In the intersection between case-based reasoning and non-monotonic reasoning, case models serve as frameworks to evaluate arguments with respect to cases. The evaluation is based on their coherence, presumptive validity, and conclusiveness, in which all valid arguments must be at least coherent, meaning that those arguments must be grounded in at least one case. This paper, on the other hand, attempts to explore new evaluations for novel arguments, which are not required to ground in any case. To develop the new evaluation, we introduce revision operators in case models and associated properties based on AGM postulates. We then de!ne a conclusively adherent evaluation, in which novel arguments can be valid. After that, we relate their application to the understanding of abstract argumentation for case-based reasoning (AA-CBR). We demonstrate how the translation of AA-CBR case bases into case models can be described through revision sequences; and analyse the properties of conclusively adherent evaluation related to evaluation and attacks in AA-CBR.
Case model [
One fundamental requirement in case models is that every valid argument must be at least coherent, meaning that the argument must be grounded in at least one existing case. However, in several applications, such as case-based reasoning, it requires to evaluate the validity of arguments even if they are not grounded in any existing cases, making the arguments always invalid in original evaluations.
Therefore, this paper introduces a crucial extension by de!ning new evaluations for novel arguments,
which are not grounded in any existing cases. To facilitate this, this paper de!nes revision operators
for case models, adapting the AGM postulates [
Furthermore, this paper explores the application of conclusively adherent evaluation to abstract
argumentation for case-based reasoning (AA-CBR) [
The contributions of this paper can be summarised as follows.
• The paper introduces families of revision operators for case models, including re!nement-based revision operators and single revision operators, and speci!cally de!nes a top revision operator and a bottom revision operator, both of which add at most one case to a case model. • The paper de!nes associated properties of these revision operators, including -success, inclusion, vacuity, and extensionality. • The paper investigates the satisfaction of these properties by di"erent types of revision operators. • The paper de!nes conclusively adherent arguments based on the satisfaction of the properties. • The paper demonstrates how to describe the translation of AA-CBR case bases into case models through bottom revision sequences of novel arguments. • The paper analyses the properties of novel arguments in the sequences related to AA-CBR concepts, stating that if a novel argument is conclusively adherent with respect to a translated case model, then the corresponding situation-outcome pair is valid with respect to the case base, and if a novel argument is not conclusively adherent with respect to a translated case model, the corresponding situation-outcome pair would attack some precedent in the AA framework if it is included in the case base, and vice versa.
The paper is structured as follows. Section 2 provides related work on the background of belief change and case-based reasoning research. Section 3 describes a formalism of case models, argument evaluations in case models, and applications of case models in case-based reasoning. Section 4 de!nes revision operators for case models with associated properties based on the AGM postulates and conclusively adherent evaluation. Section 5 presents an analysis of conclusively adherent arguments on the translation of AA-CBR case bases into case models. Section 6 discusses several implications of this paper and suggests future work. Finally, Section 7 concludes this paper.
Nonmonotonic reasoning researchers have long been interested in making inferences that go beyond the data explicitly available to a system. This interest has derived the development of various reasoning paradigms, such as belief change and case-based reasoning, that aim to simulate the kind of #exible, context-sensitive reasoning that humans routinely perform. These paradigms are especially valuable when dealing with incomplete, evolving, or con#icting information. Integrating insights from these reasoning paradigms with modern machine learning models remains a key challenge to make AI systems more reliable and explainable.
Belief change considers two main entities: beliefs and consequences, usually represented in
propositions. The main data structure in belief revision is belief sets, which is assumed to contain logical
sentences considering some beliefs and all their consequences. To explain the change in belief sets,
several operators have been formulated based on AGM framework [
Meanwhile, case-based reasoning studies how to reason through cases from prior experience.
Pioneered case-based reasoning systems, such as HYPO [10], relied mainly on analogical reasoning. Later
works have developed logical frameworks for case-based reasoning, such as abstract argumentation for
case-based reasoning [
To our knowledge, this is the !rst study to connect case models with belief change. This connection is relevant to several aspects of nonmonotonic reasoning, such as cumulative reasoning and cautious monotonicity found in conditional logic [14] and case-based reasoning with cases represented as arguments [15]. However, our focus is on case models, which speci!cally address the logical properties of cases, such as logical consistency and mutual incompatibility, with cases represented as logical sentences rather than arguments, distinguishing case models from those works in conditional logic and case-based reasoning.
In this paper, we use a classical logical language L generated from propositions in a standard way. We write ¬ for negation, → for conjunction, ↑ for disjunction, ↓ for equivalence, ↔ for tautology and ↗ for contradiction. The associated classical, deductive, and monotonic logical inference is denoted as ↘. We say L ≃ L is logically consistent i" each element of L is not a contradiction (formally, ⇐↘ ↗ for all ⇒ L). We say L ≃ L is consistent i" all elements of L can be true together (formally, L ⇐↘ ↗ ). We say L ≃ L is mutually incompatible i" two elements of L cannot be true together (formally, for all , ⇒ L such that ⇐= : → ↘ ↗ ). Examples of mutually incompatible sets are {p → q, ¬p → r} or even {p1 → q, p2 → r} if we have a background knowledge that p1 → p2 ↘ ↗ .
