Given a conditional sentence ⇒ (if then ) and respective facts, four different types of inferences are observed in human reasoning. Affirming the antecedent (AA) (or modus ponens) reasons from ; affirming the consequent (AC) reasons from ; denying the antecedent (DA) reasons ¬ from ¬ ; and denying the consequent (DC) (or modus tollens) reasons ¬ from ¬ . Among them, AA and DC are logically valid, while AC and DA are logically invalid and often called logical fallacies. Nevertheless, humans often perform AC or DA as pragmatic inference in daily life. In this paper, we realize AC, DA and DC inferences in answer set programming. Eight different types of completion are introduced and their semantics are given by answer sets. We investigate formal properties and characterize human reasoning tasks in cognitive psychology.
In the pragmatics of conditional reasoning, it is assumed that a conditional sentence is often interpreted as People use conditional sentences and reason with them in bi-conditional, that is, ‘if ’ is interpreted as ‘if and only if ’, everyday life. From an early stage of artificial intelligence and such conditional perfection produces AC or DA as (AI), researchers represent conditional sentences as if- invited inference [16, 19]. Psychological studies empirithen rules and perform deductive inference using them. cally show that a conditional sentence “ if " is rephrased Production systems or logic programming are examples into the form “ only if " with greater frequency for perof this type of systems. However, human conditional mission/obligation statements [6, 5]. For instance, the reasoning is not always logically valid. In psychology and sentence “a customer can drink an alcoholic beverage if cognitive science, it is well known that humans are more he is over 18" is rephrased into “a customer can drink likely to perform logically invalid but pragmatic inference. an alcoholic beverage only if he is over 18". It is also For instance, consider the following three sentences: reported that AA is easier than DC when a conditional is : If the team wins the first round tournament, given as “if then ". When a conditional is given as “ then it advances to the final round. only if ", on the other hand, it is rephrased as “if not : The team wins the first round tournament. then not " and this paraphrase yields a directionality op: The team advances to the final round. posite which makes DC easier than AA [3]. The fact that people do not necessarily make inferences as in standard Given the conditional sentence and the premise , af- logic brings several proposals of new interpretation of conifrming the antecedent (AA) (or modus ponens) concludes ditional sentences in cognitive psychology. Mental logic the consequence . Given and the negation of the con- [4] interprets ‘if ’ as conveying supposition and introduces sequence ¬ , denying the consequent (DC) (or modus a set of pragmatic inference schemas for if-conditionals. tollens) concludes the negation of the premise ¬ . AA Mental model theory [22], on the other hand, considers and DC are logically valid. On the other hand, people that the meanings of conditionals are not truth-functional, often infer from and or infer ¬ from and and represents the meaning of a conditional sentence by ¬ . The former is called affirming the consequent (AC) models of the possibilities compatible with the sentence. and the latter is called denying the antecedent (DA). Both A probabilistic approach interprets a conditional sentence AC and DA are logically invalid and often called logical ⇒ in terms of conditional probability ( | ), then fallacies. the acceptance rates of four conditional inferences are 21st International Workshop on Nonmonotonic Reasoning, represented by their respective conditional probabilities September 2-4, 2023, Rhodes, Greece [29]. Eichhorn . [13] use conditional logic and define ⋆A longer version of this paper is submitted to a journal and is inference patterns as combination (AA, DC, AC, DA) of currently under review. four inference rules. Given a conditional sentence “if $ hsattkpa:m//wa@ebw.waakkaayyaammaa--uu.a.acc.j.pjp(/C~s.aSkaakmaam(aC). Sakama) then ", four possible worlds (combination of truth values 0000-0002-9966-3722 (C. Sakama) of and ) are considered. An inference in each pattern © 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License is then defined by imposing corresponding constraints on CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g ACttEribUutiRon 4W.0Iontrekrnsathioonapl(CPCrBoYc4e.0e)d.ings (CEUR-WS.org)
the plausibility relation over the worlds.
