=Paper= {{Paper |id=Vol-3193/paper6ASPOCP |storemode=property |title=Assumable Answer Set Programming |pdfUrl=https://ceur-ws.org/Vol-3193/paper6ASPOCP.pdf |volume=Vol-3193 |authors=Zhizheng Zhang |dblpUrl=https://dblp.org/rec/conf/iclp/Zhang22 }} ==Assumable Answer Set Programming== https://ceur-ws.org/Vol-3193/paper6ASPOCP.pdf 
Assumable Answer Set Programming
Zhizheng Zhang
School of Computer Science and Engineering, Southeast University, No.2 Dongnandaxue Rd, Nanjing, 211198, China


                                       Abstract
                                       For modeling the assumption-based intelligent agents who make assumptions and use them to construct their
                                       belief sets, this paper proposes a logic programming language AASP (Assumable Answer Set Programming)
                                       by extending ASP (Answer Set Programming). Rational principles of assumption-based reasoning are given
                                       to define the semantics of the AASP program. The relation of AASP to some existing default formalism and
                                       extensions of ASP implies that AASP can be used to model and solve some interesting problems with incomplete
                                       information in the framework of assumption-based reasoning.

                                       Keywords
                                       assumption based reasoning, answer set, logic programming




1. Introduction
In the case of incomplete information, one framework of the mental behavior for an intelligent agent
is to make assumptions, and use them to construct belief set through deductive reasoning [1]. Var-
ious approaches for assumption-based or hypothetical reasoning have been proposed in the past.
Examples include Assumption-Based Truth Maintenance System introduced in [2], [3], [4], and [5],
Probabilistic Assumption-Based Model and language proposed in [6] and [7], Poole’s Default Theory
in [8] and [9] etc.. Some efforts are made to explore the way of identifying assumptions in reasoning,
in which assumptions are not given explicitly. For example, [10] explores the derivation of assump-
tions to explain observed events, [11] and [12] present an approach to hypothetical planning involves
generating assumptions about actions that can not be derived from the knowledge-base etc.. Besides,
many studies show that assumption-based reasoning is closely related to the topics like argumenta-
tion [13], action reasoning [14], planning [15], contextual reasoning [1], defeasible reasoning [16],
logic programming [17], default reasoning [18] [19] etc..
   Presently, Answer Set Programming (ASP) has become an increasingly popular tool for declarative
programming, knowledge representation, and nonmonotonic reasoning [20] [21] [22] [23]. Answer
sets are widely accepted as a rational reasoner’s views in an environment described by a logic program
that may contain incomplete information. The increasing success of the answer-set based reasoning
inspires us to explore how to use it in both assumption making and belief building of the framework
of the assumption-based reasoning.
   This paper presents a new logic programming formalism for assumption-based reasoning by com-
bining the idea of ASP and the framework of assumption-based reasoning. Specifically, we propose
Assumable Answer Set Programming (AASP) language, an extension of ASP, that can be used to de-
sign the assumption-based intelligent agent whose mental behaviors of assumption making and belief
building are defined on the answer-set based reasoning.

ICLP Workshops 2022, July 31, 2022, Haifa, Israel
" seu_zzz@seu.edu.cn (Z. Zhang)
 0000-0001-9851-6184 (Z. Zhang)
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   The rest of the paper will introduce AASP formally and is organized as follows. In the next sec-
tion, ASP, two extensions of ASP, and constrained default logic are briefly introduced as background
knowledge for the self-contained requirement and as objects compared with AASP. In section 3, we
introduce syntax, semantics, and some properties of the AASP program. In section 4, the relation
between AASP and some other formalism is given. We conclude in section 5 with some further dis-
cussion.
   We will restrict our discussion in this paper to propositional programs. However, as usual in answer
set programming, we admit rule schemata containing variables bearing in mind that these schemata
are just convenient representations for the set of their ground instances.


2. Preliminaries
2.1. Answer Set Program
Follow the description of ASP from [22]. A regular ASP program is a collection of rules of the form
                              𝑙1 π‘œπ‘Ÿ ... π‘œπ‘Ÿ π‘™π‘˜ ← π‘™π‘˜+1 , ..., π‘™π‘š , π‘›π‘œπ‘‘ π‘™π‘š+1 , ..., π‘›π‘œπ‘‘ 𝑙𝑛
where the 𝑙s are literals, π‘›π‘œπ‘‘ denotes negation as failure, π‘œπ‘Ÿ is epistemic disjunction. The left-hand
side of a rule is called the head and the right-hand side is called the body. A rule is called a fact if its
body is empty (equivalent to containing only a literal ⊀) and its head contains only one literal, and a
rule is called a denial if its head is empty (equivalent to containing only a literal βŠ₯). An answer set
𝑋 of a program Ξ  if it is the minimal set (in the sense of set inclusion) that satisfies Π𝑋 , where Π𝑋 is
G-L reduct of Ξ  with respect to 𝑋 achieved by two rules:
    β€’ delete all rules whose bodies are not satisfied by 𝑋 .
    β€’ delete π‘›π‘œπ‘‘ 𝑙 in the bodies of the remaining rules.
𝐴𝑆(Ξ ) is used to denote the set of all answer sets of an ASP program Ξ .