A case model [
In case models, an argument is considered as a pair of logical sentences. Given a case model ⇑C, ⇓⇔
and an argument (, ) ⇒ L ↘ L where is called a premise and is called a conclusion. An evaluation
↭ is then considered as a relation from a case model to an argument. Three evaluations, called in this
paper as basic evaluations, have been originally de!ned as follows [
Consequently, basic evaluations are ranked from the strongest to the weakest as ↭conc, ↭pres, ↭coh as ⇑C, ⇓⇔ ↭conc (, ) implies ⇑C, ⇓⇔ ↭pres (, ); and ⇑C, ⇓⇔ ↭pres (, ) implies ⇑C, ⇓⇔ ↭coh (, ). Thus, every valid argument according to basic evaluations must be at least coherent, where the premise and the conclusion can be logically implied from at least one of the cases. This property was originally designated so that a conclusion is retracted if we extend the premise to the degree that there is no grounding in any case. We call an argument such that the premise has no grounding in any case as a novel argument, formally de!ned as follows.
De!nition 1 (Novel argument). An argument (, ) is novel with respect to ⇑C, ⇓⇔ i! ′ ⇒ C, ⇐↘ . Proposition 1. A novel argument cannot be valid with respect to a basic evaluation. Proof. If ↭ is a basic evaluation, then for every case model ⇑C, ⇓⇔ and argument (, ) such that ⇑C, ⇓⇔ ↭ (, ), ⇑C, ⇓⇔ ↭coh (, ) and hence ∝ ⇒ C : ↘ . Therefore, (, ) cannot be a novel argument with respect to ⇑C, ⇓⇔ .
Since preferences in case models are total and transitive, each case model ⇑C, ⇓⇔ can be represented as a sequence of logical sentence sets L1, . . . , Ln such that, for 1 ∞ i, j ∞ n (we call each Li a layer): 1. L1, . . . , Ln is a partition of C (i.e., C = L1 ∈ . . . ∈ Ln and Li ∋ Lj = △ ); and 2. ↖ i" , ⇒ Li; and 3. > i" ⇒ Li and ⇒ Lj and i < j.
Hereafter, we write L1 > . . . > Ln instead of L1, . . . , Ln to distinguish it from a general sequence. Example 1 demonstrates a case model and some examples of basic evaluations.
Example 1. Consider the case model ⇑C, ⇓⇔ = {¬big → ¬ fly} > {wing → fly} > {big → ¬ fly}, which represents a state of knowledge concerning the distribution of animals across three distinct groups: (1) the majority group of animals which are not big and cannot "y, (2) the signi#cant group of animals which have wings and can "y, and (3) the minority group of animals which are big and cannot "y. Examples of basic evaluations and novel argument with respect to the case model are as follows: • (↔, fly) is coherent (which can be interpreted as some animals can "y) because the argument has a grounding in the case wing → fly. • (↔, ¬fly) is presumptively valid (which can be interpreted as animals presumptively cannot "y) because the argument has a grounding in the case ¬big → ¬ fly, which is ⇓ -maximal within all the cases implying the premise. • (wing, fly) is conclusive (which can be interpreted as all animals that have wings can "y, according to the current state of knowledge) because the argument has a grounding in the case wing → fly and all the cases that imply wing also imply wing → fly. • (wing → big, ¬fly) is novel (which can be interpreted as we cannot conclude whether big animals which have wings cannot "y, according to the current state of knowledge) because the argument has no grounding in any case.
In this paper, we focus particularly on the application of case models in case-based reasoning [13, 17]. In this application, the arguments are constrained so that the premises and conclusions are from disjoint domains. Firstly, we assume an outcome domain O, of which each element is called an outcome, as a !nite mutually incompatible subset of L. We call an argument (, ) an outcome-inference argument i" the premise does not imply any outcome (i.e., ′ o ⇒ O : ⇐↘ o), and the conclusion is an outcome or a special proposition nil, not included in O and not occurring in any case models. An outcome-inference argument of which the conclusion is an outcome is called an outcome-based argument while an outcome-inference argument of which the conclusion is nil is called an outcomefree argument, used when we could not !nd any outcome as a conclusion. We extend the basic evaluations by replacing “ ↘ → " in the original de!nition with “′ o ⇒ O : ↘ and ⇐↘ o” to evaluate an outcome-free argument. That is, for a case model ⇑C, ⇓⇔ and an outcome-free argument (, nil ), the basic evaluations are extended as follows: • ⇑C, ⇓⇔ ↭coh (, nil ) i" ∝ ⇒
grounding in . • ⇑C, ⇓⇔ ↭pres (, nil ) i" ∝ ⇒ C : ′ o ⇒ O
⇓ ↑. • ⇑C, ⇓⇔ ′ o ⇒ O
C : ′ o ⇒ O
: ↘ and ⇐↘ o; we then say (, nil ) has a : ↘ and ⇐↘ o, and ′ ↑ ⇒ C : if ↑ ↘ ; then ↭conc (, nil ) i" ∝ ⇒ C : ′ o ⇒ O : ↘ and ⇐↘ o; and ′ ⇒ C : if ↘ , then :<< ↘ and >> ⇐↘ o. (The statement in «...» can be omitted).