In this way, the need of considering the pragmatics of conditional reasoning has been widely recognized in In this paper, we consider logic programs with disjuncpsychology and cognitive science. On the other hand, tion, default negation, and explicit negation. A general relatively little attention has been paid for realizing such extended disjunctive program (GEDP) [25, 20] is a set of pragmatic inferences in computational logic or logic pro- rules of the form: gramming [33, 24]. From a practical perspective, however, people would expect AI to reason like humans, that is, one 1 ; · · · ; ; +1 ; · · · ; would expect AI to conclude from and , or ¬ ← +1, . . . , , +1, . . . , (1) from and ¬ in the introductory example, rather than conclude . Logic programming is a context- where ’s (1 ≤ ≤ ) are (positive or negative) independent language and has a general-purpose infer- literals and 0 ≤ ≤ ≤ ≤ . A program ence mechanism by its nature. By contrast, pragmatic might contain two types of negation: default negation inference is governed by context-sensitive mechanisms, (or negation as failure) and explicit negation ¬. For rather than context-free and general-purpose mechanisms any literal , is called an NAF-literal and define [6, 9]. As argued by [11], computational approaches to ¬¬ = . We often use the letter ℓ to mean either explain human reasoning should be cognitively adequate, a literal or an NAF-literal . The left of ← is that is, they appropriately represent human knowledge a disjunction of literals and NAF-literals (called head), (conceptually adequate) and computations behave simi- and the right of ← is a conjunction of literals and NAFlarly to human reasoning (inferentially adequate). Then if literals (called body). Given a rule of the form (1), we use logic programming for representing knowledge in define ℎ+() = {1, . . . , }, ℎ− () = daily life, it is useful to have a mechanism of automatic {+1, . . . , }, +() = {+1, . . . , }, and transformation of a knowledge base to simulate human − () = {+1, . . . , }. A rule (1) is called a reasoning depending on the context in which conditional fact if +() = − () = ∅; and it is called a consentences are used. That is, transform a program to a straint if ℎ+() = ℎ− () = ∅. A GEDP Π is conceptually adequate form in order to make computation called -free if ℎ− () = − () = ∅ for each in the program inferentially adequate. rule in Π .
In this paper, we realize human conditional reasoning A GEDP Π coincides with an extended disjunctive
proin answer set programming (ASP) [
The rest of this paper is organized as follows. Section 2 set of Π iff is a minimal set satisfying the condireviews basic notions of ASP programs considered in this tions: (i) satisfies every rule from Π , that is, for each paper. Section 3 introduces different types of completions ground rule: 1 ; · · · ; ← +1, . . . , from Π , for human conditional reasoning, and Section 4 presents {+1, . . . , } ⊆ implies {1, . . . , } ∩ ̸= ∅. their variance as default reasoning. Section 5 discusses In particular, for each constraint ← +1, . . . , from related works and Section 6 summarizes the paper. Due Π , {+1, . . . , } ̸⊆ . (ii) If contains a pair of to page limitation, proofs of propositions are omitted in complementary literals and ¬, then = . this paper. They are found in the full paper1. Second, let Π be any GEDP and ⊆ . The reduct Π of Π by is a -free EDP obtained as follows: a 1http://web.wakayama-u.ac.jp/~sakama/hcr2023.pdf rule of the form: 1 ; · · · ; ← +1, . . . , is in Π iff there is a ground rule of the form (1) from Π Proposition 2.2. Let Π be a program and Ψ = such that ℎ− () ⊆
and − () ∩ = ∅. For Π , ℎ+() = ∅ and ℎ− () ∩ − () ̸= ∅}. from Π [20]. An answer set is consistent if it is not . constraint in Π , then Π is not contradictory. { | ∈ programs of the form Π , their answer sets have already been defined. Then, is an answer set of Π iff is an answer set of Π .
answer sets coincides with that given in [
|= (resp. Π |= ) if a literal is equivalent for any program Π . included in some (resp. every) consistent answer set of Π .
same set of answer sets. Two programs Π 1 and Π 2 are strongly equivalent if Π 1 ∪ Π and Π 2 ∪ Π are equivalent for any program Π [26]. In particular, two rules 1 and 2 are strongly equivalent if Π ∪ {1} and Π ∪ {2} are
An answer set of a GEDP is not always minimal, i.e., a program Π
may have two answer sets and such that [25, 20]. This is in contrast with the case of EDPs where every answer set is minimal.
; ← , ; ← .
Suppose a rule such that ℎ+() = ∅:
+1, . . . , , +1, . . . , . (2) ←
← +1 ; · · · ; Define a rule () of the form: +1, . . . , , +1, . . . , ,
+1, . . . , that is obtained by shifting +1 ; · · · ℎ− () to +1, . . . , in +( ()). The two ; in rules (2) and (3) are strongly equivalent under the answer set semantics. { | ∈ Π (Π ∖ Φ) answer sets.
Proposition 2.1 ([20]). Let Π be a program and Φ = ∪ { () | ∈ Φ }. Then Π and Π ′ have the same and ℎ+() = ∅}. Also, let Π ′ = Then, Π and Π ∖
Ψ have the same answer sets.
Example 2.3. For any Π , Π is equivalent to Π ∪ { ← , , } (Proposition 2.1), ∪ { ← , } which is further simplified to Π (Proposition 2.2). Proposition 2.3. Let Π be a not-free GEDP. If there is a
3.1. AC Completion
We first introduce a framework for reasoning by is firming the consequent (AC) in ASP. In GEDPs, a afconditional sentence sented by the rule “1 ; · · · ; ; +1 ; · · · ←
⇒ (if then ) is reprewhere is a disjunction ; " and is a conjunction “+1, . . . , , +1, . . . , ". To realize reasoning backward from
to , we extend a program Π by introducing new rules.
Definition 3.1 (AC completion).
