Weak Completion Semantics and its Applications in Human Reasoning

Abstract

I present a logic programming approach based on the weak completions semantics to model human reasoning tasks, and apply the approach to model the suppression task, the selection task as well as the belief-bias e↵ ect, to compute preferred mental models of spatial reasoning tasks and to evaluate indicative as well as counterfactual conditionals.

Introduction

Observing the performance of humans in cognitive tasks like the suppression [ 3 ] or the selection task [ 31 ] it is apparent that human reasoning cannot be adequately modeled by classical two-valued logic. Whereas there have been many approaches to develop a normative model for human reasoning which are not based on logic like the mental model theory [ 22 ] or probabilistic approaches [ 15 ], Keith Stenning and Michiel von Lambalgen have developed a logic-based approach [ 30 ] where, in a first step, they reason towards an appropriate representation of some aspects of the world as logic program and, in a second step, reason with respect to the least model of the program. Their approach is based on the three-valued (strong) Kripke-Kleene logic [ 23 ], is non-monotonic, and utilizes some form of completion as well as abduction. Most interestingly, the results developed within the fields of logic programming and computational logic within the last decades could not be immediately applied to adequately model human reasoning tasks but rather some modifications were needed. As a consequence, theorems, propositions and lemmas formally proven for a theory without these modification cannot be readily applied but their proofs must be adapted as well.

Unfortunately, some of the formal results stated in [ 30 ] are not correct. Somewhat surprisingly, we were able to show in [ 19 ] that the results do hold if the Kripke-Kleene logic is replaced by the three-valued Lukasiewicz logic [ 25 ]. We have called our approach weak completion semantics (WCS) because in the completion of a program, undefined relations are not identified with falsehood but rather are left unknown. Whereas our original emphasis was on obtaining formally correct results, WCS has been applied to many di↵ erent human reasoning tasks in the meantime: the suppression task, the abstract as well as the social selection task, the belief-bias e↵ ect, the computation of preferred mental models in spational reasoning tasks as well as the evaluation of conditionals.

This paper gives an overview on WCS as well as its applications to human reasoning tasks.

Logic Programs

Weak Completion Semantics

We assume the reader to be familiar with logic programming, but we repeat basic notions and notations. A (logic) program is a finite set of (program) clauses of the form A > , A ? or A B1 ^ . . . ^ Bn, n > 0 where A is an atom, Bi, 1  i  n, are literals and > and ? denote truth and falsehood, resp. A is called head and >, ? as well as B1 ^ . . .^ Bn are called body of the corresponding clause. Clauses of the form A > and A ? 1 are called positive and negative facts, resp. In this paper, P denotes a program, A a ground atom and F a formula. We assume that each non-propositional program contains at least one constant symbol. We also assume for each program that the underlying alphabet consists precisely of the symbols mentioned in the program, if not indicated di↵ erently. When writing sets of literals we omit curly brackets if a set has only one element.

gP denotes the set of all ground instances of clauses occurring in P. A ground atom A is defined in gP i↵ gP contains a clause whose head is A; otherwise A is said to be undefined. def (S, P) = {A body 2 gP | A 2 S _ ¬A 2 S} is called definition of S in P, where S is a set of ground literals. Such a set S is said to be consistent i↵ it does not contain a pair of complementary literals.

A level mapping for P is a function ` which assigns to each atom occurring in gP a natural number. Let `(¬A) = `(A). P is acyclic i↵ there exists a level mapping ` such that for each A L1 ^ . . . ^ Ln 2 gP we find that `(A) > `(Li), 1  i  n. 2.2

Weak Completion For a given P, consider the following transformation: (1) For each defined atom A, replace all clauses of the form A body1, . . . , A bodym occurring in gP by A body1 _ . . . _ bodym. (2) Replace all occurrences of by $ . The obtained ground program is called weak completion of P or wcP.2 2.3

Lukasiewicz Logic An interpretation is a mapping from the set of formulas into the set of truth values. A model for F is an interpretation which maps F to true. We consider the three-valued Lukasiewicz (or L-) logic [ 25 ] (see Table 1) and represent each interpretation I by hI>, I? i, where I> = {A | I(A) = >}, I? = {A | I(A) = ? }, I> \ I? = ; , and each ground atom A 62 I> [ I? is mapped to U. Hence, under the empty interpretation h; , ;i all ground atoms are unknown. Let hI>, I? i and hJ >, J ? i be two interpretations. We define hI>, I? i ✓ h J >, J ? i i↵ I> ✓ J > and I? ✓ J ? , hI>, I? i [ h J >, J ? i = hI> [ J >, I? [ J ? i. 1 Under WCS a clause of the form A

the only clause in the definition of A. 2 Note that undefined atoms are not identified with ? as in the completion of P [ 5 ].