Proposition 2. Two outcome-inference arguments (, ) and ( ↑, ↑) with di!erent conclusions ⇐= ↑ cannot have a grounding in the same case (i.e., a non-contradiction logical sentence). Proof. Following the extended de!nition of basic evaluations, if one conclusion is nil, then the proposition clearly holds. If two conclusions are included in O, then the case implies → ↑ which leads to contradiction as O are mutually incompatible but the case is non-contradiction.
To develop a new evaluation for novel arguments, we !rst introduce revision operators and their
associated properties based on AGM postulates [
if ⇑C, ⇓⇔ ↭ (˜, ˜ ) then ⇑C, ⇓⇔ ▽ (, ) ↭ (˜, ˜ ). 3. vacuity i! for every outcome-inference argument (˜, ˜ ),
if ⇑C, ⇓⇔ ⇐ ↭coh (˜, ˜ ) and ˜ ⇐= then ⇑C, ⇓⇔ ▽ (, ) ⇐↭coh (˜, ˜ ). 4. extensionality i! for every ↑ ⇒ L such that ↓ ↑ and every outcome-inference argument (˜, ˜ ), if ⇑C, ⇓⇔ ▽ (, ) ↭ (˜, ˜ ) then ⇑C, ⇓⇔ ▽ ( ↑, ) ↭ (˜, ˜ ), and vice versa.
In De!nition 2, we classify success and inclusion into three levels according to three basic evaluations using a label “ -" . Intuitively, -success means that a revision argument should be valid in -evaluation with respect to the revised case model. -inclusion means that the valid arguments in -evaluation should be preserved their validity after revision. Vacuity means that every incoherent argument with di"erent conclusion from a revision argument should remain incoherent with respect to the revised case model. Finally, extensionality means that if two revision arguments are logically equivalent, the revision should result two revised case models that evaluate any argument equivalently. We then de!ne a family of revision operators based on case model re#nement [18], de!ned as follows. De!nition 3 (Case Model Re!nement [18]). A case model re#nement is an order ̸ over the case model space M such that for ⇑C, ⇓⇔ , ⇑C↑, ⇓ ↑⇔ ⇒ M , ⇑C, ⇓⇔ ̸ ⇑ C↑, ⇓ ↑⇔ i! for all ⇒ C, there exists ↑ ⇒ C↑ such that ↑ ↘ . We then say ⇑C↑, ⇓ ↑⇔ re#nes ⇑C, ⇓⇔ .
It has been studied that ̸ is a re#exive and transitive order [18], meaning that it is an order over the case model space, starting from the case model with the empty case set ⇑△ , △⇔ to more re!ned case models. A re#nement-based revision operator is de!ned as follows.
De!nition 4 (Re!nement-based revision operator). A revision operator ▽ is re!nement-based i! for every ⇑C, ⇓⇔ ⇒ M and every outcome-inference argument (, ), ⇑C, ⇓⇔ ▽ (, ) ↭coh (, ) (coh-success); and ⇑C, ⇓⇔ ̸ ⇑ C, ⇓⇔ ▽ (, ) (revision-as-re#nement).
Consequently, re!nement-based revision operators are assured to satisfy coh-inclusion. Proposition 3. Every re#nement-based revision operator satis#es coh-inclusion.
Proof. Let (, ) be an outcome-inference argument, ▽ be a re!nement-based revision operator, and ⇑C↑, ⇓ ↑⇔ = ⇑C, ⇓⇔ ▽ (, ). Since ⇑C, ⇓⇔ ̸ ⇑ C↑, ⇓ ↑⇔, for every outcome-inference argument (˜, ˜ ), if there is ⇒ C such that ↘ ˜ → ˜ then there is ↑ ⇒ C↑ such that ↑ ↘ and hence ↑ ↘ ˜ → ˜ . Therefore, if ⇑C, ⇓⇔ ↭coh (˜, ˜ ) then ⇑C, ⇓⇔ ▽ (, ) ↭coh (˜, ˜ ).