2.1.1. CR-Prolog
CR-Prolog extends the regular ASP with a purpose of representing indirect exceptions to defaults
([22]). Follow the description of CR-Prolog from [24] and [22], a CR-Prolog program is a collection of
regular ASP rules or consistency-restoring rules (CR-rule) of the form
                                               +
                              𝑙1 π‘œπ‘Ÿ ... π‘œπ‘Ÿ π‘™π‘˜ ← π‘™π‘˜+1 , ..., π‘™π‘š , π‘›π‘œπ‘‘ π‘™π‘š+1 , ..., π‘›π‘œπ‘‘ 𝑙𝑛
where the 𝑙s are literals, π‘›π‘œπ‘‘ denotes negation as failure, π‘œπ‘Ÿ is epistemic disjunction. And, ≀ is a partial
order defined on sets of CR-rules in the program. This partial order is often referred to as a preference
relation based on the set-theoretic inclusion (𝑅1 ≀ 𝑅2 iff 𝑅1 βŠ‚ 𝑅2 ) or defined by the cardinality of the
corresponding sets (𝑅1 ≀ 𝑅2 iff |𝑅1 | βŠ‚ |𝑅2 |).
   The set of regular ASP rules of a CR-Prolog program Ξ© is denoted by Ξ©π‘Ÿ ; By 𝛼(π‘Ÿ) we denote a regular
                                                                     +
rule obtained from a consistency-restoring rule π‘Ÿ by replacing ← by ←, and 𝛼 can be expanded in a
standard way to a set 𝑅 of CR-rules, i.e., 𝛼(𝑅) = {𝛼(π‘Ÿ)|π‘Ÿ ∈ 𝑅}.
   A minimal (with respect to the preference relation of the program) collection 𝑅 of CR-rules of Ξ©
such that Ξ©π‘Ÿ βˆͺ 𝑅 is consistent (i.e., has an answer set) is called an abductive support of Ξ©. Then, a set
𝑀 is called an answer set of Ξ© if it is an answer set of a regular program Ξ© βˆͺ 𝛼(𝑅) for some abductive
support 𝑅 of Ξ©. We use π΄π‘†βŠ‚ (Ξ©) and 𝐴𝑆♯ (Ξ©) to denote the collection of answer sets of Ξ© w.r.t. the
preference relation on set-theoretic inclusion and the cardinality respectively.
2.1.2. Abductive ASP
We consider two versions of the abductive answer set programs. In [25], an abductive logic program
(ALP93) Ξ“ is defined as a pair < 𝑃,  > where 𝑃 is a regular ASP program and  is a set of literals
from the language of 𝑃 called abducibles. 𝐺 a ground literal represents a positive observation. A set
𝑆 is a belief set of Ξ“ with respect to 𝐸 if 𝑆 is an answer set of 𝑃 βˆͺ 𝐸 where 𝐸 βŠ† . 𝑆 is called  minimal
if there is no belief set 𝑇 of Ξ“ such that 𝑇 ∩  βŠ‚ 𝑆 ∩ . A set 𝐸 is an explanation of 𝐺 with respect
to Ξ“ if 𝐺 is true in a belief set 𝑆 of Ξ“ such that 𝐸 = 𝑆 ∩ 𝐴. An explanation 𝐸 of 𝐺 is minimal if no
𝐸 β€² βŠ‚ 𝐸 is an explanation of 𝐺. 𝐸 is a minimal explanation of 𝐺 iff 𝑆 is an  minimal belief set of
< 𝑃 βˆͺ {← π‘›π‘œπ‘‘ 𝐺},  >.
   In [26], an abductive logic program (ALP95) Ξ“ is defined as a pair < 𝑃,  > where both 𝑃 and 
are regular ASP programs. 𝐺 a ground literal represents a positive observation. A pair (𝐸, 𝐹 ) is a
explanation of 𝐺 with respect to Ξ“ if
    1. 𝐺 βŠ† 𝑀 for βˆ€π‘€ ∈ 𝐴𝑆((𝑃 βˆ’ 𝐹 ) βˆͺ 𝐸)
    2. (𝑃 βˆ’ 𝐹 ) βˆͺ 𝐸) is consistent
    3. 𝐸 βŠ† ( βˆ’ 𝑃) and 𝐹 βŠ†  ∩ 𝑃
On the other hand, a pair (𝐸, 𝐹 ) is an anti-explanation of 𝐺 with respect to Ξ“ if
  1. 𝐺 ⊈ 𝑀 for βˆ€π‘€ ∈ 𝐴𝑆((𝑃 βˆ’ 𝐹 ) βˆͺ 𝐸)
  2. (𝑃 βˆ’ 𝐹 ) βˆͺ 𝐸) is consistent
  3. 𝐸 βŠ† ( βˆ’ 𝑃) and 𝐹 βŠ†  ∩ 𝑃
  An (anti-)explanation (𝐸, 𝐹 ) of 𝐺 is called minimal if for any (anti-)explanation (𝐸 β€² , 𝐹 β€² ) of 𝐺, 𝐸 β€² βŠ† 𝐸
and 𝐹 β€² βŠ† 𝐹 imply 𝐸 β€² = 𝐸 and 𝐹 β€² = 𝐹 .