? is turned into A $ ? provided that it is F ¬F ^ > U ? _ > U ? > U ? $ > U ? > ? > > U ? > > > > > > > > > > U ? ? > U U U ? U > U U U U > > U U > U

U U ? ? ? ? ? > U ? ? ? U > ? ? U > Table 1. Truth tables for the L-semantics, where we have used >, ? and U instead of true, false and unknown, resp., in order to shorten the presentation.

Theorem 1. (Model Intersection Property) For each program P, the intersection of all L-models of P is an L-model of P.

This result was formally proven in [ 19 ] for programs not containing negative facts, but it holds also for programs with negative facts. 2.4

A Semantic Operator The following operator was introduced by Stenning and van Lambalgen [ 30 ], where they also showed that it admits a least fixed proint: P (hI>, I? i) = hJ >, J ? i, where

J > = {A | A body 2 gP and body is true under hI>, I? i}, J ? = {A | def (A, P) 6= ; and body is false under hI>, I? i for all A body 2 def (A, P)}.

The P operator di↵ ers from the semantic operator defined by Fitting in [ 13 ] in the additional condition def (A, P) 6= ; required in the definition of J ? . This condition states that A must be defined in order to be mapped to false, whereas in the (strong) Kripke-Kleene-semantics considered by Fitting an atom is mapped to false if it is undefined. This reflects precisely the di↵ erence between the weak completion and the completion semantics. The (strong) Kripke-Kleene-semantics was also applied in [ 30 ]. However, as shown in [ 19 ] this semantics is not only the cause for a technical bug in one theorem of [ 30 ], but it does also lead to a nonadequate model of some human reasoning tasks. Both, the technical bug as well as the non-adequate modeling, can be avoided by using WCS.

Theorem 2. The least fixed point of

pletion of P. [ 19 ]

P is the least L-model of the weak com

In the remainder of this paper, MP denotes the least L-model of wcP. 2.5

Contraction It was Fitting’s idea [ 14 ] to apply metric methods to compute least fixed points of semantic operators and, in particular, he showed that for so-called acceptable3 3 Please see [ 14 ] for a definition of acceptable programs. The class of acyclic programs is a proper subset of the class of acceptable programs. programs the semantic operator defined in [ 13 ] is a contraction.4 Consequently, Banach’s contraction mapping theorem [ 2 ] can be applied to compute the least fixed point of the semantic operator.

As shown in [ 18 ], P may not be a contraction if P is acceptable. But the following weaker result holds for programs not containing any cycles.

Theorem 3. If P is an acyclic program, then

P is a contraction. [18]

As a consequence, the computation of the least fixed point of initialized with an arbitrary interpretation.

P can be 2.6

A Connectionist Realization Within the core-method [ 1, 17 ] semantic operators of logic programs are computed by feed-forward connectionist networks, where the input and the output layer represent interpretations. By connecting the output with the input layer, the networks are turned into recurrent ones and can now be applied to compute the least fixed points of the semantic operators.

Theorem 4. For each datalog program P there exists a recurrent connectionist

network which will converge to a stable state representing MP if initialized with the empty interpretation.

The theorem was proven in [ 20 ] for propositional programs but extends to datalog programs. From the discussion in the previous paragraph we conclude that the network may be initialized by some interpretation if P is a contraction. 2.7

Weak Completion Semantics The weak completion semantics (WCS) is the approach to consider weakly completed logic programs and to reason with respect to the least L-models of these programs. We write P |=wcs F i↵ formula F holds in MP . WCS is non-monotonic. 2.8

Relation to Well-Founded Semantics WCS is related to the well-founded semantics (WFS) as follows: Let P+ = P \ {A ? | A ? 2 P} and u be a new nullary relation symbol not occurring in P. Furthermore, let P⇤ = P+ [ {B u | def (B, P) = ; } [ {u ¬u}.

Theorem 5. If P is a program which does not contain a positive loop, then MP and the well-founded model for P⇤ coincide. [11]

4 A mapping f : M ! M on a metric space (M, d) is a contraction i↵ there exists a k 2 (0, 1) such that for all x, y 2 M we find d(f (x), f (y))  k ⇥ d(x, y).