Let Π
be a program and ∈ Π a rule of the form (1). First, for each disjunct in ℎ+() and ℎ− (), converse the implication: (4) and (5) in () represent converse implications respectively. We do not include this type of rules for confrom the disjunction in the head of to the conjunction in structing (Π) because it would disable AC reasoning. the body of . (Π) collects rules Σ ← ℓ (1 ≤ ≤ ) For instance, transforming Π = { ← , ← } into having the same (NAF-)literal ℓ on the right of ← , and Π ∪ { ; ← } produces the answer set {}, then constructs “Σ 1 ; · · · ; Σ ← ℓ ", which we call an is not obtained. With this reason, constraints and facts are extended rule. Introducing (Π) to Π realizes reasoning not completed at the first step of Definition 3.1. Note also by AC in Π . that the result of AC completion is syntax-dependent in
The set (Π) contains an extended rule having a dis- general. That is, two (strongly) equivalent programs may junction of conjunctions in its head, while it is transformed produce different AC completions. to rules of a GEDP. That is, the extended rule: (ℓ11, . . . , ℓ11 ) ; · · · ; (ℓ1, . . . , ℓ ) ←
ℓ is identified with the set of 1 × · · · × rules of the form: ℓ11 ; · · · ; ℓ ← ℓ (1 ≤ ≤ ; 1 ≤ ≤ ).
By this fact, (Π) is viewed as a GEDP and we do not distinguish extended rules and rules of a GEDP hereafter.
The semantics of (Π) is defined by its answer sets.
sentence which is subject to AC inference, while “← , " is a constraint which is not subject to AC inference by Example 3.1. Let Π = { ; ← , , ← }. definition. For instance, given the conditional sentence “if Then, (Π) = { (, ) ; ← , , ← it is sunny, the grass is not wet" and the fact “the grass is } where the 1st rule “(, ) ; ← " is identi- not wet", people would infer “it is sunny" by AC inference. ifed with “ ; ← " and “ ; ← "; and the 2nd On the other hand, given the constraint “it does not happen rule “, ← " is identified with “ ← " that wet-grass and sunny-weather at the same time" and and “ ← ". (Π) ∪ { ←} has two answer the fact “the grass is not wet", the number of people who sets {, } and {, }. infer “it is sunny" by AC would be smaller because the cause-effect relation between “sunny" and “not wet" is not explicitly expressed in the constraint.
Reasoning by AC is nonmonotonic in the sense that Π |= (or Π |= ) does not imply (Π) |= (or (Π) |= ) in general.
By definition, if there is more than one rule having the same (NAF-)literal in the heads, they are collected to produce a single converse rule. For instance, Π = { ← , ← } produces (Π) = { ; ← } but not Λ = { ← , ← }. Then, (Π) ∪ { ←} has two answer sets {, } and {, }. Suppose that the new Example 3.3. Π = { ← , ← , ← Tfahcetn¬← (Π ′i)s ∪ad{ded←} to hΠa.s tPheutanΠs′we=r sΠet {∪,{¬},w← h}ich. {} hasth←e an,swe←r set}{ha,st}h,e wanhsiwleerset({Π), =}. Π ∪ represents the result of AC reasoning in Π ′. If Λ is used instead of (Π) , however, Π ′ ∪ Λ ∪ { ←} has the In Example 3.3, reasoning by AC produces which answer set . The result is too strong because is con- blocks deriving using the first rule in Π . As a concrete sistently inferred from Π ′ ∪ { ←} by AC reasoning. As example, an online-meeting is held on time if no network a concrete example, put = -, = , and trouble arises. However, it turns that the web browser = -. Then (Π ′) ∪ { - ← } is unconnected and one suspects that there is some trouhas the answer set { -, ¬ , - }, ble on the network. Put =“online-meeting is held on while Π ′ ∪ Λ ∪ { - ← } has the answer set . time", =“network trouble", =“the web browser is unAC completion derives an antecedent from a consequent, connected". In this case, one may withdraw the conclusion but it does not derive negation of antecedent by its na- after knowing . As such, additional rules (Π) may ture. For instance, given Π = { ; ← , ← } , change the results of Π . One can see the effect of AC (Π) |= but (Π) ̸|= ¬ . reasoning in a program Π by comparing answer sets of Π
Note that in Definition 3.1 the converse of con- and (Π) . straints and facts are not considered. When A consistent program Π may produce an inconsistent ℎ+() = ℎ− () = ∅, is considered a (Π) . In converse, an inconsistent Π may produce a rule with false in the head, then (4) and (5) become consistent (Π) . “+1, . . . , , +1, . . . , ← false" which has no effect as a rule. On the other hand, when Example 3.4. Π 1 = { ← ¬ , ← } is consis+() = − () = ∅, is considered a rule with tent, but (Π 1) = Π 1 ∪ { ¬ ← } is contradictory. true in the body, then (4) and (5) become “ ← Π 2 = { ← , ← , ← } is incoherent, but (1 ≤ ≤ ) and “ ← ( + 1 ≤ ≤ ), (Π 2) = Π 2 ∪ { ← } is consistent. Example 3.2. Let Π 1 = { ← } and Π 2 = { ← , }. By Proposition 2.1, Π 1 and Π 2 are strongly equivalent, but (Π 1) = Π 1 ∪ { ← } and (Π 2) = Π 2. As a result, (Π 1) has the answer set {} while (Π 2) has the answer set ∅.
is given below.