2.2. Constrained Default Logic
As defined by Reiter in [27], a default theory Ξ” is a pair (𝐷, π‘Š ) where π‘Š is a set of first-order formulas
and 𝐷 is a set of defaults of the form
                                                 𝛼 ∢ M𝛽
                                                     𝛾
where 𝛼, 𝛽, and 𝛾 are quantifier-free first order formulas. 𝛼 is called the prerequisite, 𝛽 the justifi-
cations, and 𝛾 the consequent. Formally, For any set 𝑆 of first order logic formulas, Let Ξ“(𝑆) be the
smallest set of formulas such that
    β€’ π‘Š βŠ† Ξ“(𝑆)
    β€’ Ξ“(𝑆) = 𝑇 β„Ž(Ξ“(𝑆))
         𝛼 ∢ M𝛽
    β€’ if          ∈ 𝐷 and 𝛼 ∈ Ξ“(𝑆) and ¬𝛽 βˆ‰ 𝑆 then 𝛾 ∈ Ξ“(𝑆)
            𝛾
A set of formulas 𝐸 is an extension of Ξ” iff 𝐸 = Ξ“(𝐸)
   [28] proposes a variant of the extension of Reiter’s default logic, for any set 𝑇 of first order logic
formulas, let Ξ₯(𝑇 ) be the pair of smallest sets (𝑆 β€² , 𝑇 β€² ) of formulas such that
    β€’ π‘Š βŠ† 𝑆′ βŠ† 𝑇 β€²
    β€’ 𝑆 β€² = 𝑇 β„Ž(𝑆 β€² ) and 𝑇 β€² = 𝑇 β„Ž(𝑇 β€² )
                𝛼 ∢ M𝛽
    β€’ For any           ∈ 𝐷, if 𝛼 ∈ 𝑆 β€² and 𝑇 βˆͺ {𝛽, 𝛾 } ⊬ βŠ₯ then 𝛾 ∈ 𝑆 β€² and 𝛽 ∧ 𝛾 ∈ 𝑇 β€² .
                   𝛾
A pair of sets of formulas (𝐸, 𝐢) is an constrained extension of Ξ” iff Ξ₯(𝐢) = (𝐸, 𝐢).
3. Assumable Answer Set Program
3.1. Syntax
An AASP rule π‘Ÿ is written as
                                      𝑙1 π‘œπ‘Ÿ ... π‘œπ‘Ÿ π‘™π‘˜ ← 𝑒1 , ..., π‘’π‘š ∢ π‘™π‘˜+1 , ..., 𝑙𝑛 .
where the 𝑙s are literals in propositional logic language and 𝑒s are literals possibly preceded by nega-
tion as failure π‘›π‘œπ‘‘, : is called assumption operator. β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) is used to denote the left-hand side of
π‘Ÿ where π‘œπ‘Ÿ is an epistemic disjunction, and β„Žπ‘’π‘Žπ‘‘π‘™π‘–π‘‘(π‘Ÿ) is used to denote the set {𝑙1 , ..., π‘™π‘˜ } of literals
in the head of π‘Ÿ. π‘π‘œπ‘‘π‘¦(π‘Ÿ) is used to denote the right-hand side of π‘Ÿ, and π‘π‘œπ‘‘π‘¦π‘™π‘–π‘‘(π‘Ÿ) is used to de-
note the set {𝑒1 , ..., π‘’π‘š , π‘™π‘˜+1 , ..., 𝑙𝑛 } of literals in the body of π‘Ÿ. 𝑒1 , ..., π‘’π‘š is called the precondition of π‘Ÿ
and denoted by π‘π‘π‘œπ‘‘π‘¦(π‘Ÿ). Let π‘π‘π‘œπ‘‘π‘¦π‘™π‘–π‘‘(π‘Ÿ) denote the set {𝑒1 , ..., π‘’π‘š } of literals in the precondition of
π‘Ÿ. π‘™π‘˜+1 , ..., 𝑙𝑛 is called the assumption of π‘Ÿ and denoted by π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ). Let π‘Žπ‘π‘œπ‘‘π‘¦π‘™π‘–π‘‘(π‘Ÿ) denote the set
{π‘™π‘˜+1 , ..., 𝑙𝑛 } of literals in the assumption of π‘Ÿ. As in usual logic programming, a rule is called a fact if
its body is empty (equival to containing only a literal ⊀) and its head contains only one literal, and
a rule is called a constraint if its head is empty (equival to containing only a literal βŠ₯). An AASP
rule is called assumption-free if its assumption is empty, otherwise it is called an assumption rule.
Sometimes, we use β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) ← π‘π‘œπ‘‘π‘¦(π‘Ÿ) or β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) ← π‘π‘π‘œπ‘‘π‘¦(π‘Ÿ) ∢ π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ) to denote π‘Ÿ. 𝑙𝑖𝑑(π‘Ÿ) is used
to denote the set of propositional logic literals appearing in π‘Ÿ. π‘Ÿ can be read as On the assumption of
π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ), β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) is believed if π‘π‘π‘œπ‘‘π‘¦(π‘Ÿ) is believed or β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) is believed if π‘π‘π‘œπ‘‘π‘¦(π‘Ÿ) is believed and
it is consistent to assume π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ).
    An AASP program is a collection of AASP rules. 𝑙𝑖𝑑(Ξ ) is used to denote the set of propositional
logic literals appearing in Ξ . For convenient description, sometimes an AASP program Ξ  is written
as a pair (Π𝐷 , Ξ π‘Š ) in which Ξ π‘Š is the set of assumption-free AASP rules in Ξ  and Π𝐷 is the set of
assumption rules in Ξ .
    It is clear that an assumption-free AASP rule is a regular ASP rule, and an assumption-free AASP
program is an ASP program that can be dealt with by ASP solvers like DLV ([29]), CLASP ([30]).

3.2. Semantics
3.2.1. Satisfiability
Let 𝑀 be a collection of literals, π‘Ÿ be an AASP rule, the notion of satisfiability denoted by ⊧AASP is
defined below.
    β€’ 𝑀 ⊧AASP 𝑙 if 𝑙 ∈ 𝑀
    β€’ 𝑀 ⊧AASP π‘›π‘œπ‘‘ 𝑙 if 𝑙 βˆ‰ 𝑀
    β€’ 𝑀 ⊧AASP β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) if βˆƒπ‘™ ∈ β„Žπ‘’π‘Žπ‘‘π‘™π‘–π‘‘(π‘Ÿ), 𝑀 ⊧AASP 𝑙
    β€’ 𝑀 ⊧AASP π‘π‘π‘œπ‘‘π‘¦(π‘Ÿ) if βˆ€π‘’ ∈ π‘π‘π‘œπ‘‘π‘¦π‘™π‘–π‘‘(π‘Ÿ), 𝑀 ⊧AASP 𝑒
    β€’ 𝑀 ⊧AASP π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ) if βˆ€π‘’ ∈ π‘Žπ‘π‘œπ‘‘π‘¦π‘™π‘–π‘‘(π‘Ÿ), 𝑀 ⊧AASP 𝑒
    β€’ 𝑀 ⊧AASP π‘π‘œπ‘‘π‘¦(π‘Ÿ) if 𝑀 ⊧AASP π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ) and 𝑀 ⊧AASP π‘π‘π‘œπ‘‘π‘¦(π‘Ÿ).
    β€’ 𝑀 ⊧AASP π‘Ÿ if whenever 𝑀 ⊧AASP π‘π‘œπ‘‘π‘¦(π‘Ÿ), 𝑀 ⊧AASP β„Žπ‘’π‘Žπ‘‘(π‘Ÿ).
  We say 𝑀 is a model of an AASP program Ξ , denoted by 𝑀 ⊧AASP Ξ , if we have 𝑀 ⊧AASP π‘Ÿ for
βˆ€π‘Ÿ ∈ Ξ . A set 𝑀 of literals is inconsistent if it contains a literal 𝑙 and its contrary 𝑙̄ .
3.2.2. Assumable Answer Set
We first introduce the notion of Assumption Set of an AASP program that is viewed as the result of
assumption making in the framework of assumption-based reasoning.

