The weak completion semantics is a novel computational theory based on logic programs. It is extended to deal with equalities, which is a prerequisite to represent and reason about actions and causality as in the fluent calculus. This is discussed in the context of ethical decision making. In order to decide questions about the moral permissibility of actions, counterfactuals need to be considered. Somewhat surprisingly, this can be straightforwardly done in the extended approach.
The weak completion semantics (WCS) is a novel cognitive theory. Its original idea is based on [Stenning and van Lambalgen2008] who proposed to model human reasoning tasks by, firstly, reasoning towards a normal logic program to represent the reasoning task and, secondly, by reasoning with respect to the least model of the normal logic program. Unfortunately, Stenning and van Lambalgen’s approach contained a technical bug which was corrected in [Ho¨lldobler and Kencana Ramli2009].
The WCS is based on many techniques and methods from
logic programming and computational logic. However, these
techniques and methods are usually tweaked a little bit in
order to model human reasoning tasks adequately. For
example, programs are not completed in the sense of [Clark1978],
but only weakly completed. Instead of the semantic operator
introduced in [Fitting1985], a modified operator introduced
in [Stenning and van Lambalgen2008] is used. Instead of the
three-valued Kripke-Kleene logic used in [Fitting1985,
Stenning and van Lambalgen2008], the three-valued Łukasiewicz
logic [Łukasiewicz1920] is used. Because of the latter,
normal logic programs admit a least model and reasoning is
performed with respect to this model
The approach has been applied to various human reasoning tasks like the suppression task [Byrne1989, Dietz et al.2012], the selection task [Wason1968, Dietz et al.2013], and human syllogistic reasoning [Khemlani and JohnsonLaird2012, Oliviera da Costa et al.2017]. In fact, WCS performed better on the human syllogistic reasoning tasks than all 12 cognitive theories discussed in [Khemlani and JohnsonLaird2012]. As all human reasoning tasks are solved within one framework, the WCS is an integrated and computational cognitive theory. We are unaware of any other theory of this kind and with such a wide variety of applications.
Recently, ethical decision making has received much attention as autonomous agents become part of our daily life. In particular, we were inspired by [Pereira and Saptawijaya2016], who studied computational models of machine ethics. Various ethical problems are implemented as logic programs and these programs can be queried for moral permissibility. 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.
The problems studied in [Pereira and Saptawijaya2016] were trolley problems or variants thereof like the bystander case. In these problems, actions with direct and indirect effects must be considered. Hence, in order to model and reason about these problems within the WCS, the WCS must be extended to deal with actions and causality. We have chosen the fluent calculus [Ho¨lldobler and Schneeberger1990] for modeling actions and causality because it treats fluents as resources which can be consumed and produced. This property is shared with Petri networks [Ho¨lldobler and Jovan2014], the latter of which have already been used in computational models for human reasoning [Barrett2010].
In the fluent calculus [Ho¨lldobler and Schneeberger1990] states are represented as multisets of fluent. Multisets are represented with the help of a binary function symbol written infix and a constant 1 such that is commutative, associative, and 1 is its unit element. For example, the multisets f_ g_ and f_a; b; b g_ are represented by the fluent terms 1 and a b b, respectively. In order to deal with function symbols like in the WCS, we need to extend WCS to handle equality. Luckily, as shown in [Dietz Saldanha et al.2018] the key properties of the WCS, viz. the existence of a least model and the fact that this model can be computed as the least fixed point of an appropriate semantic operator, hold also for logic programs with equality.
In this paper, we will focus on the representation of the bystander case. We will show how to represent this problem in the extended approach. In particular, we formalize a purely utilitarian view [Bentham2009] and the doctrine of double effect [Aquinas1988]. In order to decide which action is morally permissible in the bystander case we need to reason about a counterfactual [Nickerson2015]. It turns out, that this can be straightforwardly done in the extended approach. 2
We assume the reader to be familiar with the WCS as
presented in [Ho¨lldobler2015, Dietz Saldanha et al.2017]. In
the weak completion semantics with equality (WCSE) a logic
program P is considered together with a set E of equations.
As shown in [Jaffar et al.1984], E defines a finest
congruence relation on the set of ground terms. Let [t] denote the
congruence class defined by the ground term t. For example,
[a b b] = [b a b] = [b b a 1]. Furthermore,
let [p(t1; : : : ; tn)] be an abbreviation for p([t1]; : : : ; [tn]),
where p is an n-ary relation symbol and all ti, 1 i n,
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
consider E -interpretations and E -models as usual
As shown in [Dietz Saldanha et al.2018], a logic program P together with a set E of equation has a least E -model under the three-valued Łukasiewicz logic [Łukasiewicz1920]. This model is the least fixed point of the following semantic operator: Let I be an E -interpretation. We define E (I) = P hJ >; J ?i where
J > J ? = = f[A] j there exists A Body 2 gP
and I(Body) = >g; f[A] j there exists A Body 2 gP and for all A0 Body 2 gP with [A] = [A0] we find I(Body) = ?g; and gP denotes the set of all ground instances of clauses occurring in P.