Proposition 3.1. If a PDP Π contains no constraint, then (Π) is consistent. Moreover, for any answer set of Π , there is an answer set of (Π) such that ⊆ .
Proposition 3.2. If a program Π is contradictory, then (Π) is contradictory. 3.2. DC Completion We next introduce a framework for reasoning by denying the consequent (DC) in ASP. There are two ways for negating a literal—one is using explicit negation and the other is using default negation. Accordingly, there are two ways of completing a program for the purpose of reasoning by DC.
Example 3.5 shows that WDC is nonmonotonic as Π |= but (Π) ̸|= . SDC is also nonmonotonic (see Example 3.8). The result of DC completion is syntax-dependent in general.
Example 3.6. Let Π 1 = { ← } and Π 2 = { ← , }. Then, (Π 1) = Π 1 ∪ { ¬ ← } and (Π 2) = Π 2 ∪ { ¬ ; ¬ ← } . As a result, (Π 1) has the answer set ∅ while (Π 2) has two answer sets {¬ } and {¬ }.
WDC keeps the consistency of the original program.
Proposition 3.3. If a program Π has a consistent answer set , then is an answer set of (Π) .
The converse of Proposition 3.3 does not hold in general.
Example 3.7. The program Π = Definition 3.2 (DC completion). Let Π be a program. answer set, while (Π) = For each rule ∈ Π of the form (1), define () as the the answer set {}. rule: { ← { ←
} has no , ← } has +1; · · · ; ; +1; · · · ; and define () as the rule: ¬ +1; · · · ; ¬ ; +1; · · · ; ←
1, . . . , , +1, . . . , (6) ← ¬ 1, . . . , ¬ , +1, . . . , . (7)
Proposition 3.4. Let Π be a consistent program such that every constraint in Π is -free (i.e., ℎ+() = ℎ− () = ∅ implies − () = ∅ for any ∈ Π ).
Then (Π) is not contradictory.
In particular, (6) or (7) becomes a fact if ℎ+() = Example 3.8. The program Π = { ← , ← ¬ ℎ− () = ∅; and it becomes a constraint if ¬ ← } has the answer set {¬ }, but (Π) = Π +() = − () = ∅. The weak DC comple- { ¬ ← ¬ , ← ¬ , ← } is incoherent. tion and the strong DC completion of Π are respectively defined as: Proposition 3.5. If a program Π is contradictory, then both (Π) and (Π) are contradictory.
, ∪ (Π) (Π) = Π = Π ∪ { () | ∈ Π }, ∪ { () | ∈ Π }.
By definition, (Π) and (Π) introduce contrapositive rules in two different ways. In (6), literals (1 ≤ ≤ ; + 1 ≤ ≤ ) are negated using default negation and NAF-literals ( + 1 ≤ ≤ ; + 1 ≤ ≤ ) are converted to . In (7), on the other hand, literals (1 ≤ ≤ ; + 1 ≤ ≤ ) are negated using explicit negation ¬ and NAF-literals ( + 1 ≤ ≤ ; + 1 ≤ ≤ ) are converted to . (Π) and (Π) are GEDPs and their semantics are defined by their answer sets. In particular, (Π) becomes an EDP if Π is an EDP. The WDC and SDC produce different results in general.
In GEDPs contraposition of a rule does not hold in general, so the program Π = { ← , ¬ ← } does not deduce ¬ . completes the program as (Π) = Π ∪ { ¬ ← ¬ , ← } and makes ¬ deducible. In this sense, SDC has the effect of making explicit negation closer to classical negation in GEDP. 3.3. DA Completion As a third extension, we introduce a framework for reasoning by denying the antecedent (DA) in ASP. As in the case of DC completion, two different ways of completion are considered depending on the choice of negation.