Definition 1. Given an AASP program Ξ , an arbitrary set 𝐴 βŠ† 𝑙𝑖𝑑(Ξ ), 𝐴 is an assumption set of Ξ  if
and only if
                                                              ←
                                              𝐴 ∈ 𝐴𝑆 (Ξ (𝐴) βˆͺ 𝐴 )
where Π(𝐴) is a regular ASP program obtained by

                      Ξ (𝐴) = {β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) ← π‘π‘π‘œπ‘‘π‘¦(π‘Ÿ)|π‘Ÿ ∈ Ξ  and 𝐴 ⊧AASP π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ)}
    ←
and 𝐴 is used to denote the rules set {𝑙 ← |𝑙 ∈ 𝐴}.

𝐴𝑆𝑆(Ξ ) is used to denote the collection of all assumption sets of an AASP program Ξ .

Example 1. Consider Ξ 1 :
                                                  𝑝 β†βˆΆ π‘ž
𝐴1 = βˆ…, 𝐴2 = {𝑝}, 𝐴3 = {π‘ž}, and 𝐴4 = {𝑝, π‘ž}. We have

                                          Ξ 1(𝐴1 ) = βˆ…     Ξ (𝐴
                                                           1
                                                             2)
                                                                =βˆ…

                                      Π(𝐴
                                       1
                                         3)
                                            = {𝑝 ←}       Ξ (𝐴
                                                           1
                                                             4)
                                                                = {𝑝 ←}
Therefore,
                                          ←                    ←
                                 Π(𝐴
                                  1
                                    1)
                                       βˆͺ 𝐴1 = βˆ…         Ξ (𝐴
                                                         1
                                                           2)
                                                              βˆͺ 𝐴2 = {𝑝 ←}
             ←              ←
both Π(𝐴
      1
        3)
           βˆͺ 𝐴3 and Ξ (𝐴
                     1
                       4)
                          βˆͺ 𝐴4 are:
                                                     𝑝←
                                                     π‘žβ†
Then,
                                          ←                          ←
                           𝐴𝑆 (Ξ (𝐴
                                1
                                  1)
                                     βˆͺ 𝐴1 ) = {βˆ…}        𝐴𝑆 (Ξ 1(𝐴2 ) βˆͺ 𝐴2 ) = {{𝑝}}
                                      ←                               ←
                        𝐴𝑆 (Ξ (𝐴
                             1
                               3)
                                  βˆͺ 𝐴3 ) = {{𝑝, π‘ž}}       𝐴𝑆 (Ξ (𝐴
                                                               1
                                                                 4)
                                                                    βˆͺ 𝐴4 ) = {{𝑝, π‘ž}}
Thus, we have
                                                 ←                          ←
                              𝐴1 ∈ 𝐴𝑆 (Ξ (𝐴
                                        1
                                          1)
                                             βˆͺ 𝐴1 )       𝐴2 ∈ 𝐴𝑆 (Ξ (𝐴
                                                                    1
                                                                      2)
                                                                         βˆͺ 𝐴2 )
                                                 ←                          ←
                              𝐴3 βˆ‰ 𝐴𝑆 (Ξ (𝐴
                                        1
                                          3)
                                             βˆͺ 𝐴3 )       𝐴4 ∈ 𝐴𝑆 (Ξ (𝐴
                                                                    1
                                                                      4)
                                                                         βˆͺ 𝐴4 )
Hence, βˆ…, {𝑝}, and {π‘ž, 𝑝} are assumption sets of Ξ 1 , {π‘ž} is not.

Definition 2. Given an AASP program Ξ , an arbitrary set 𝑀 βŠ† 𝑙𝑖𝑑(Ξ ), 𝑀 is an assumable answer set of
Π if and only if there exists an assumption set 𝐴 of Π such that
   1. 𝑀 ∈ 𝐴𝑆(Ξ (𝐴) ), and
   2. 𝑀 βŠ† 𝐴
We say that 𝑀 is an assumable answer set of Ξ  on the assumption set 𝐴, and that (𝑀, 𝐴) is a view of Ξ .
   𝐴𝐴𝑆(Ξ ) is used to denote the collection of all assumable answer sets of an AASP program Ξ .
𝑉 𝐼 πΈπ‘Š (Ξ ) is used to denote the collection of all views of an AASP program Ξ .

Example 2. Continue Ξ 1 mentioned in the above example. Consider 𝑀1 = βˆ… and 𝑀2 = {𝑝}, obviously,
we have
                    𝐴𝑆(Ξ (𝐴1)
                        1 ) = {βˆ…}      𝐴𝑆(Ξ (𝐴 2)
                                            1 ) = {βˆ…}    𝐴𝑆(Ξ (𝐴  4)
                                                               1 ) = {𝑝}

Then,
                                       𝑀1 ∈ 𝐴𝑆(Ξ 1(𝐴1 ) ) and 𝑀1 βŠ† 𝐴1
                                       𝑀1 ∈ 𝐴𝑆(Ξ 1(𝐴2 ) ) and 𝑀1 βŠ† 𝐴2
                                       𝑀2 ∈ 𝐴𝑆(Ξ 1(𝐴4 ) ) and 𝑀2 βŠ† 𝐴4
Hence, βˆ… is an assumable answer set of Ξ 1 on 𝐴1 , βˆ… is also an assumable answer set of Ξ 1 on 𝐴2 , and {𝑝}
is an assumable answer set of Π1 on 𝐴4 .