Abduction An abductive framework consists of a logic program P, a set of abducibles AP = {A > | def (A, P) = ; } [ {A ? | def (A, P) = ; }, a set of integrity constraints IC, i.e., expressions of the form ? B1^ . . .^ Bn, and the entailment relation |=wcs ; it is denoted by hP, AP , IC, |=wcs i.

By Theorem 1, each program and, in particular, each finite set of positive and negative ground facts has an L-model. For the latter, this can be obtained by mapping all heads occurring in this set to true. Thus, in the following definition, explanations as well as the union of a program and an explanation are satisfiable.

An observation O is a set of ground literals; it is explainable in the framework hP, AP , IC, |=wcs i i↵ there exists a minimal E ✓ AP called explanation such that MP[ E satisfies IC and P [ E |=wcs L for each L 2 O. F follows creduluously from P and O i↵ there exists an explanantion E such that P [ E |=wcs F . F follows skeptically from P and O i↵ for all explanantions E we find P [ E |=wcs F . 2.10

Revision 3 3.1 3. Mrev(P,S)(S) = >.

Applications

The Suppression Task Let S be a finite and consistent set of ground literals in rev (P, S) = (P \ def (S, P)) [ {A > | A 2 S} [ {A ? | ¬A 2 S}, where A denotes an atom. rev (P, S) is called the revision of P with respect to S. The following result was formally proven in [ 7 ].

Proposition 6. 1. rev is non-monotonic,

i.e., there exist P, S and F such that P |=wcs F and rev (P, S) 6|=wcs F . 2. If MP (L) = U for all L 2 S, then rev is monotonic.

Ruth Byrne has shown in [ 3 ] that graduate students with no previous exposure to formal logic did suppress previously drawn conclusions when additional information became available. Table 2 shows the abbreviations that will be used in this subsection, whereas Table 3 gives an account of the findings of [ 3 ]. Interestingly, in some instances the previously drawn conclusions were valid (cases AE and ACE in Table 3) whereas in other instances the conclusions were invalid (cases AL and ABL in Table 3) with respect to classical two-valued logic.

Following [ 30 ] conditionals are encoded by licences for implications using abnormality predicates. In the case AE no abnormalities concerning the library are known. However, in the case ACE it becomes known that one can visit the library only if it is open and, thus, not being open becomes an abnormality for the first implication. Likewise, one may argue that there must be a reason for studying in the library. In the case ACE the only reason for studying in A If she has an essay to finish, then she will study late in the library. B If she has a textbook to read, then she will study late in the library. C If the library stays open, she will study late in the library.

E She has an essay to finish.

E She does not have an essay to finish.

L She will study late in the library.

L She will not study late in the library. the library is to finish an essay and, consequently, not having to finish an essay becomes an abnormality for the second implication. Alltogether, for the cases AE and ACE we obtain the programs

PAE PACE = {` = {` e ^ ¬ab1, e e ^ ¬ab1, e > , ab1 ? }, > , ab1 ¬o, ` o ^ ¬ab2, ab2 ¬e} with MPAE = h{e, `}, {ab1}i and MPACE = h{e}, {ab2}i, where `, e, o and ab denote that she will study late in the library, she has an essay to finish, the library stays open and abnormality, resp. Hence, MPAE (`) = > and MPACE (`) = U. Thus, WCS can model the suppression of a previously drawn conclusion.

For the examples in the second column of Table 3 abduction is needed. E.g., for the case ABL we obtain the program

PAB = {` e ^ ¬ab1, ab1 ? , ` t ^ ¬ab3, ab3 ? } with MPAB = h; , {ab1, ab3}i, where t denotes that she has a textbook to read. The observation O = ` can be explained by E1 = {e > } and E2 = {t > }. In order to adequately model Byrne’s selection task, we have to be skeptical as otherwise–being credoluous–we would conclude that she has an essay to finish.

A complete account of Byrne’s selection task under WCS is given in [ 10, 21 ]. D F 3 7 89% 16% 62% 25% beer coke 22yrs 95% 0.025% 0.025% 16yrs 80% In the original (abstract) selection task [ 31 ] participants were given the conditional if there is a D on one side of the card, then there is 3 on the other side and four cards on a table showing the letters D and F as well as the numbers 3 and 7. Furthermore, they know that each card has a letter on one side and a number on the other side. Which cards must be turned to prove that the conditional holds?

Griggs and Cox [ 16 ] adapted the abstract task to a social case. Consider the conditional if a person is drinking beer, then the person must be over 19 years of age and again consider four cards, where one side shows the person’s age and on the other side shows the person’s drink: beer, coke, 22yrs and 16yrs. Which drinks and persons must be checked to prove that the conditional holds?