Definition 3.3 (weak DA completion). Let Π be a proExample 3.5. Let Π = { ← }. Then, gram and ∈ Π a rule of the form (1). First, inverse the (Π) = { ← , ← } and implication: (Π) = { ← , ← ¬ }. (Π) has two answer sets {} and {}, while (Π) has the ← single answer set {}. (1 ≤ ≤ ), +1 ; · · · ; ; +1 ; · · · ; (8) ← +1 ; · · · ; ; +1 ; · · · ; (9) are collected to produce a single inverse rule. For in( + 1 ≤ ≤ ). stance, Π = { ← , ← } produces (Π) = { ← , } but not Λ = { ← In (8) and (9), the disjunction +1 ; · · · ; ; , ← }. Suppose the new fact ← +1 ; · · · ; appears on the right of ← . The is added to Π . Put Π ′ = Π ∪ { ←} . Then (Π ′) produced (8) (resp. (9)) is considered an abbrevia- has the answer set {, }. If Λ is used instead of (Π) , tion of the collection of ( − ) rules: ( ← however, Π ′ ∪ Λ is incoherent because the first rule of Λ is +1), . . . , ( ← ) (resp. ( ← not satisfied. The result is too strong because is deduced +1), . . . , ( ← )), hence we abuse the term by ← and ← , and it has no direct connection to DA ‘rule’ and call (8) or (9) a rule. In particular, (8) is not pro- inference in the first rule of Λ . Hence, we conclude duced if ℎ+() = ∅ or +() = − () = ∅; if both and are negated in (Π) . and (9) is not produced if ℎ− () = ∅ or +() = The strong DA completion is defined in a similar way. − () = ∅. The set of rules (8)–(9) is denoted as (). Next, define (Π) =
{ ℓ ← Γ 1, . . . , Γ | ℓ ← Γ (1 ≤ ≤ ) is in ⋃︁ () } ∈Π where ℓ is either a literal ( + 1 ≤ ≤ ) or an NAF literal (1 ≤ ≤ ), and each Γ (1 ≤ ≤ ) is a disjunction of literals and NAF literals. The weak DA completion of Π is defined as: (Π) = Π
∪ (Π) .
(8) and (9) in () represent inverse implication from the (default) negation of the conjunction in the body of to the (default) negation of the disjunction in the head of . (Π) collects rules ℓ ← Γ (1 ≤ ≤ ) having the same (NAF-)literal ℓ on the left of ← , and constructs “ℓ ← Γ 1, . . . , Γ ", which we call an extended rule. Introducing (Π) to Π realizes reasoning by weak DA.
An extended rule has a conjunction of disjunctions in its body, while it is transformed to rules of a GEDP as the case of AC completion. That is, the extended rule: ℓ ←
(ℓ11 ; · · · ; ℓ11 ) , . . . , (ℓ1 ; · · · ; ℓ ) is identified with the set of 1 × · · · × rules of the form: ℓ ← ℓ11 , . . . , ℓ (1 ≤ ≤ ; 1 ≤ ≤ ).
By this fact, (Π) is viewed as a GEDP and we do not distinguish extended rules and rules of a GEDP hereafter. The semantics of (Π) is defined by its answer sets.
Example 3.9. Let Π = { ; ← , , ; ← , ← } . Then. (Π) = { ← ; , ← ( ; ), , ← } where the first rule “ ← ; " is identified with “ ← " and “ ← "; and the second rule “ ← ( ; ), " is identified with “ ← , " and “ ← , ". Then, (Π) has the answer set {, }.
As in the case of AC completion, if there is more than one rule having the same (NAF-)literal in the heads, they Definition 3.4 (strong DA completion). Let Π be a program and ∈ Π a rule of the (1). First, inverse the implication: ¬ ← ¬
+1 ; · · · ; ¬ ; +1 ; · · · ; (10) ← ¬ +1 ; · · · ; ¬ ; +1 ; · · · ; (1 ≤ ≤ ),
(11) ( + 1 ≤ ≤ ).
As in the case of WDA, the produced (10) (resp. (11)) is considered an abbreviation of the collection of ( − ) rules: (¬ ← ¬ +1), . . . , (¬ ← ) (resp. ( ← ¬ +1), . . . , ( ← )), hence we call (10) or (11) a rule. In particular, (10)–(11) are not produced when their heads or bodies are empty. The set of rules (10)–(11) is denoted as (). Next, define (Π) =
{ ℓ ← Γ 1, . . . , Γ | ℓ ← Γ (1 ≤ ≤ ) is in ⋃︁ () } ∈Π where ℓ is either a literal ( + 1 ≤ ≤ ) or ¬ (1 ≤ ≤ ), and each Γ (1 ≤ ≤ ) is a disjunction of positive/negative literals. The strong DA completion of Π is defined as: (Π) = Π
∪ (Π) .
formed to rules of a GEDP. Then (Π) is viewed as a GEDP and its semantics is defined by its answer sets. In particular, (Π) becomes an EDP if Π is an EDP.
The result of DA completion is syntax-dependent in general.
Example 3.10. Let Π 1 = { ← } and Π 2 = { ← , }. Then, (Π 1) = Π 1 ∪ { ← } and (Π 2) = Π 2. As a result, (Π 1) has the answer set {} while (Π 2) has the answer set ∅.
Both WDA and SDA are nonmonotonic in general. Example 3.11. (1) Π 1 = { ← produces (Π 1) = Π 1 ∪ { ←
AC and DA are logically invalid and additional rules for AC and DA often make a program inconsistent. In this section, we relax the effects of the AC or DA completion by introducing additional rules as default rules in the sense of [31]. More precisely, we capture AC and DA as the following default inference rules: (default AC) (default DA) ( ( ⇒ ) ∧ : ⇒ ) ∧ ¬ : ¬ ¬
and the fact , conclude as a default consequence. The default DA rule is read in a similar manner. We encode ⇒ these rules in ASP. 4.1. Default AC completion The AC completion is modified for default AC reasoning. () as the set of rules: Definition 4.1 (default AC completion).