The intuitions implied in the definitions are as the following principles based on the ASP-based rea-
soning:
   1. Rationality of Assumption Making. This principle tells that, in assuming a formula, the agent
      reasons and behaves as if it is a fact.
   2. Rationality of Belief Building on Assumptions. This principle tells that the agent’s beliefs are
      obtained by reasoning within the scope of assumptions.
   3. Consistency between Assumptions and Beliefs. This principle tells that an agent’s assumptions
      and beliefs must be consistent.
The notion of Assumption Set is defined by the first principle. By the second and the third principles,
the notion of Assumable Answer Set is defined.
   In the scenario of incomplete information, different assumption sets may be generated by a different
set of assumption rules in the program. Some of them are from more assumption rules and some of
them from less. Just as the example above shows, the assumption set βˆ… satisfies no assumption of
the rule in the program Ξ 1 , and {𝑝, π‘ž} satisfies the assumption of one rule in the program. Based
on this observation, we define several strategies of assumption making while keeping the principles
unchanged.

Definition 3. Given an AASP program Π, 𝐴 is an assumption set of Π,
   1. 𝐴 is a maxβŠ‚ assumption set of Ξ  if there is no assumption set 𝐴′ of Ξ  such that Ξ (𝐴) βŠ‚ Ξ (𝐴 ) .
                                                                                                    β€²


   2. 𝐴 is a minβŠ‚ assumption set of Ξ  if there is no assumption set 𝐴′ of Ξ  such that Ξ (𝐴) βŠƒ Ξ (𝐴 ) .
                                                                                                   β€²


   3. 𝐴 is a maxβ™― assumption set of Ξ  if there is no assumption set 𝐴′ of Ξ  such that |Ξ (𝐴) | < |Ξ (𝐴 ) |.
                                                                                                      β€²


   4. 𝐴 is a minβ™― assumption set of Ξ  if there is no assumption set 𝐴′ of Ξ  such that |Ξ (𝐴) | > |Ξ (𝐴 ) |.
                                                                                                     β€²




Definition 4. Given an AASP program Ξ , 𝑀 is called maxβŠ‚ (minβŠ‚ or maxβ™― or minβ™― ) assumable answer set
of Ξ  if (𝑀, 𝐴) is a view of Ξ  and 𝐴 is a maxβŠ‚ (minβŠ‚ or maxβ™― or minβ™― ) assumption set of Ξ . Correspondingly,
the pair (𝑀, 𝐴) is called a maxβŠ‚ (minβŠ‚ or maxβ™― or minβ™― ) view of Ξ .

   maxβŠ‚ (minβŠ‚ or maxβ™― or minβ™― )-𝐴𝐴𝑆(Ξ ) is used to denote the collection of all maxβŠ‚ (minβŠ‚ or maxβ™― or
minβ™― ) assumable answer sets of an AASP program Ξ . maxβŠ‚ (minβŠ‚ or maxβ™― or minβ™― )-𝑉 𝐼 πΈπ‘Š (Ξ ) is used
to denote the collection of all maxβŠ‚ (minβŠ‚ or maxβ™― or minβ™― ) views of an AASP program Ξ .
Example 3. Continue Ξ 1 mentioned in the above examples. Obviously, both βˆ… and {𝑝} are not only
minβŠ‚ but also minβ™― assumption sets, while {𝑝, π‘ž} is both maxβŠ‚ and maxβ™― assumption set. Thus, βˆ… is a
minβŠ‚ assumable answer set and also a minβ™― assumable answer set of Ξ 1 . {𝑝} is a maxβŠ‚ assumable answer
set and also a maxβ™― assumable answer set of Ξ 1 .

The intuitions of π‘šπ‘Žπ‘₯ and π‘šπ‘–π‘› strategies of assumption making are direct: π‘šπ‘Žπ‘₯ means that the
reasoner is positive/optimistic/ credulous in making assumptions. π‘šπ‘–π‘› is just the opposite. Let us
consider a common advice β€œWork hard and you will succeed” that can be represented by an AASP
program with one rule
                                       𝑠𝑒𝑐𝑐𝑒𝑒𝑑 β†βˆΆ π‘€π‘œπ‘Ÿπ‘˜β„Žπ‘Žπ‘Ÿπ‘‘
Then, a positive reasoner’s view is the π‘šπ‘Žπ‘₯βŠ‚ (π‘šπ‘Žπ‘₯β™― ) view ({𝑠𝑒𝑐𝑐𝑒𝑒𝑑}, {π‘€π‘œπ‘Ÿπ‘˜β„Žπ‘Žπ‘Ÿπ‘‘, 𝑠𝑒𝑐𝑐𝑒𝑒𝑑}) of the
program, a passive reasoner’s view is the π‘šπ‘–π‘›βŠ‚ (π‘šπ‘–π‘›β™― ) view (βˆ…, βˆ…) of the program.

3.3. Properties
In this subsection, some properties of AASP are given.
Theorem 1. For a assumption-free AASP program Ξ :

                                   𝐴𝐴𝑆(Ξ ) = X⋆ βˆ’ 𝐴𝐴𝑆(Ξ ) = 𝐴𝑆(Ξ )

where 𝑋 ∈ {π‘šπ‘Žπ‘₯, π‘šπ‘–π‘›} and ⋆ ∈ {βŠ‚, β™―}.

Theorem 2. For an AASP program Ξ , an assumption set of Ξ  is a model of Ξ .