One should observe that the set E of equations is built into the computation of the E -operator: In the computation
P of J >, if a ground atom A is mapped to true because it is the 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 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 members of the congruence class containing A are mapped to false. 3
A trolley, whose conductor has fainted, is headed towards two people walking on the main track.1 The banks of the track are so steep that these two people will not be able to get off the track in time. Hank is standing next to a switch, which can turn the trolley onto a side track, thereby preventing it from killing the two people. However, there is a man standing on 1Note that in the original trolley problem, five people are on the main track. For the sake of simplicity, we assume that only two people are on the main track. (trolley moving to track 1) (trolley killing first human) (trolley killing second human) the side track. Hank can change the switch, killing him. Or he can refrain from doing so, letting the two die. Is it morally permissible for Hank to change the switch?
The case is illustrated in Figure 1 (initial state). The tracks are divided into segments 0, 1, and 2, the arrow represents that the trolley t is moving forward and that the track is clear (c), the switch is in position m (main) but can be changed into position s (side), and a bullet above a track segment represents a human (h) on this track. t; c, and h may be indexed to denote the track to which they apply. In addition, we need a fluent d denoting a dead human.
We choose to represent a state by a pair of multisets consisting of the casualties in its second element and all other fluents in its first element. Multisets are represented by socalled fluent terms in the fluent calculus, i.e., the initial state of the bystander case is the pair (t0 c0 m h1 h1 codes the multiset f_t0; c0; m; h1; h1; h2 g_. of fluent terms. The casualties are represented in the second element of (1) by the constant 1 encoding the empty multiset. Initially, there are no casualties, but casualties will play a special role when preferring one action over another as will be discussed later in this section. The first element of (1) en
There are two kinds of actions, the ones which can be performed by Hank (the direct actions donothing and change), and the actions which are performed by the trolley (the indirect actions downhill and kill ). We will represent the actions by the trolley explicitly with the help of a five-place relation symbol action specifying the preconditions, the name, and the immediate effects of an action. As a state is represented by two multisets, the preconditions anf the immediate effects have also two parts: action(t0 c0 action(t0 c0 m; 1; downhill ; t1 c0 s; 1; downhill ; t2 c0 m; 1) s; 1) > > action(t1 action(t2 > > If the trolley is on track 0, this track is clear, and the switch is in position m, then it will run downhill onto track 1 whereas track 0 remains clear and the switch will remain in position m; if, however, the switch is in position s, the trolley will 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 leading to a casualty.
The possible actions of Hank are the base cases in the definition of causality:2 causes(donothing ; t0 c0 m causes(change; t0 c0 s h1 It checks whether in a given state (P1 Z1; P2 Z2) an action A0 is applicable, which is the case if the preconditions (P1; P2) are contained in the given state. If this holds, then the action is executed leading to the successor state (E1 Z1; E2 Z2), where (E1; E2) are the direct effects of the action A0. In other words, if an action is applied, then its preconditions are consumed and its direct effects are produced. Such an action application is considered to be a ramification [Thielscher2003] with respect to the initial, direct action performed by Hank. Hence, the first argument A of causes is not changed. The execution of an action is also conditioned by :ab(A0), where ab is an abnormality predicate. Such abnormalities were introduced in [Stenning and van Lambalgen2008] to represent conditionals as licenses for inference. In this example, there is nothing abnormal known with respect to the actions downhill and kill and, consequently, the assumptions ab(downhill ) ? ab(kill ) ? are added to the program. But we can imagine situations, where the trolley will only cross the switch if the switch is not broken.3
2In the original version of the fluent calculus, causes is a ternary predicate stating that the execution of a plan transfers an initial into a goal state. Its base case is of the form causes(X; [ ]; X), i.e., the empty plans transforms arbitrary states X into X. Generating models bottom up using a semantic operator one has to consider all ground instances of this atom, which is usually too large to consider as a base case for human reasoning episodes. The solution presented in this paper overcomes this problem in that we only have a small number of base cases depending on the number of options an agent like Hank may consider.
3If the switch is broken, the trolley may derail. Such a scenario can be modeled in WCSE as well, but it is beyond the scope of this paper to discuss it in detail. 0 0 m m m s s s 1 2
Let P be the program consisting of the clauses mentioned in this section so far and E be the set of equations specifying that is associative, commutative, and 1 being its unit element. Hank has the choice to do nothing or to change the switch. Depending on his decision, the trolley will execute its actions which are computed as ramifications in the fluent calculus [Thielscher2003]. If Hank is doing nothing, then the least E -model of P – which is equal to the least fixed point of E – is computed by iterating E starting with the empty interpretation h;; ;i. The followingPequivalence classes will
P be mapped to true in subsequent iterations:4 [causes(donothing ; t0 c0 [causes(donothing ; t1 c0 [causes(donothing ; t1 c0 [causes(donothing ; t1 c0 m m m m h1 h1 h1 h2; d h1 h2; 1)] h1 h2; 1)] h2; d)] d)] They correspond precisely to the four states shown in Figure 1. No further action is applicable to the elements of the final congruence class. The two people on the main track will be killed.