When confronted with both tasks, participants reacted quite di↵ erently as shown in Table 4. Moreover, if the conditionals are modeled as implications in classical two-valued logic, then some of the drawn conclusions are not valid. The Abstract Case This case is artificial and there is no common sense knowledge about the conditional. Let D, F , 3, and 7 be propositional variables denoting that the corresponding symbol or number is on one side of a card. Following [ 24 ], we assume that the given conditional is viewed as a belief and represented as a clause in

Pac = {3

D ^ ¬ab1, ab1 ? }, where the negative fact was added as there are no known abnormalities. We obtain MPac = h; , ab1i and find that this model does not explain any symbol on the cards. Let Aac = {D > , D ? , F > , F ? , 7 > , 7 ? } in the abductive framework hPac, Aac, ; , |=wcs i. Table 5 shows the explanations for the cards with respect to this framework.

In case D was observed, the least model maps also 3 to >. In order to be sure that this corresponds to the real situation, we need to check if 3 is true. case beer coke 22yrs 16yrs

Psc {ab2 ? , b > } {ab2 ? , b ? } {ab2 ? , o > } {ab2 ? , o ? }

MPsc hb, ab2i h; , {b, ab2}i

ho, ab2i h; , {o, ab2}i |=wcs o

b ^ ¬ab2 no yes yes no turn yes no no yes Therefore, the card showing D is turned. Likewise, in case 3 is observed, D is also mapped to >, which can only be confirmed if the card is turned. The Social Case In this case most humans are quite familiar with the conditional as it is a standard law. They are also aware–it is common sense knowledge– that there are no exceptions or abnormalities. Let o represent a person being older than 19 years and b a person drinking beer. The conditional can be represented by o b ^ ¬ab2 and is viewed as a social constraint which must follow logically from the given facts. Table 6 shows the four di↵ erent cases.

One should observe that in the case 16yrs the least model of the weak completion of Psc, i.e. h; , {o, ab2}i, assigns U to b and, consequently, to both, b ^ ¬ab2 and o b ^ ¬ab2, as well. Overall, in the cases beer and 16yrs the social constraint is not entailed by the least L-model of the weak completion of the program. Hence, we need to check these cases out and, hopefully, find that the beer drinker is older than 19 and that the 16 years old is not drinking beer.

A complete account of the selection task under WCS is given in [ 6 ]. 3.3

The Belief-Bias E↵ ect Evans et. al. [ 12 ] made a psychological study showing possibly conflicting processes in human reasoning. Participants were confronted with syllogisms and had to decide whether they are logically valid. Consider the following syllogism:

No addictive things are inexpensive.

Some cigarettes are inexpensive.

Therefore, some addictive things are not cigarettes.

(Premise1) (Premise2) (Conclusion) The conclusion does not follow from the premises in classical logic: If there are inexpensive cigarettes but no addictive things, then the premises are true, but the conclusion is false. Nevertheless, most participants considered the syllogism to be valid. Evans et. al. explained the answers by an unduly influence of the participants’ own beliefs.

Before we can model this line of reasoning under WCS, we need to tackle the problem that the head of a program clause must be an atom, whereas the conclusion of the rule if something is inexpensive, then it is not addictive5 is a 5 (Premise1) can be formalized in many syntactically di↵ erent, but semantically equivalent ways in classical logic. We have selected a form which allows WCS to adequately model the belief-bias e↵ ect. negated atom. If the relation symbol add is used to denote addiction, then this technical problem can be overcome by introducing a new relation symbol add 0, specifying by means of the clause add (X) ¬add 0(X) that add 0 is the negation of add under WCS and requiring by means of the integrity constraint

ICadd = {?

add (X) ^ ¬add 0(X)} that add and add 0 cannot be simultaneously true.

We can now encode (Premise1) following Stenning and van Lambalgen’s idea to represent conditionals by licences for implications [ 30 ]: (1) (2) (3) (4) (5) (Bias1) (Bias2) add 0(X) inex (X) ^ ¬ab1(X), ab1(X) ? .

As for (Premise2), Evans et. al. have argued that it includes two pieces of information. Firstly, there exists something, say a, which is a cigarette: cig(a)

> .

Cigarettes are addictive,

Secondly, it contains the following belief that humans seem to have:

Cigarettes are inexpensive.

This belief implies (Premise2) and biases the process of reasoning towards a representation such that we obtain: inex (X) cig(X) ^ ¬ab2(X), ab2(X) ? .

Additionally, it is assumed that there is a second piece of background knowledge, viz. it is commonly known that which in the context of (1) and (2) can be specified by stating that cigarettes are abnormalities regarding add 0: ab1(X) cig(X).