Theorem 3. For an AASP program Ξ  and a literal subset 𝐴 βŠ† 𝑙𝑖𝑑(Ξ )
                                          ←                               ←
                           𝐴 ∈ 𝐴𝑆(Ξ (𝐴) βˆͺ 𝐴) if and only if 𝐴 ∈ 𝐴𝑆(Ξ  ↓𝐴 βˆͺ 𝐴)

where
                Ξ  ↓𝐴 = {β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) ← π‘π‘π‘œπ‘‘π‘¦(π‘Ÿ), π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ)|π‘Ÿ ∈ Ξ  and 𝐴 ⊧AASP π‘Žπ‘π‘œπ‘‘π‘¦(π‘Ÿ)}

Theorem 3 tells that an alternative definition of the notion of Assumption Set is: 𝐴 is an assumption set
                      ←
of Ξ  if 𝐴 ∈ 𝐴𝑆(Ξ  ↓𝐴 βˆͺ 𝐴), that can be viewed as another way of formalizing the principle of Rationality
of Assumption Making.
Theorem 4. For an AASP program Ξ  = (Π𝐷 , Ξ π‘Š ), let 𝐴 be an assumption set of Ξ 

                                               Ξ π‘Š βŠ† Ξ (𝐴)

Theorem 5. For an AASP program Ξ  = (Π𝐷 , Ξ π‘Š ), let 𝑀 be an assumable answer set of Ξ 

                                              𝑀 ⊧AASP Ξ π‘Š

Theorem 5 tells that an assumable answer set of an AASP program always satisfies the assumption-
free rules in the program.
Theorem 6. AASP is constraint monotonic under the assumable answer set semantics.

Theorem 7. AASP is not constraint monotonic under the X⋆ assumable answer set semantics, where
𝑋 ∈ {π‘šπ‘Žπ‘₯, π‘šπ‘–π‘›} and ⋆ ∈ {βŠ‚, β™―}.
Theorem 7 can be demonstrated by the following example.
Example 4. Ξ 2 is an AASP program containing one rule:
                                                𝑝 β†βˆΆ 𝑝
Ξ 2 has only one maxβŠ‚ (also maxβ™― ) assumable answer set {𝑝} and only one minβŠ‚ (also minβ™― ) assumable
answer set βˆ…. Consider Ξ β€²2 :
                                             𝑝 β†βˆΆ 𝑝
                                                  ←𝑝
Clearly, Ξ β€²2 has only one maxβŠ‚ (maxβ™― ) assumable answer set βˆ… that is not a maxβŠ‚ (or maxβ™― ) assumable
answer set of Ξ 2 . Consider Ξ β€²β€²
                             2:
                                              𝑝 β†βˆΆ 𝑝
                                                ← π‘›π‘œπ‘‘ 𝑝
            2 has only one minβŠ‚ (or minβ™― ) assumable answer set {𝑝} that is not a minβŠ‚ (or minβ™― ) as-
Obviously, Ξ β€²β€²
sumable answer set of Ξ 2 .


4. Relations
For exploring the power of AASP in commonsense reasoning, this section presents the relationship
between AASP and several well-known knowledge representation languages, including constrained
default logic [28], CR-Prolog [31], and abductive logic programming [25] [26]. First of all, we compare
AASP and ASP in representing assumptions.

4.1. Comparison of AASP and ASP in Representing Assumptions
It seems that the assumption rule On the assumption of 𝛼, 𝛽 is believed if 𝛾 is believed can also be coded
into an ASP rule 𝛽 ← 𝛾 , π‘›π‘œπ‘‘ ¬𝛼 where π‘›π‘œπ‘‘ ¬𝛼 is used to express 𝛼 is assumable or it is consistent to
assume 𝛼. However, the following cases demonstrate the difference between ASP encodings and AASP
encodings of the assumptions.
   Firstly, let us consider a case containing:
    β€’ An assumption: 𝑝 if it is consistent to assume 𝑝.
    β€’ A constraint: p is impossible.
Intuitively, the constraint is a denial of 𝑝 such that the assumption is blocked, thus the result is βˆ…. If
the case is modeled by an ASP program
                                              𝑝 ← π‘›π‘œπ‘‘ ¬𝑝
                                                  ←𝑝
there is no solution because the ASP program is unsatisfiable. If the case is represented by an AASP
program
                                            𝑝 β†βˆΆ 𝑝
                                                  ←𝑝
there is an assumable answer set βˆ… as expected.
  Now, let us consider another case with two assumptions:
    β€’ p if it is consistent to assume r, and

    β€’ q if it is consistent to assume Β¬π‘Ÿ.

If they are represented as an ASP program

                                                𝑝 ← π‘›π‘œπ‘‘ Β¬π‘Ÿ

                                                 π‘ž ← π‘›π‘œπ‘‘ π‘Ÿ
The result is its answer set {𝑝, π‘ž}. Meanwhile, if they are represented as an AASP program

                                                   𝑝 β†βˆΆ π‘Ÿ

                                                  π‘ž β†βˆΆ Β¬π‘Ÿ
There are three assumable answer sets {𝑝}, {π‘ž}, and βˆ…. Among them, both {𝑝} and {π‘ž} are π‘šπ‘Žπ‘₯βŠ‚
(and π‘šπ‘Žπ‘₯β™― ) assumable answer sets, and βˆ… is a π‘šπ‘–π‘›βŠ‚ (and π‘šπ‘–π‘›β™― ) assumable answer set. Consider that
π‘Ÿ and its contrary Β¬π‘Ÿ cannot appear in one world, the results given by the AASP program should be
more praised than that of the ASP program.
   Another seeming ASP-based encoding of it is consistent to assume 𝛼 is π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ 𝛼. But, it is easy to
verify that the encoding of the second case in this way

                                               𝑝 ← π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ π‘Ÿ

                                               π‘ž ← π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ Β¬π‘Ÿ
has only one answer set βˆ….
  The above two examples demonstrate that it is hard to represent assumptions using the regular
ASP language, and that AASP provides an easy approach to handling assumptions by extending ASP
with a new operator :.