On the other hand, if Hank is changing the switch, then the least fixed point of PE contains
[causes(change; t2 c0 s h1 h1; d)]: The two people on the main track will be saved but the person on the side track will be killed. This case is illustrated in Figure 2.
The two cases can be compared by means of a prefer clause: prefer (A1; A2) causes(A1; Z1; D1) ^ causes(A2; Z2; D1 d :abprefer (A1) abprefer (change) ? abprefer (donothing )
D2) ^ ? Comparing D1 and D1 d D2, action A2 leads to at least one more dead person than action A1. Hence, A1 is preferred over A2 if nothing abnormal is known about A1.
4The first two iterations of PE are shown in detail in the Appendix.
Under an utilitarian point of view [Bentham2009], the change action is preferable to the donothing action as it will kill fewer humans. On the other hand, we know that a purely utilitarian view is not allowed in case of human casualties. Hank may ask himself: Would I still save the humans on the main track if there were no human on the side track and I changed the switch? This is a counterfactual. But we can easily deal with it in WCSE by starting a new computation with the additional fact causes(change; t0 c0 s h1 h1 c2; 1) >: (4) Comparing (2) and (4), h2 has been replaced by c2. There is no human on track 2 anymore and, hence, this track is clear. This is a minimal change necessary to satisfy the precondition of the counterfactual. In this case, the least E -model of the extended program will contain [causes(change; t0 c0 s h1 h1 c2; 1)]: This case is illustrated in Figure 3. Using permissible(change) prefer (change; donothing ) ^ causes(change; t2 c0 s h1 :abpermissible (change) abpermissible (change) ? allows Hank to conclude that changing the switch is permissible within the doctrine of double effect [Aquinas1988]. 4
We have extended the WCS to WCSE and we have shown how the bystander case can be modeled in the extended approach. We believe that the methods and techniques can be applied to all ethical decision problems discussed in [Pereira and Saptawijaya2016]. In [Dietz Saldanha et al.2018] we have already considered the footbridge and the loop case. Moreover, we have applied the doctrine of triple effect [Kamm2006] to distinguish between direct and indirect intentional killings. Currently, we are working out the details for all problems. For us it is important that all these problems can be discussed within the presented framework and are compatable to our solutions for other human reasoning tasks like the suppression and the selection task.
On the other hand, there are many open questions. The examples discussed in this paper are hand-crafted and we would like to develop an extension, where examples taken from the h1 c2; 1) ^ being true under I1 is moral machine project (moralmachine.mit.edu) can be automatically treated under WCSE. We also would like to generalize the reasoning such that if an action does something good and nothing abnormal is known, then it is permissible. This, however, requires a formalization of ‘something good’ and very likely a formalization of ‘something bad’. And, we should have a closer look at counterfactuals and minimal change.
Acknowledgements I’d like to thank Dominic Deckert, Emmanuelle-Anna Dietz Saldanha, Sybille Schwarz, and Lim Yohanes Stefanus for jointly developing the weak completion semantics with equality.
Let P be the program developed in Section 3 and E be the set of equations specifying that is associative, commutative, and 1 being its unit element. Let I0 = h;; ;i be the empty interpretation. Suppose Hank has decided to do nothing. Then,
PE (I0) = I1 = hI1>; I1?i; where I1> = f [causes(donothing ; t0 c0 m h1 h1 h2; 1)]; [action(t0 c0 m; 1; downhill ; t1 c0 m; 1)]; [action(t0 c0 s; 1; downhill ; t2 c0 s; 1)]; [action(t1 h1; 1; kill ; t1; d)]; [action(t2 h2; 1; kill ; t2; d)] g; I1? = f [ab(downhill )];
[ab(kill )] g: Considering the body of (3) we find that both possible ground instances of ab(A0), viz. ab(downhill ) and ab(kill ), are false under I1 and, consequently, their negations are true under I1. The only ground instance of causes(A; P1
Z1; P2
Z2) causes(donothing ; t0 c0 m h1 h1 h2; 1): Hence, we are searching for a ground instance of action(P1; P2; A0; E1; E2) being true under I1 such that the ground instance of P1 is contained 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 possible ground instance of an action meeting the conditions is action(t0 c0 m; 1; downhill ; t1 c0 m; 1): (7) Comparing the second arguments of (5) and (6) with the first argument of (7) we find that
P1 = t0 c0 m and
Z1 = h1 h1 h2: Likewise, comparing the third arguments of (5) and (6) with the second argument of (7) we find that P2 = 1 and Z2 = 1. Combining Z1 with the fourth argument of (7) and, likewise, combining Z2 with the fifth argument of (7) we learn that causes(donothing ; t1 c0 m h1 h1 must be true under PE (I1). (5) (6)