Since every re!nement-based revision operator satis!es coh-success and coh-inclusion, it cannot satisfy conc-success and conc-inclusion, showing that maintaining conclusiveness is challenging for re!nement-based revision operators (see [18] for details on the hardness of the validity change). Proposition 4. Every re#nement-based revision operator cannot satisfy conc-success. Proof. Let ⇑C, ⇓⇔ be a case model with non-empty case set C (i.e., C ⇐= △ ). Consider an outcomeinference argument (, ) such that ⇑C, ⇓⇔ ↭coh (, ), and (, ↑) be an outcome-inference argument such that ↑ ⇐= . Since ⇑C, ⇓⇔ ↭coh (, ), there is a case ⇒ C that (, ) has a grounding in. Let ▽ be a re!nement-based revision operator and ⇑C↑, ⇓ ↑⇔ = ⇑C, ⇓⇔ ▽ (, ↑). Then, there is ↑ ⇒ C↑ such that ↑ ↘ (according to De!nition 3) and hence (, ) has a grounding in ↑, ↑ ↘ , and (, ↑) has no grounding in ↑. Thus, ⇑C, ⇓⇔ ▽ (, ↑) ⇐↭conc (, ↑). Therefore, no re!nement-based operator can satisfy conc-success in this setting.
Proposition 5. Every re#nement-based revision operator cannot satisfy conc-inclusion. Proof. Let (, ) be an outcome-inference argument and (˜, ˜ ) be an outcome-inference argument with ↘ ˜ and ⇐= ˜ . Consider a case model ⇑C, ⇓⇔ such that ⇑C, ⇓⇔ ↭conc (˜, ˜ ). Let ▽ be a re!nement-based revision operator and (C↑, ⇓ ↑) = ⇑C, ⇓⇔ ▽ (, ). Since ▽ satis!es coh-success and ↘ ˜, there is ↑ ⇒ C↑ such that (, ) has a grounding in ↑, ↑ ↘ ˜, and (˜, ˜ ) has no grounding in ↑. Thus, ⇑C, ⇓⇔ ↭conc (˜, ˜ ) but ⇑C, ⇓⇔ ▽ (, ) ⇐↭conc (˜, ˜ ). Therefore, no re!nement-based operator can satisfy conc-inclusion in this setting.
Next, we de!ne a sub-family of re!nement-based revision operators, called single revision operators, which add at most one case to an original case model so that a revision argument has a grounding in the added case. To simplify the de!nition, we assume that, in a case model ⇑C, ⇓⇔ , each case i ⇒ C contains a unique proposition pi not occurring in arguments such that pi → pj ↘ ↗ for two distinct cases i, j ⇒ C (i.e. i ⇐= j ). This pi works as a naming proposition to ensure the requirement of mutual incompatibility in case models. We then say a single revision operator is simple if the added case is minimally constructed from the revision argument. Speci!cally, we de!ne two simple revision operators. One is a top revision operator, where the new case is added to the new highest layer. The second is a bottom revision operator, where the new case is added to the new lowest layer. The formal de!nition is as follows.
De!nition 5 (Single, top, and bottom revision operator). A revision operator ▽ is single i! • if ⇑C, ⇓⇔ ↭coh (, • if ⇑C, ⇓⇔ ⇐ ↭coh (, ), ⇑C, ⇓⇔ ▽ ), ⇑C, ⇓⇔ ▽ (, (, ) = ⇑C, ⇓⇔ ; and ) = ⇑C ∈ { n}, ⇓ ↑⇔ where (, ) has a grounding in n and the preference ⇓ is preserved in ⇓ ↑, meaning that for every case , ⇒ C, ⇓ i! ⇓ ↑ .
De!nition 6 (Simple, top, and bottom revision operator). A single revision operator ▽ is simple i! the new case n in De#nition 5 satis#es the following conditions: • if is an outcome (i.e., ⇒ O ), n = pn → → ; and • if is nil, n = pn → , where pn is a new naming proposition. A top revision operator ↫ is a simple revision operator such that ′ ⇒ C : n >↑ and a bottom revision operator ↬ is a simple revision operator such that ′ ⇒ C : > ↑ n.
Intuitively, the top revision operator gives preference to new cases, whereas the bottom one gives preference to existing cases. Example 2 demonstrates examples of revisions using the bottom revision operator.
Example 2. Let {p0, p1, p2, p3} be a mutually incompatible set of naming propositions. Consider an outcome domain O = {fly, ¬fly} and a case model ⇑C, ⇓⇔ = {p0 → ¬ big → ¬ fly} > {p1 → wing → fly} > {p2 → big → ¬ fly} (similar to Example 1 but with naming propositions).
• ⇑C, ⇓⇔ ↫ (wing → big, ¬fly) = {p3 → wing → big → ¬ fly} > {p0 → ¬ big → ¬ fly} > {p1 → wing → fly} > {p2 → big → ¬ fly}. Since the top revision operator gives preference to the new case, the revision does not preserve the presumptive validity of some arguments, such as (↔, ¬fly). • ⇑C, ⇓⇔ ↬(wing→ big, ¬fly) = {p0→¬ big→¬ fly} > {p1→ wing→ fly} > {p2→ big→¬ fly} > {p3 → wing → big → ¬ fly}. Since the bottom revision operator gives preference to the existing cases, the revision preserves the validity of any presumptively valid arguments, such as (↔, ¬fly).