Let Π be a program. For each rule ∈ Π of the form (1), define
not resolve incoherency in general. does Example 4.3. Let Π = { ← , ← , ← }. Then, (Π) = Π ∪ { ← } is incoherent. (Π) = Π ∪ { ← , ¬ } is still incoherent.
does not change it.
Proposition 4.4. Let Π be a program. If (Π) (resp. (Π) ) has a consistent answer set , then is an answer set of (Π) (resp. (Π) ).
Thus, default AC/DA completion is used for avoiding contradiction in programs containing explicit negation.
Proposition 4.2. Let Π be a program. If (Π) has There is a number of studies on human conditional reasona consistent answer set , then is an answer set of ing in psychology and cognitive science. In this section, (Π) . we focus on related work based on logic programming. 4.2. Default DA completion 5.1. Completion The DA completion is modified for default DA reasoning. The idea of interpreting if-then rules in logic programs as bi-conditional dates back to [7]. Clark introduces Definition 4.2 (default DA completion). Let Π be a predicate completion in normal logic programs (NLPs), program. Define which introduces the only-if part of each rule to a program. Given a propositional program Π , Clark comple(Π) = { ℓ ← Γ 1, . . . , Γ , | tion (Π) is obtained by two steps: (i) all rules ℓ ← Γ (1 ≤ ≤ ) is in ⋃︁ () }, ← 1,. . ., ← in Π having the same head are ∈Π replaced by ↔ 1 ∨ · · · ∨ , where (1 ≤ ≤ ) is (Π) = { ℓ ← Γ 1, . . . , Γ , | a conjunction of literals; and (ii) for any atom appearing in the head of no rule in Π , add ↔ false. The AC comℓ ← Γ (1 ≤ ≤ ) is in ⋃︁ () } pletion introduced in this paper extends the technique to ∈Π the class of GEDP, while the result is generally different where ℓ, Γ , (), and () are the same as those from Clark completion in NLPs. For instance, given the in Defs. 3.3 and 3.4. In addition, = ¬ if ℓ = program: Π 1 = { ← , ← } , Clark completion be, and = if ℓ = ; = ¬ if comes (Π 1) = { ↔ ∨ ⊤, ↔ ⊥ } where ⊤ ℓ = , and = if ℓ = ¬ . The weak default and ⊥ represent true and false, respectively. (Π 1) DA completion and the strong default DA completion of has the single completion model {} called a supported Π are respectively defined as: model [2]. In contrast, (Π 1) = Π 1 ∪ { ← } has the answer set {, }. The difference comes from the fact (Π) = Π ∪ (Π) , that in (Π 1), is identified with false but this is (Π) = Π ∪ (Π) . not the case in (Π 1). In Clark completion undefined atoms (i.e., atoms appearing in the head of no rule) are
Rules in (Π) and (Π) are converted into interpreted false. We do not use this type of completion the form of a GEDP, so (Π) and (Π) because it disturbs the basic type of AC reasoning that are viewed as GEDPs. Like the DAC completion, both infers from and ← . Clark completion is extended WDDA and SDDA introduce an additional NAF literal to to normal disjunctive programs by several researchers each rule. [27, 1, 28]. Those extensions reduce to Clark completion in NLPs, so that they are different from the AC compleProposition 4.3. Let Π be a consistent program. If tion. We also introduce the DC completion and the DA (Π) (or (Π) ) has an answer set , then completion. When Π 2 = { ← }, (Π 2) = ̸= . { ↔ ¬, ↔ ⊥ } has the supported model {} while
The WDDA/SDDA eliminates contradiction but does (Π 2) = Π 2 ∪ { ← } has two answer sets not resolve incoherency. {} and {}. When Π 3 = { ← , ← , ← }, (Π 3) = { ↔ ∨ ¬, ↔ } has the supExample 4.4 (cont. Example 3.12). Let Π 1 = ported model {, } while (Π 3) = Π 3∪{ ← { ← } where (Π 1) is incoherent. , , ← } has no answer set. Thus, com (Π 1) = Π 1 ∪ { ← , ¬ } is pletion introduced in this paper is generally different from still incoherent. Let Π 2 = { ¬ ← , ¬ ← } Clark completion in NLPs. where (Π 2) is contradictory. (Π 2) = The weak completion [18] leaves undefined atoms unΠ 2 ∪ { ← ¬ , ¬ } has the answer set {¬}. known under 3-valued logic. In Π 1 = { ← , ← } , the weak completion becomes (Π 1) = { ↔ planation for the observation = ¬. The problem ⊤ }. Then is true but is unknown in (Π 1), is resolved by reasoning by DC. For the rule in Π 3, which is again different from the result of (Π 1) that () = { ← ¬ __ }. Then has the answer set {, }. In Π 2 = { ← }, (Π 3) ∪ {} computes the explanation { }. (Π 2) = { ↔ ¬ } then both and are un- In contrast to , is used for abduction from known. In contrast, (Π 2) has two answer sets {} negative observations. A