4.2. Relation to Constrained Default Logic
Constrained default logic meets some properties that are in line with human intuition in common-
sense reasoning. The following theorem shows that AASP is closely related to the constrained default
logic.
  Define a mapping πœ™ from a disjunction-free and NAF-free program of AASP to default theories,
identifies an AASP rule π‘Ÿ:
                                     𝑙0 ← 𝑙1 , ..., π‘™π‘˜ ∢ π‘™π‘˜+1 , ..., 𝑙𝑛                          (1)
with the default πœ™(π‘Ÿ):
                                        𝑙1 ∧ ... ∧ π‘™π‘˜ ∢ Mπ‘™π‘˜+1 ∧ ... ∧ 𝑙𝑛
                                                                                                   (2)
                                                        𝑙0
Then we have

Theorem 8. For any disjunction-free and NAF-free AASP program Ξ , if (𝑆, 𝐴) is a maxβŠ‚ view of Ξ , then
(𝑇 β„Ž(𝑆), 𝑇 β„Ž(𝐴)) is a constrained extension of πœ™(Ξ ).
Example 5. Consider Ξ 3 :
                                                    𝑐 β†βˆΆ 𝑏
                                                   𝑑 β†βˆΆ ¬𝑏
has two maxβŠ‚ views: ({𝑐}, {𝑏, 𝑐}), and ({𝑑}, {𝑑, ¬𝑏}). {𝑐, 𝑑} is not an assumable answer set of Ξ 3 be-
cause {𝑏, ¬𝑏, 𝑐, 𝑑} is not an assumption set of the program. Correspondingly, πœ™(Ξ 3 ) is a default theory
containing two defaults:
                                           ∢ M𝑏         ∢ M¬𝑏
                                              𝑐            𝑑
that has two extensions:
                                            (𝑇 β„Ž(𝑐), 𝑇 β„Ž(𝑏, 𝑐))
and
                                             (𝑇 β„Ž(𝑑), 𝑇 β„Ž(¬𝑏, 𝑑))

4.3. Relation to CR-Prolog
CR-Prolog is an ASP extension that greatly extends the expression ability of ASP language. The
following theorem provides an approach to converting a CR-Prolog program into an AASP program.
   Define a mapping πœ‚ from a CR-prolog to AASP, identifies a CR-rule π‘Ÿ:
                                              +
                             𝑙1 π‘œπ‘Ÿ ... π‘œπ‘Ÿ π‘™π‘˜ ← π‘™π‘˜+1 , ..., π‘™π‘š , π‘›π‘œπ‘‘ π‘™π‘š+1 , ..., π‘›π‘œπ‘‘ 𝑙𝑛               (3)

with an AASP rule πœ‚(π‘Ÿ):

                        𝑙1 π‘œπ‘Ÿ ... π‘œπ‘Ÿ π‘™π‘˜ ← π‘™π‘˜+1 , ..., π‘™π‘š , π‘›π‘œπ‘‘ π‘™π‘š+1 , ..., π‘›π‘œπ‘‘ 𝑙𝑛 ∢ π‘Žπ‘π‘π‘™π‘¦π‘Ÿ           (4)

where π‘Žπ‘π‘π‘™π‘¦π‘Ÿ is used to denote the fresh atom obtained from a CR-rule π‘Ÿ. Besides, πœ‚ identifies a regular
ASP rule in the CR-prolog program with itself.

Theorem 9. For any CR-Prolog program Ξ©

                                       𝐴𝑆⋆ (Ξ©) = min⋆ βˆ’ 𝐴𝐴𝑆(πœ‚(Ξ©))

where ⋆ ∈ {βŠ‚, β™―}.

Example 6. Consider a simple CR-Prolog program Ξ© that contains only one CR-rule π‘Ÿ:
                                                          +
                                                      π‘Žβ†

It is easy to see Ξ© has only one answer set βˆ…. Then, an AASP program πœ‚(Ξ©) is

                                                  π‘Ž β†βˆΆ π‘Žπ‘π‘π‘™π‘¦π‘Ÿ

whose minβ™― and minβŠ‚ assumable answer set is also βˆ….
4.4. Relation to Abductive ASP
Define a mapping πœƒ, for an ALP93 program Ξ“ =< 𝑃,  >, πœƒ(Ξ“) is an AASP program:

                                         𝑃 βˆͺ {𝑙 β†βˆΆ 𝑙|𝑙 ∈ }

We have
Theorem 10. For the ALP93 program Ξ“ =< 𝑃,  >, 𝑆 is a minβŠ‚ assumable answer set of πœƒ(Ξ“) if and only
if 𝑆 is a -minimal belief set of Ξ“.

Example 7. Consider an abductive logic program Ξ“1 =< 𝑃,  > in [32]:
    β€’ 𝑃: 𝑝 ← π‘›π‘œπ‘‘ π‘Ž

    β€’ : π‘Ž
The program has one -minimal belief set {𝑝}. πœƒ(Ξ“1 ) is an AASP

                                             𝑝 ← π‘›π‘œπ‘‘ π‘Ž

                                               π‘Ž β†βˆΆ π‘Ž
Both {π‘Ž} and {𝑝} are its assumable answer sets, only {𝑝} is its minβŠ‚ assumable answer set as expected.

Theorem 11. For the ALP93 program Ξ“ =< 𝑃,  > and a positive observation 𝐺.
   1. If 𝑆 is a minβŠ‚ assumable answer set of the AASP program πœƒ(Ξ“) βˆͺ {← π‘›π‘œπ‘‘ 𝐺}, then 𝑆 ∩  is a
      credulous explanation of 𝐺 with respect to Ξ“.
   2. If 𝐸 is a credulous explanation of 𝐺 with respect to Ξ“, then there exists a minβŠ‚ assumable answer
      set 𝑆 of the AASP program πœƒ(Ξ“) βˆͺ {← π‘›π‘œπ‘‘ 𝐺} such that 𝐸 = 𝑆 ∩ .