Proof. Let ⇑C, ⇓⇔ be a case model, (, ) be an outcome-inference argument, ▽ be a single revision operator, and ⇑C↑, ⇓ ↑⇔ = ⇑C, ⇓⇔ ▽ (, ). From De!nition 6, C ≃ C↑. Therefore, for every ⇒ C, ⇒ C↑ logically implies itself and hence a single revision operator is re!nement-based.
Proof. Let ⇑C, ⇓⇔ be a case model, (, ) be an outcome-inference argument, ▽ be a single revision operator, and ⇑C↑, ⇓ ↑⇔ = ⇑C, ⇓⇔ ▽ (, ). For every outcome-inference argument (˜, ˜ ) such that ⇐= ˜ , if ⇑C, ⇓⇔ ⇐ ↭coh (˜, ˜ ) and hence there is no ⇒ C that (˜, ˜ ) has a grounding in, then there is no ↑ ⇒ C↑ such that (˜, ˜ ) has a grounding in either because n is only a case that is added and (, ) has a grounding in n. Therefore, for every outcome-inference argument (˜, ˜ ) such that ⇐= ˜ , if ⇑C, ⇓⇔ ⇐ ↭coh (˜, ˜ ) then ⇑C, ⇓⇔ ▽ (, ) ⇐↭coh (˜, ˜ ).
Proof. Let ⇑C, ⇓⇔ be a case model, (, ) be an outcome-inference argument, and ▽ be a simple revision operator. Since it is trivial in the case ⇑C, ⇓⇔ ↭coh (, ) then we prove in the case ⇑C, ⇓⇔ ⇐ ↭coh (, ). From De!nition 6, if ⇒ O , consider n = pn → → and n↑ = p↑n → ↑ → ; and if = nil, consider n = pn → and n↑ = p↑n → ↑. Then, every outcome-inference argument (˜, ˜ ) that has a grounding in n, also has a grounding in n↑, and vice versa (naming propositions do not occur in arguments). Therefore, ⇑C, ⇓⇔ ▽ (, ) ↭ (˜, ˜ ) i" ⇑C, ⇓⇔ ▽ ( ↑, ) ↭ (˜, ˜ ) for every ⇒ { coh, pres, conc}.
Proof. Let ⇑C, ⇓⇔ be a case model and (, ) be an outcome-inference argument. From De!nition 6, ⇑C, ⇓⇔ ↫ (, ) = ⇑C ∈ { n}, ⇓ ↑⇔ and ′ ⇒ C : n >↑ . Then, ′ ⇒ C ∈ { n} : if ↘ then n ⇓ ↑ . Therefore, ⇑C, ⇓⇔ ↫ (, ) ↭pres (, ).
Proof. Let ⇑C, ⇓⇔ be a case model and (, ) be an outcome-inference argument. From De!nition 6, ⇑C, ⇓⇔ ↬(, ) = ⇑C ∈ { n}, ⇓ ↑⇔ such that ′ ⇒ C : > ↑ n. Then, for every outcome-inference argument (˜, ˜ ), if ⇑C, ⇓⇔ ↭pres (˜, ˜ ) then there is ⇒ C that (˜, ˜ ) has a grounding in, and for every ↑ ⇒ C if ↑ ↘ then ⇓ ↑. Since > ↑ n and > ↑ ↑, for every ↑↑ ⇒ C ∈ { n} if ↑↑ ↘ then ⇓ ↑↑. Therefore, ⇑C, ⇓⇔ ↬(, ) ↭pres (˜, ˜ ).
Interestingly, if a revision argument is novel, every simple revision operator satis!es conc-success (and hence pres-success).
Proposition 11. If a revision argument is novel with respect to the original case model, every simple revision operator satis#es conc-success.
Proof. Let ⇑C, ⇓⇔ be a case model, (, ) be a novel argument with respect to ⇑C, ⇓⇔ , and ▽ be a simple revision operator. From De!nition 1 and De!nition 6, ⇑C, ⇓⇔ ▽ (, ) = ⇑C ∈ { n}, ⇓ ↑⇔ and for every ⇒ C, ⇐↘ . Therefore, there is only n ⇒ C ∈ { n} such that n ↘ and (, ) has a grounding in n. Therefore, ⇑C, ⇓⇔ ▽ (, ) ↭conc (, ).
Consequently, if a revision argument is novel with respect to the original case model, the bottom revision operator becomes satisfying all the properties de!ned except conc-inclusion. We then de!ne conclusively adherent evaluation as follows.