Alltogether, let Padd be the program consisting of the clauses (1)-(5). Because (Conclusion) is about an object which is not necessarily a we need to add another constant, say b, to the alphabet underlying Padd . We obtain

MPadd = h{cig(a), inex (a), ab1(a), add (a)}, {ab2(a), ab2(b), add 0(a)}i.

Turning to (Conclusion) we consider its first part as the observation O = add (b) which needs to be explained with respect to the abductive framework hPadd , {cig(b)

> , cig(b) ? }, ICadd , |=wcs i. We find two minimal explanations E? = {cig(b) leading to the minimal models ? } and E> = {cig(b) > } MPadd [ E? = h {cig(a), inex (a), ab1(a), add (a), add (b)},

{ab2(a), ab2(b), add 0(a), cig(b), inex (b), ab1(b), add 0(b)} i, MPadd [ E> = h {cig(a), inex (a), ab1(a), add (a), cig(b), inex (b), ab1(b), add (b)}, {ab2(a), ab2(b), add 0(a), add 0(b)} i, respectively. Because under MPadd [ E> all known addictive objects (a and b) are cigarettes and under MPadd [ E? the addictive object b is not a cigarette, (Conclusion) follows creduluously, but not skeptically.

On the other hand, the two explanations E? and E> do not seem to be equally likely given (Premise1) and (Bias1). Rather, E? seems to be the main explanation whereas E> seems to be the exceptional case. Pereira and Pinto [ 26 ] have introduced so-called inspection points which allow to distinguish between main and exceptional explanations in an abductive framework. Formally, they introduce a meta-predicate inspect and require that if inspect (A) > or inspect (A) ? are elements of an explanation E for some literal or observation L, then either A > or A ? must be in E as well and, moreover, A ? or A > must be elements of explanations for some literal or observation L0 6= L, where A is a ground atom.

With the help of inspection points, the program Padd can be rewritten to Pa0dd = (Padd \ {ab1(X) cig(X)}) [ {ab1(X) inspect (cig(X))} framework hPa0dd , A0add , ICadd , |=wcs i, where and the explanation O = add (b) is to be explained with respect to the abductive A0add = { cig(b) > , cig(b) ? , inspect (cig(b)) > , inspect (cig(b)) ? , inspect (cig(a)) > , inspect (cig(a)) ? }.

Now, E? is the only explanation for add (b) and, hence, (Conclusion) follows skeptically in the revised approach.

More details about our model of the belief-bias e↵ ect and abduction using inspection points can be found in [ 27, 28 ]. 3.4

Spatial Reasoning Consider the following spatial reasoning problem. Suppose it is known that a ferrari is left of a porsche, a beetle is right of the porsche, the porsche is left of a hummer, and the hummer is left of a dodge. Is the beetle left of the hummer?

The mental model theory [ 22 ] is based on the idea that humans construct socalled mental models, which in case of a spatial reasoning problem is understood as the presentation of the spatial arrangements between objects that correspond to the premises. In the example, there are three mental models: ferrari porsche beetle hummer dodge ferrari porsche hummer beetle dodge ferrari porsche hummer dodge beetle Hence, the answer to the above mentioned question depends on the construction of the mental models.

In the preferred model theory [ 29 ] it is assumed that humans do not construct all mental models, but rather a single, preferred one, and that reasoning is performed with respect to the preferred mental model. The preferred mental model is believed to be constructed by considering the premises one by one in the order of their occurrence and to place objects directly next to each other or, if this impossible, in the next available space. For the example, the preferred mental model is constructed as follows: ferrari porsche ferrari porsche beetle ferrari porsche beetle hummer ferrari porsche beetle hummer dodge Hence, according to the preferred model theory, the beetle is left of the hummer.

In [ 8 ] we have specified a logic program P taking into account the premises of a spatial reasoning problem such that MP corresponds to the preferred mental model. Moreover, within the computation of MP as the least fixed point of P , the preferred mental model is constructed step by step as in [ 29 ]. 3.5

Conditionals Conditionals are statements of the form if condition then consequence. In this paper we distinguish between indicative and subjunctive (or counterfactual) conditionals. Indicative conditionals are conditionals whose condition is either true or unknown; the consequence is asserted to be true if the condition is true. On the contrary, the condition of a subjunctive or counterfactual conditional is either false or unknown; in the counterfactual circumstance of the condition being true, the consequence is asserted to be true.6 We assume that the condition and the consequence of a conditional are finite and consistent sets of literals.