negative observation represents and {}, and (Π 2) has the single answer set {}. that some evidence is not observed and it is represented as = , which should be distinguished from the 5.2. Abductive Logic Programming (positive) observation = ¬ meaning that ¬ is observed. In the abductive program ⟨ Π 3, Γ 1 ⟩, the negative An abductive logic program [23] is defined as a pair observation = __ is explained us⟨ Π , Γ ⟩ where Π is a program and Γ ( ⊆ ) is a set ing () = { ← __ }. of literals called abducibles. It is assumed that ab- Then (Π 3) ∪ {} has the answer set { }. ducibles appear in the head of no rule in Π . Given an In this way, both AC and DC are used for computing observation as a ground literal, the abduction prob- explanations deductively, and DC is used for computing lem is to find an explanation (⊆ Γ) satisfying (i) explanations that are not obtained using the framework ∪ |= and (ii) ∪ is consistent, where |= of [23]. Console . [8] and Fung . [15] compute is either |= or |= depending on the problem. Here abduction by deduction using Clark completion. Abducwe consider |= that realizes credulous abduction. Con- tion using AC/DC completion is close to those approaches, sider (Π 1, Γ 1) where Π 1 = { __ ← while the approach based on Clark completion is restricted , ¬ __ ← } to normal logic programs (NLPs). As argued above, a and Γ 1 = {}. Π 1 represents that a train ar- (positive) observation = ¬ is distinguished from a rives on time unless there is an accident. Then the negative observation = , but such a distinction observation = ¬ __ has the expla- is not considered in NLPs handling only default negation. nation = {}. Since abduction reasons Inoue . [21] introduce transaction programs for combackward from an observation, it is computed using puting extended abduction, which computes explanations the AC completion. Let ⟨ Π , Γ ⟩ be an abductive pro- for both positive/negative observations. A transaction gram and an observation. Then, a set ⊆ Γ program is a meta-level specification for computing the is an explanation of if ∈ ℎ+() for some converse of conditionals, and is defined for NLPs only. ∈ Π and (Π) ∪ {} has a consistent answer set such that ∩ Γ = . In the above example, 5.3. Human Conditional Reasoning (Π 1) ∪ {} is Π 1 ∪ (Π 1) ∪ {} where (Π 1) = { ← __, ← Stenning . [33] formulate human conditional reason¬ __ }. (Π 1)∪{} has the answer set ing using Clark’s program completion under the three = { ¬ __, }. Then, ∩ Γ 1 = valued logic of [14]. They represent a conditional sentence { } is the explanation. Note that (Π) in- “if then " as a logic programming rule:“ ← ∧ ¬" troduces converse of every rule, while explanations are where represents an abnormal atom. In this setting, DA computed using the AC completion of a subset Π ′ ⊆ Π is represented as Π 1 = { ← ⊥ , ← ∧¬, ← in general. For instance, consider Π 2 = { ← , ← ⊥ }. The rule “ ← ⊥ " means that is a proposition ¬, ← } and Γ 2 = {, ¬}. Then = has the to which the closed world assumption [30] is applied. explanation = {} in ⟨ Π 2, Γ 2 ⟩, while (Π 2) ∪ If a program does not contain ← ⊥ , nor any other {} = Π 2 ∪ { ← , ¬ ← , ← } is contradic- rule in which occurs in its head, then is interpreted tory. By putting Π ′2 = { ← }, (Π ′2) ∪ {} has unknown. Then its completion (Π 1) = { ↔ the consistent answer set = {, } where ∩ Γ = {}. ⊥, ↔ ∧ ¬, ↔ ⊥ } derives ↔ ⊥. On As such, abduction and AC completion produce different the other hand, completion does not realize AC or DC results in general. inference by itself. In their framework, AC is represented
Abductive logic programs of [23] cannot compute as Π 2 = { ← ⊤ , ← ∧ ¬, ← ⊥ } , while explanations when contrary to the consequent is ob- (Π 2) = { ↔ ⊤ ∨ ( ∧ ¬), ↔ ⊥ } served. For instance, consider ⟨ Π 3, Γ 1 ⟩ where Π 3 = does not derive . Likewise, DC is represented as { __ ← }. Given the ob- Π 3 = { ← ⊥ , ← ∧ ¬, ← ⊥ } , while servation = ¬ __, no explanation is (Π 3) = { ↔ ⊥ ∨ ( ∧ ¬), ↔ ⊥ } does obtained from ⟨ Π 3, Γ 1 ⟩. Generally, a program Π does not derive ↔ ⊥. They then interpret ← ∧ ¬ as not necessarily contain a pair of rules and ′ that define an integrity constraint meaning that “if succeeds (resp. and ¬, respectively. When there is a rule defining fails) then ∧ ¬ succeeds (resp. fails)" to get the AC but no rule defining ¬, abduction computes no ex- consequence (resp. DC consequence ¬).