De!nition 7 (Conclusively adherent evaluation). Let ⇑C, ⇓⇔ be a case model and (, ) be an outcomeinference argument. (, ) is conclusively adherent with respect to ⇑C, ⇓⇔ , denoted as ⇑C, ⇓⇔ ↭conc↓ ad (, ), i! for every conclusive outcome-inference argument ( ↑, ↑) with ↘ ↑, ↑ = . Example 3. Continuing from Example 2, let O = {fly, ¬fly} and ⇑C, ⇓⇔ = {p0 → ¬ big → ¬ fly} > {p1 → wing → fly} > {p2 → big → ¬ fly}. Examples of conclusively adherent evaluations are as follows: • ⇑C, ⇓⇔ ↭conc↓ ad (wing → australia, fly) because its conclusion is consistent with (wing, fly), which is the only conclusive outcome-inference argument with the premise that can be implied from wing → australia. • ⇑C, ⇓⇔ ⇐ ↭conc↓ ad (wing→ big, fly) because its conclusion is not consistent with (big, ¬fly), which is a conclusive outcome-inference argument with the premise that can be implied from wing → big.
Following De!nition 7, it is impossible to have two conclusively adherent arguments with the same premise but di"erent conclusions. Interestingly, conclusively adherent arguments can be either coherent arguments or novel arguments. For example, (wing → australia, fly) in Example 3 demonstrates a conclusively adherent argument which is novel. Following the proof in Proposition 5, if a revision argument is conclusively adherent, the bottom revision operator becomes satisfying conc-inclusion, making it satisfy all properties de!ned in De!nition 2.
Corollary 1. If a revision argument is conclusively adherent with respect to the original case model, the bottom revision operator (↬) satis#es conc-inclusion.
In this section, we apply conclusively adherent evaluation studied in the previous section to understand
abstract argumentation for case-based reasoning (AA-CBR) [
A case base ! ≃ 2F ↘ O is a !nite set of situation-outcome pairs representing precedents. In AA-CBR, it assumes that a case base ! is default-consistent (that is, (△ , ¬od) ⇒ / !) and outcome-consistent (that is, (X, o) ⇒ ! implies (X, o↑) ⇒ / ! for any o↑ ⇐= o).
A case-based evaluation |↖ is a relation from a case base to a situation-outcome pair. Let ! be a case base and (X, o) be a situation-outcome pair. ! |↖ (X, o) intuitively determines whether a situationoutcome pair (X, o) is valid with respect to !. Analogous to what we have de!ned, if every (X↑, o↑) ⇒ !, X ⊋ X↑, then we say (X, o) is a novel situation-outcome pair with respect to !. As it is named, an AA-CBR evaluation is based on abstract argumentation framework [7], recapped as follows. De!nition 8 (abstract argumentation framework). An abstract argumentation framework (AA framework) is a pair (A, ≿). Each element of A represents an argument and ≿ is a binary relation over A. For x, y ⇒ A , if x ≿ y then we say x attacks y. For a set of arguments E ≃ A and an argument x ⇒ A , E defends x if, for every y ⇒ A that attacks x, there is an argument z ⇒ E attacks y. Then, the grounded extension of (A, ≿) can be constructed inductively as G = [︃ i↗ 0 Gi, where G0 is the set of unattacked arguments, and for i ⇓ 0, Gi+1 is the set of arguments that Gi defends.
The AA-CBR evaluation is determined by the default argument (△ , od) and relevant precedents,
forming an AA-framework in which precedents with di"erent outcomes and concisely speci!c situations
attack the others. We denote the AA-CBR evaluation as |↖ aacbr, formally de!ned as follows.
De!nition 9 (AA-CBR evaluation). An AA framework corresponding to a case base !, a default outcome
od, and a situation N is (A, ≿) satisfying the following conditions [
If (△ , od) is in the grounded extension of (A, ≿) then ! |↖ aacbr (N, od). Otherwise, ! |↖ aacbr (N, ¬od).
Hence, the AA-CBR evaluation is consistent and complete, meaning that for every situation X, either ! |↖ aacbr (X, od) or ! |↖ aacbr (X, ¬od).
Example 4. Let O = {fly, ¬fly} where ¬fly is a default outcome and a case base ! = {({wing, thin}, fly), ({big}, ¬fly)}. Figure 1 depicts the AA framework corresponding to !, ¬fly, and {wing, big}. Consequently, ! |↖ aacbr ({wing, big}, ¬fly) as (△ , ¬fly) is in the grounded extension of the AA framework. anTyhAeAp-rCeBviRoucasswe obrakse[s13in]tsohcoawsse tmhoatdewlse. cGainvetrnanasclaastee (△ , ¬fly) base ! and a case-based evaluation |↖ , a translated case model ⇑C, ⇓⇔ is de!ned as any case model that evalu- ({wing, thin}, fly) ates every non-novel argument as presumptively valid i" the corresponding situation-outcome pair is valid with ({wing, big}, ?) respect to !. That is, for every non-novel situationoutcome pair (X, o) with respect to !, ! |↖ (X, o) i" Figure 1: Corresponding AA framework ⇑C, ⇓⇔ ↭pres (con(X), o).