Conditionals are evaluated with respect to some background information specified as a program and a set of integrity constraints. More specifically, as the weak completion of each program admits a least L-model, conditionals are evaluated under the least L-model of a program. In the reminder of this section let P be a program, IC be a finite set of integrity constraints, and MP be the least L-model of wcP such that MP satisfies IC. 6 In the literature the case of a condition being unknown is usually not explicitely considered; there also seems to be no standard definition for indicative and counterfactual conditionals.

In this setting we propose to evaluate a conditional cond (C, D) as follows, where C and D are finite and consistent sets of literals: 1. If MP (C) = > and MP (D) = >, then cond (C, D) is true. 2. If MP (C) = > and MP (D) = ? , then cond (C, D) is false. 3. If MP (C) = > and MP (D) = U, then cond (C, D) is unknown. 4. If MP (C) = ? , then evaluate cond (C, D) with respect to Mrev(P,S), where S = {L 2 C | MP (L) = ? }. 5. If MP (C) = U, then evaluate cond (C, D) with respect to MP0 , where – P0 = rev (P, S) [ E, – S is a smallest subset of C and E ✓ Arev(P,S) is a minimal explanation for C \ S such that MP0 (C) = >.

In words, if the condition of a conditional is true, then the conditional is an indicative one and is evaluated as implication in L-logic. If the condition is false, then the conditional is a counterfactual conditional. In this case, i.e., in case 4, non-monotonic revision is applied to the program in order to reverse the truth value of those literals, which are mapped to false.

The main novel contribution concerns the final case 5. If the condition C of a conditional is unknown, then we propose to split C into two disjoint subsets S and C \ S, where the former is treated by revision and the latter by abduction. In case C contains some literals which are true and some which are unknown under MP , then the former will be part of C \ S because the empty explanation explains them. As we assume S to be minimal this approach is called minimal revision followed by abduction (MRFA). Furthermore, because revision as well as abduction are only applied to literals which are assigned to unknown, case 5 is monotonic.

As an example consider the forest fire scenario taken from [ 4 ]: The conditional cond (¬dl , ¬↵ ), if there had not been so many dry leaves on the forest floor, then the forest fire would not have occurred, is to be evaluated with respect to P↵ = {↵ l ^ ¬ab1, l > , ab1 ¬dl , dl > }, which states that lightning (l ) causes a forest fire (↵ ) if nothing abnormal (ab1), is taking place, lightning happened, the absence of dry leaves (dl ) is an abnormality, and dry leaves are present. We obtain MP↵ = h{dl , l , ↵ }, {ab1}i and find that the condition ¬dl is false. Hence, we are dealing with a counterfactual conditional. Following Step 4 we obtain S = {¬dl }, rev (P↵ , ¬dl ) = {↵ l ^ ¬ab1, l > , ab1 ¬dl , dl ? } and Mrev(P↵ ,¬dl) = h{l , ab1}, {dl , ↵ }i. Because ↵ is mapped to false under this model, the conditional is true.

Let us extend the example by adding arson (a) causes a forest fire: P↵ a = P↵ [ {↵

a ^ ¬ab2, ab2 ? }.

We find MP↵ a = h{dl , l , ↵ }, {ab1, ab2}i and Mrev(P↵ a ,¬dl) = h{l , ab1}, {dl , ab2}i. Under this model ↵ is unknown and, consequently, cond (¬dl , ¬↵ ) is unknown as well.

As final example consider P↵ a and the conditional cond ({↵ , ¬dl }, a): if a forest fire occurred and there had not been so many dry leaves on the forest floor, then arson must have caused the fire. Because the condition {↵ , ¬dl } is false under MP↵ a we follow Step 4 and obtain S = {¬dl }, rev (P↵ a , ¬dl ) = (P↵ a \ {dl > }) [ {dl ? } and Mrev(P↵ a ,¬dl) = h{l , ab1}, {dl , ab2}i. One should observe that ↵ as well as the condition {↵ , ¬dl } are unknown under this model. Hence, we follow Step 5, consider the abductive framework hrev (P↵ a , ¬dl ), {a > , a ? }, ; , |=wcs i and learn that {↵ , ¬dl } can be explained by {a > }. Hence, by MRFA we obtain as final program rev (P↵ a , ¬dl ) [ {a > } and find

Mrev(P↵ a ,¬dl)[ {a > } = h{l , ab1, ↵ , a}, {dl , ab2}i.

Because a is mapped to true under this model, the conditional is true as well.