Dietz . [11] point out a technical flaw in the for- tional is factual. In the former case, the observation mulation by [33]. Suppose a conditional sentence ← does not imply because ← ⊤ can make explainwhere and are unknown U. Under the Fitting seman- able by itself. As a result, is not a skeptical explanation tics, however, the truth value of the rule U ← U is U, then of . In the latter case, the observation ¬ does not imit does not represent the truth of the sentence. To remedy ply ¬ because if one employs the explanation ← ⊤ , the problem, they employ Łukasiewicz’s 3-valued logic ← ∧ ¬ does not produce ↔ . which maps U ← U to ⊤. Dietz . [12] use logic programming rules to repre
Comparing the above mentioned two studies with our sent different types of conditionals. For instance, the rule: approach, there are several differences. First, they trans- “concl ← prem(x), sufficient(x)" represents MP that late a conditional sentence “if then " into the rule concl follows if a sufficient premise is asserted to be true. ← ∧ ¬ . However, it is unlikely that people who By contrast, “not_concl ← not_prem(x), necessary(x)" commit logical fallacies, especially younger children [32], represents DA that concl does not follow if a necessary translate the conditional sentence into the rule of the above premise is asserted to be false. In the current study, complex form in their mind. We represent the conditional we do not distinguish different types of conditionals as sentence directly as ← , and assume that people would in [10, 12]. However, completion is done for individual interpret it as bi-conditional depending on the context it rules, so we could realize partial completion by selecting is used. Second, in order to characterize AC or DC infer- rules Π ′ ⊆ Π that are subject to be completed in practice. ence, [33] interpret a conditional sentence as an integrity More precisely, if a program Π consists of rules 1 havconstraint, while [11] uses abductive logic programs. Our ing necessary antecedents and 2 having non-necessary framework does not need a specific interpretation of rules antecedents, apply AC completion to 1 while keep 2 (such as integrity constraints) nor need an extra mecha- as they are. The resulting program then realizes AC innism of abductive logic programs. Third, they use a single ference using 1 only. Likewise, if a program Π consists (weak) completion for all AC/DA/DC inferences, while of rules 3 having obligatory consequents and 4 having we introduce different types of completions for each in- factual consequents, apply DC completion to 3 while ference. By separating respective completions, individual keep 4 as they are. The resulting program then realizes inferences are realized in a modular way and freely com- DC inference using 3 only. bined depending on their application context. Fourth, they handle normal logic programs, while our framework can handle a more general class of logic programs as GEDPs. 6. Conclusion
Cramer . [10] represent conditionals as in [33] and use the weak completion and abductive logic programs This paper studies a method of realizing human condias in [11]. They formulate different types of conditionals tional reasoning in ASP. Different types of completion based on their contexts and argue in which case AC or DC are introduced to realize logically invalid inferences AC is more likely to happen. More precisely, a conditional and DA as well as a logically valid inference DC. In psysentence whose consequent appears to be obligatory given chology and cognitive science, empirical studies show the antecedent is called an obligation conditional. An that people perform AC, DA or DC inference depending example of an obligation conditional is that “if Paul rides on the context in which a conditional sentence is used. a motorbike, then he must wear a helmet". If the conse- We could import the results of those studies and encode quence of a conditional is not obligatory, then it is called knowledge in a way that people are likely to use it. The a factual conditional. The antecedent of a conditional proposed theory is used for such a purpose to realize pragsentence is said to be necessary iff its consequent can- matic inferences in ASP and produce results that are close not be true unless is true. For example, the library being to human reasoning in practice. open is a necessary antecedent for studying in the library. Completions introduced in this paper are defined in a Cramer . argue that AA and DA occur independently modular way, so one can apply respective completion to of the type of a conditional. On the other hand, in AC specific rules of a program according to their contexts. most people will conclude from ⇒ and , while They are combined freely and can be mixed in the same the number of people who conclude nothing will increase program. Those completions are general in the sense that if is a non-necessary antecedent. In DC, most people they are applied to logic programs containing disjunction, will conclude ¬ from ⇒ and ¬ , while the num- explicit and default negation. Since a completed program ber of people who conclude nothing will increase if the is still in the class of GEDPs and a GEDP is transformed conditional is factual. Those assumptions are verified by to a semantically equivalent EDP [20], answer sets of comquestioning participants who do not receive any education pleted programs are computed using existing answer set in logic beyond high school training. They then formulate solvers. In the full paper, the proposed theory is applied to the situation by introducing the abducible ← ⊤ if the representing human reasoning tasks in the literature, and antecedent is non-necessary, and ← ⊤ if the condi- is used for computing common sense reasoning in AI.