Again, we cannot evaluate novel arguments in the translated case model using ↭pres. For instance, given ! in Example 4, since ({wing, australia}, ¬fly) is a novel situation-outcome pair with respect to !, the argument (wing → australia, ¬fly) can be novel with respect to the translated case model and hence it cannot be presumptively valid. To evaluate novel arguments, we describe a translated case model discussed in [12, 13], as a bottom revision sequence in which each argument in a revision sequence is novel with respect to the prior revised case model. Proposition 12 (adapted from [12, 13]). Given a AA-CBR case base !, there exists a translated case model ⇑C, ⇓⇔ which can be written as a revision sequence: ⇑C, ⇓⇔ = ⇑△ , △⇔ ↬(con(X1), o1)↬ . . . ↬(con(Xn), on) such that if Xi ⫅̸ Xj then i < j for every 1 ∞ i, j ∞ n.
A revision sequence in Proposition 12 can be obtained by enumerating all non-novel situationoutcome pairs that are valid with respect to the case base. Example 5 demonstrates a translated case model from a bottom revision sequence, given the case base ! in Example 4.
Example 5. Given ! in Example 4, ⇑C, ⇓⇔ = ⇑△ , △⇔ ↬(↔, ¬fly)↬(wing, ¬fly)↬(thin, ¬fly)↬ (wing → thin, fly)↬(big, ¬fly), which is equivalent to {p0 → ¬ fly} > {p1 → wing → ¬ fly} > {p2 → thin → ¬ fly} > {p3 → wing → thin → fly} > {p4 → big → ¬ fly}, is a translated case model.
In Example 5, each revision argument is also novel with respect to the prior revised case model in the sequence. Now, we consider whether each revision argument is conclusively adherent with respect to the prior revised case model as follows.
• ⇑△ , △⇔ ↭conc↓ ad (↔, ¬fly) • ⇑△ , △⇔ ↬(↔, ¬fly) ↭conc↓ ad (wing, ¬fly) • ⇑△ , △⇔ ↬(↔, ¬fly)↬(wing, ¬fly) ↭conc↓ ad (thin, ¬fly) • ⇑△ , △⇔ ↬(↔, ¬fly)↬(wing, ¬fly)↬(thin, ¬fly) ⇐↭conc↓ ad (wing → thin, fly) • ⇑△ , △⇔ ↬(↔, ¬fly)↬(wing, ¬fly)↬(thin, ¬fly)↬(wing → thin, fly) ↭conc↓ ad (big, ¬fly) We have that if a novel revision argument is not conclusively adherent with respect to the prior revised case model, such as (wing → thin, fly) in the example, then it contributes a precedent attack in AA-CBR in the corresponding AA-framework.
Proposition 13. Let ! be an AA-CBR case base with default outcome od and ⇑C, ⇓⇔ be a translated case model. For every novel situation-outcome pair (X, o) with respect to !, ⇑C, ⇓⇔ ⇐ ↭conc↓ ad (con(X), o) i! (X, o) attacks an argument in the AA-framework corresponding to ! ∈ {(X, o)}, od and any situation N . Proof. (Left-to-right) If ⇑C, ⇓⇔ ⇐ ↭conc↓ ad (con(X), o), then there exists a conclusive argument (, ¬o) with con(X) ↘ . It implies that (1) there is (Y, ¬o) ⇒ ! ∈ {(△ , od)} such that (con(Y ), ¬o) is conclusive with respect to ⇑C, ⇓⇔ and ↘ con(Y ) (2) there is no (Z, o) ⇒ ! ∈ {(△ , od)} such that (con(Z), o) is coherent and con(Z) ↘ ; otherwise, (con(Y ), ¬o) cannot be conclusive. It implies that there is no (Z, o) ⇒ ! ∈ {(△ , od)} such that Y ⫅̸ Z ⫅̸ X. Therefore, (X, o) attacks (Y, ¬o) in the corresponding AA framework.
(Right-to-left) If (X, o) attacks (Y, ¬o) ⇒ ! ∈ {(△ , od)} in the corresponding AA framework, then Y ⫅̸ X and there is no (Z, o) ⇒ ! ∈ {(△ , od)} such that Y ⫅̸ Z ⫅̸ X. Thus, there is a conclusive argument (, ¬o) such that con(X) ↘ and ↘ con(Y ). Therefore, ⇑C, ⇓⇔ ⇐ ↭conc↓ ad (con(X), o).
On the other hand, if a novel revision argument is conclusively adherent, then the corresponding situation-outcome pair is valid with respect to the case base.
Proposition 14. Let ! be an AA-CBR case base and ⇑C, ⇓⇔ be a translated case model. For every novel
situation-outcome pair (X, o) with respect to !, if ⇑C, ⇓⇔ ↭conc↓ ad (con(X), o) then ! |↖ aacbr (X, o).