More details about the evaluation of conditionals under WCS can be found in [ 7, 9 ]. 4

Conclusion

I have presented the weak completion semantics (WCS) and have demonstrated how various human reasoning tasks can be adequately modeled under WCS. To the best of my knowledge, WCS is the computational logic based approach which can handle most human reasoning tasks within a single framework. For example, [ 30 ] discusses only the selection task in detail and mentions the selection task, whereas [ 24 ] discusses the selection task in detail and mentions the suppression task.

But there are many open questions. I only claim that conditionals are adequately evaluated as shown in Section 3.5; this claim must be thoroughly tested. We may also consider scenarios, where abduction needs to be applied to satisfy the consequent of a conditional. The connectionist model reported in 2.6 does not yet include abduction and we are unaware of any connectionist realization of sceptical abduction.

Acknowledgements I like to thank Michiel van Lambalgen for the discussions

at the ICCL summer school 2008 which initialized this research. Carroline Dewi Puspa Kencana Ramli wrote an outstandings master’s thesis in which she developed the formal framework of the WCS including the connectionist realization; she has received the EMCL best master’s thesis award 2009. The relationship between WCS and WFS was established jointly with Emmanuelle-Anna Dietz and Christoph Wernhard. Abduction was added to the framework with the help of Emma, Christoph and Tobias Philipp. The ideas underlying the revision operator were developed jointly with Emma and Lu´ıs Moniz Pereira.

The suppression task was the running example throughout the development of WCS involving Carroline, Tobias, Christoph, Emma and Marco Ragni. The solution for the selection task was developed with Emma and Marco. The approach to model spatial reasoning problems is a revised version of the ideas first developed by Raphael H¨ops in his bachelor thesis under the supervision of Emma; many thanks to Marco who introduced us to this problem. Emma and Lu´ıs proposed the solution for the belief bias e↵ ect. The procedure to evaluate conditionals is the result of many discussions with Emma, Lu´ıs and Bob Kowalski. Finally, I like to thank the referees of the paper for many helpful comments.

References

1. S. Bader and S. Ho¨lldobler. The Core method: Connectionist model generation . In S. Kollias, A. Stafylopatis , W. Duch , and E. Ojaet, editors, Proceedings of the 16th International Conference on Artificial Neural Networks (ICANN) , volume 4132 of Lecture Notes in Computer Science, pages 1 - 13 . Springer-Verlag, 2006 .

2. S. Banach . Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales . Fund. Math., 3 : 133 - 181 , 1922 .

3. R. Byrne . Suppressing valid inferences with conditionals . Cognition , 31 : 61 - 83 , 1989 .

4. R. M. J. Byrne . The Rational Imagination: How People Create Alternatives to Reality . MIT Press, Cambridge, MA, USA, 2007 .

5. K. Clark . Negation as failure . In H. Gallaire and J. Minker, editors, Logic and Databases , pages 293 - 322 . Plenum, New York, 1978 .

6. E.-A. Dietz , S. Ho¨lldobler, and M. Ragni . A computational logic approach to the abstract and the social case of the selection task . In Proceedings Eleventh International Symposium on Logical Formalizations of Commonsense Reasoning , 2013 . commonsensereasoning.org/2013/proceedings.html.

7. E.-A. Dietz and S. H¨olldobler. A new computational logic approach to reason with conditionals . In F. Calimeri, G. Ianni, and M. Truszczynski, editors, Logic Programming and Nonmonotonic Reasoning , 13th International Conference, LPNMR, volume 9345 of Lecture Notes in Artificial Intelligence . Springer, 2015 .

8. E.-A. Dietz , S. Ho¨lldobler, and R. Ho ¨ps. A computational logic approach to human spatial reasoning . Technical Report KRR-2015-02 , TU Dresden, International Center for Computational Logic , 2015 .

9. E.-A. Dietz , S. Ho¨lldobler, and L. M. Pereira . On indicative conditionals . In S. H¨olldobler and Y. Liang, editors, Proceedings of the First International Workshop on Semantic Technologies , volume 1339 of CEUR Workshop Proceedings , pages 19 - 30 . CEUR-WS.org, 2015 . http://ceur-ws. org/ Vol- 1339 /.

10. E.-A. Dietz, S. Ho¨lldobler, and M. Ragni . A computational logic approach to the suppression task . In N. Miyake, D. Peebles , and R. P . Cooper, editors, Proceedings of the 34th Annual Conference of the Cognitive Science Society , pages 1500 - 1505 . Cognitive Science Society , 2012 .

