanha, Sybille Schwarz, and
Lim Yohanes Stefanus for jointly developing the weak com-
pletion semantics with equality.
Under an utilitarian point of view [Bentham2009], the
change action is preferable to the donothing action as it will Appendix
kill fewer humans. On the other hand, we know that a purely Let P be the program developed in Section 3 and E be the
utilitarian view is not allowed in case of human casualties. set of equations specifying that ◦ is associative, commutative,
Hank may ask himself: Would I still save the humans on the and 1 being its unit element. Let I0 = h∅, ∅i be the empty in-
main track if there were no human on the side track and I terpretation. Suppose Hank has decided to do nothing. Then,
changed the switch? This is a counterfactual. But we can E
easily deal with it in WCSE by starting a new computation ΦP (I0 ) = I1 = hI1> , I1⊥ i,
with the additional fact where
causes(change, t0 ◦ c0 ◦ s ◦ h1 ◦ h1 ◦ c2 , 1) ← >. (4) I1> = { [causes(donothing, t0 ◦ c0 ◦ m ◦ h1 ◦ h1 ◦ h2 , 1)],
[action(t0 ◦ c0 ◦ m, 1, downhill , t1 ◦ c0 ◦ m, 1)],
Comparing (2) and (4), h2 has been replaced by c2 . There is [action(t0 ◦ c0 ◦ s, 1, downhill , t2 ◦ c0 ◦ s, 1)],
no human on track 2 anymore and, hence, this track is clear. [action(t1 ◦ h1 , 1, kill , t1 , d)],
This is a minimal change necessary to satisfy the precondition [action(t2 ◦ h2 , 1, kill , t2 , d)] },
of the counterfactual. In this case, the least E-model of the I1⊥ = { [ab(downhill )],
extended program will contain [ab(kill )] }.
[causes(change, t0 ◦ c0 ◦ s ◦ h1 ◦ h1 ◦ c2 , 1)]. Considering the body of (3) we find that both possible ground
instances of ab(A0 ), viz. ab(downhill ) and ab(kill ), are false
This case is illustrated in Figure 3. Using under I1 and, consequently, their negations are true under I1 .
permissible(change) ← The only ground instance of
prefer (change, donothing) ∧ causes(A, P1 ◦ Z1 , P2 ◦ Z2 ) (5)
causes(change, t2 ◦ c0 ◦ s ◦ h1 ◦ h1 ◦ c2 , 1) ∧ being true under I1 is
¬abpermissible (change)
causes(donothing, t0 ◦ c0 ◦ m ◦ h1 ◦ h1 ◦ h2 , 1). (6)
abpermissible (change) ← ⊥
Hence, we are searching for a ground instance of
allows Hank to conclude that changing the switch is permis- action(P1 , P2 , A0 , E1 , E2 )
sible within the doctrine of double effect [Aquinas1988].
being true under I1 such that the ground instance of P1 is con-
4 Discussion tained in t0 ◦c0 ◦m◦h1 ◦h1 ◦h2 and the ground instance of P2
is contained in 1. There are four candidates in I1 . The only
We have extended the WCS to WCSE and we have shown possible ground instance of an action meeting the conditions
how the bystander case can be modeled in the extended is
approach. We believe that the methods and techniques
action(t0 ◦ c0 ◦ m, 1, downhill , t1 ◦ c0 ◦ m, 1). (7)
can be applied to all ethical decision problems discussed
in [Pereira and Saptawijaya2016]. In [Dietz Saldanha et Comparing the second arguments of (5) and (6) with the first
al.2018] we have already considered the footbridge and the argument of (7) we find that
loop case. Moreover, we have applied the doctrine of triple P1 = t0 ◦ c0 ◦ m
effect [Kamm2006] to distinguish between direct and indirect and
intentional killings. Currently, we are working out the details Z1 = h1 ◦ h1 ◦ h2 .
for all problems. For us it is important that all these prob- Likewise, comparing the third arguments of (5) and (6) with
lems can be discussed within the presented framework and the second argument of (7) we find that P2 = 1 and Z2 = 1.
are compatable to our solutions for other human reasoning Combining Z1 with the fourth argument of (7) and, likewise,
tasks like the suppression and the selection task. combining Z2 with the fifth argument of (7) we learn that
On the other hand, there are many open questions. The ex-
amples discussed in this paper are hand-crafted and we would causes(donothing, t1 ◦ c0 ◦ m ◦ h1 ◦ h1 ◦ h2 , 1)
E
like to develop an extension, where examples taken from the must be true under ΦP (I1 ).
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