Proof. If ⇑C, ⇓⇔ ↭conc↓ ad (con(X), o), then for every conclusive argument (, ) with con(X) ↘ ,
= o. It implies that (1) there is (Xn, o) ⇒ ! ∈ {(△ , od)} with ↘ con(Xn) and hence Xn ≃ X (2)
there does not exist (Y, ¬o) such that X ⫅̸ Y ; otherwise, such (, ) cannot be conclusive. Therefore,
all such (Xn, o) are nearest cases [
The converse of Proposition 14 is not true. That is, !|↖ aacbr(X, o) does not imply that ⇑C, ⇓⇔ ↭conc↓ ad (con(X), o). The reason is that, for any novel situation outcome pair (X, o), there are possibly two conclusive arguments with opposite conclusions (, o ) and ( ↑, ¬o) such that con(X) ↘ and con(X) ↘ ↑. For instance, from Example 5, we have !|↖ aacbr({wing, thin, big}, ¬fly) but the translated case model ⇑C, ⇓⇔ ⇐ ↭conc↓ ad (wing → thin → big, ¬fly) because there are two conclusive arguments with opposite conclusions, namely (wing → thin, fly) and (big, ¬fly).
The analysis in the previous section highlights two key observations. First, we showed that novel arguments which are not conclusively adherent are related to precedent attacks in AA-CBR. Second, we showed that novel arguments which are conclusively adherent are related to the AA-CBR evaluation. Interestingly, these observations are parallel to precedential constraint [11]. In precedential constraint, certain cases are constrained – i.e., the outcome must be decided in a speci!c way – otherwise, they cannot be consistently added as new precedents. In contrast, unconstrained cases can be consistently added as new precedents regardless of their outcome. This similarity suggests a promising direction for future work of translating precedential constraint case bases into case models through revision sequences.
Since AA-CBR employs argumentation to infer an outcome for every situation, it allows arguments that are not conclusively adherent to be considered valid. When such valid arguments are added as new precedents, they introduce new attacks that alter the grounded extensions of the corresponding AA frameworks. This observation is consistent with previous !ndings on AA-CBR [15], which show that the addition of valid arguments can change the validity of other arguments. Formally, it is possible that !|↖ aacbr(X1, o1) and !|↖ aacbr(X2, o2) do not imply ! ∈ {(X1, o1)}|↖ aacbr(X2, o2). This implies the same conclusion to [15] that AA-CBR lacks cautious monotonicity and does not satisfy all the postulates of iterated belief change.
Cumulative AA-CBR (cAA-CBR) [15] has been developed as a cautious monotonic version of AA-CBR that includes a new argument into its case base only if the argument is invalid with respect to the existing case base. Thus, translated case models from the cAA-CBR case bases can be described as revision sequences of novel arguments that are not conclusively adherent with respect to the preceding revised case model. This motivates further research to identify more evaluations for novel arguments in order to examine other CBR models, such as precedential constraint and cAA-CBR.
This paper investigated new evaluations in case models for novel arguments, which are not required to have a grounding in any existing cases. To facilitate this, we introduced revision operators for case models and proposed a set of associated properties –adapted from the AGM postulates – including -success, -inclusion, vacuity, and extensionality. We then systematically examined whether these properties are satis!ed by various types of revision operators.
A key contribution of this work is the introduction of the conclusively adherent evaluation. An outcome-inference argument is conclusively adherent i" every conclusive arguments with more general premises shares the same outcome. We showed that the bottom revision operator satis!es all the de!ned properties only if the revision argument is conclusively adherent. Furthermore, we investigated translated case models from abstract argumentation for case-based reasoning (AA-CBR) case bases as revision sequences of novel arguments. The investigation revealed two !ndings: 1. Novel arguments that are not conclusively adherent correspond to precedent attacks in AA-CBR.
This results from the presence of a conclusive argument with a di"ering outcome, triggering an attack in the corresponding AA framework. 2. Novel arguments that are conclusively adherent correspond to the AA-CBR evaluation. That is, if a novel argument is conclusively adherent with respect to the translated case model, then the corresponding situation-outcome pair is valid with respect to the AA-CBR case base.
Future research can explore how to generalise the translation of case bases into case models via revision sequences for other case-based reasoning models and explore suitable evaluations for them.
This work was supported by the “R&D Hub Aimed at Ensuring Transparency and Reliability of Generative AI Models” project of the Ministry of Education, Culture, Sports, Science and Technology and JSPS KAKENHI Grant Numbers, JP22H00543. GPP was partially funded by the ERC under the EU’s Horizon 2020 research and innovation programme, grant agreement No. 101020934. GPP conducted part of this work as a JSPS International Research Fellow, under the FY2023 JSPS Postdoctoral Fellowship for Research in Japan (Short-term), at the National Institute of Informatics, Tokyo, Japan. We would like to thank all the reviewers for their valuable and insightful comments.
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