11. E.-A. Dietz, S. Ho¨lldobler, and C. Wernhard . Modelling the suppression task under weak completion and well-founded semantics . Journal of Applied Non-Classical Logics , 24 : 61 - 85 , 2014 .

12. J. Evans , J. Barston , and P. Pollard . On the conflict between logic and belief in syllogistic reasoning . Memory & Cognition , 11 : 295 - 306 , 1983 .

13. M. Fitting . A Kripke-Kleene semantics for logic programs . Journal of Logic Programming , 2 ( 4 ): 295 - 312 , 1985 .

14. M. Fitting . Metric methods - three examples and a theorem . Journal of Logic Programming , 21 ( 3 ): 113 - 127 , 1994 .

15. G. Gigerenzer and D. Murray . Cognition ad Intuitive Statistics. Hillsdale, NJ: Erlbaum, 1987 .

16. R. Griggs and J. Cox . The elusive thematic materials e↵ ect in the wason selection task . British Journal of Psychology , 73 : 407 - 420 , 1982 .

17. S. Ho¨lldobler and Y. Kalinke . Towards a new massively parallel computational model for logic programming . In Proceedings of the ECAI94 Workshop on Combining Symbolic and Connectionist Processing , pages 68 - 77 . ECCAI, 1994 .

18. S. Ho¨lldobler and C. D. P. Kencana Ramli . Contraction properties of a semantic operator for human reasoning . In L. Li and K. K . Yen, editors, Proceedings of the Fifth International Conference on Information , pages 228 - 231 . International Information Institute, 2009 .

19. S. Ho¨lldobler and C. D. P. Kencana Ramli . Logic programs under three-valued Lukasiewicz's semantics . In P. M. Hill and D. S. Warren, editors, Logic Programming , volume 5649 of Lecture Notes in Computer Science, pages 464 - 478 . Springer-Verlag Berlin Heidelberg, 2009 .

20. S. H¨olldobler and C. D. P. Kencana Ramli . Logics and networks for human reasoning . In C. Alippi, M. M. Polycarpou , C. G. Panayiotou , and G. Ellinasetal, editors, Artificial Neural Networks - ICANN , volume 5769 of Lecture Notes in Computer Science, pages 85 - 94 . Springer-Verlag Berlin Heidelberg, 2009 .

21. S. H¨olldobler, T. Philipp, and C. Wernhard . An abductive model for human reasoning . In Proc. Tenth International Symposium on Logical Formalizations of Commonsense Reasoning , 2011 . commonsensereasoning.org/2011/proceedings.html.

22. P. Johnson-Laird. Mental Models: Towards a Cognitive Science of Language, Inference, and Consciousness . Cambridge University Press, Cambridge, 1983 .

23. S. Kleene . Introduction to Metamathematics. North-Holland, 1952 .

24. R. Kowalski . Computational Logic and Human Thinking: How to be Artificially Intelligent . Cambridge University Press, 2011 .

25. J. Lukasiewicz . O logice tro´jwarto´sciowej . Ruch Filozoficzny , 5 : 169 - 171 , 1920 . English translation: On Three-Valued Logic . In: Jan Lukasiewicz Selected Works. (L. Borkowski, ed.), North Holland, 87 - 88 , 1990 .

26. J. Pereira and A. Pinto . Inspecting side-e↵ ects of abduction in logic programming . In M. Balduccini and T. Son, editors, Logic Programming , Knowledge Representation, and Nonmonotonic Reasoning: Essays in Honour of Michael Gelfond , volume 6565 of Lecture Notes in Artificial Intelligence , pages 148 - 163 . Springer, 2011 .

27. L. M. Pereira , E.-A. Dietz , and S. Ho¨lldobler. An abductive reasoning approach to the belief-bias e↵ ect . In C. Baral, G. D. Giacomo , and T. Eiter, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the 14th International Conference , pages 653 - 656 , Cambridge, MA, 2014 . AAAI Press.

28. L. M. Pereira , E.-A. Dietz , and S. Ho¨lldobler. Contextual abductive reasoning with side-e↵ ects . In I. Niemela¨, editor, Theory and Practice of Logic Programming (TPLP) , volume 14 , pages 633 - 648 , 2014 . Cambridge University Press.

29. M. Ragni and M. Knau ↵ . A theory and a computational model of spatial reasoning . Psychological Review , 120 : 561 - 588 , 2013 .

30. K. Stenning and M. van Lambalgen . Human Reasoning and Cognitive Science . MIT Press, 2008 .

31. P. Wason . Reasoning about a rule . The Quarterly Journal of Experimental Psychology , 20 : 273 - 281 , 1968 .