Example 8. Continue to consider Ξ“1 mentioned in Example 7, given a positive observation 𝐺 = 𝑝.
Clearly, βˆ… is a credulous explanation of 𝑝 with respect to Ξ“1 . Now, we have πœƒ(Ξ“1 ) βˆͺ {← π‘›π‘œπ‘‘ 𝐺}:

                                             𝑝 ← π‘›π‘œπ‘‘ π‘Ž

                                               π‘Ž β†βˆΆ π‘Ž
                                              ← π‘›π‘œπ‘‘ 𝑝
that has a minβŠ‚ assumable answer set βˆ… such that 𝐸 = βˆ….

  Define a mapping πœƒ β€² , for the ALP95 program Ξ“ =< 𝑃,  >, πœƒ β€² (Ξ“) is an AASP program:

                        (𝑃 βˆ’ ) βˆͺ {β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) ← π‘π‘œπ‘‘π‘¦(π‘Ÿ), π‘Žπ‘π‘π‘™π‘¦π‘Ÿ |π‘Ÿ ∈ ( βˆ’ 𝑃)}βˆͺ

                                 {π‘Žπ‘π‘π‘™π‘¦π‘Ÿ β†βˆΆ π‘Žπ‘π‘π‘™π‘¦π‘Ÿ |π‘Ÿ ∈ ( βˆ’ 𝑃)}βˆͺ
                           {β„Žπ‘’π‘Žπ‘‘(π‘Ÿ) ← π‘π‘œπ‘‘π‘¦(π‘Ÿ), π‘›π‘œπ‘‘ π‘π‘™π‘œπ‘π‘˜π‘Ÿ |π‘Ÿ ∈ ( ∩ 𝑃)}βˆͺ
                                  {π‘π‘™π‘œπ‘π‘˜π‘Ÿ β†βˆΆ π‘π‘™π‘œπ‘π‘˜π‘Ÿ |π‘Ÿ ∈ ( ∩ 𝑃)}
where both π‘Žπ‘π‘π‘™π‘¦π‘Ÿ and π‘π‘™π‘œπ‘π‘˜π‘Ÿ are used to denote the fresh atoms obtained from π‘Ÿ ∈ . We have
Theorem 12. For the ALP95 program Ξ“ =< 𝑃,  > and a positive observation 𝐺.
    1. If 𝑆 is a minβŠ‚ assumable answer set of the AASP program πœƒ β€² (Ξ“) βˆͺ {← π‘›π‘œπ‘‘ 𝐺}, then there exists a
       minimal explanation of 𝐺 with respect to Ξ“ is

                                        ({π‘Ÿ|π‘Žπ‘π‘π‘™π‘¦π‘Ÿ ∈ 𝑆}, {π‘Ÿ|π‘π‘™π‘œπ‘π‘˜π‘Ÿ ∈ 𝑆})
    2. If (𝐸, 𝐹 ) is a minimal explanation of 𝐺 with respect to Ξ“, then there exists a minβŠ‚ assumable answer
       set 𝑆 of the AASP program πœƒ β€² (Ξ“) βˆͺ {← π‘›π‘œπ‘‘ 𝐺} such that

                                               𝐸 = {π‘Ÿ|π‘Žπ‘π‘π‘™π‘¦π‘Ÿ ∈ 𝑆}

                                                𝐹 = {π‘Ÿ|π‘π‘™π‘œπ‘π‘˜π‘Ÿ ∈ 𝑆}

Example 9. Consider an abductive logic program Ξ“2 =< 𝑃,  > and a positive observation 𝐺:
     β€’ 𝑃:
                                                   𝑓 𝑙𝑦 ← π‘π‘–π‘Ÿπ‘‘
                                                π‘π‘–π‘Ÿπ‘‘ ← 𝑝𝑒𝑛𝑔𝑒𝑖𝑛
                                                    𝑝𝑒𝑛𝑒𝑖𝑛 ←

     β€’ :
                                                 (1). 𝑓 𝑙𝑦 ← π‘π‘–π‘Ÿπ‘‘
                                              (2). ¬𝑓 𝑙𝑦 ← 𝑝𝑒𝑛𝑔𝑒𝑖𝑛

     β€’ 𝐺: ¬𝑓 𝑙𝑦
Thus, πœƒ β€² (Ξ“2 ) is:
                                             π‘π‘–π‘Ÿπ‘‘ ← 𝑝𝑒𝑛𝑔𝑒𝑖𝑛
                                                 𝑝𝑒𝑛𝑒𝑖𝑛 ←
                                         𝑓 𝑙𝑦 ← π‘π‘–π‘Ÿπ‘‘, π‘›π‘œπ‘‘ π‘π‘™π‘œπ‘π‘˜1
                                        ¬𝑓 𝑙𝑦 ← 𝑝𝑒𝑛𝑔𝑒𝑖𝑛, π‘Žπ‘π‘π‘™π‘¦2
                                            π‘π‘™π‘œπ‘π‘˜1 β†βˆΆ π‘π‘™π‘œπ‘π‘˜1
                                            π‘Žπ‘π‘π‘™π‘¦2 β†βˆΆ π‘Žπ‘π‘π‘™π‘¦2
Then, πœƒ β€² (Ξ“2 ) βˆͺ {← π‘›π‘œπ‘‘ ¬𝑓 𝑙𝑦} has a minβŠ‚ assumable answer set

                                 {π‘Žπ‘π‘π‘™π‘¦2 , π‘π‘™π‘œπ‘π‘˜1 , ¬𝑓 𝑙𝑦, 𝑝𝑒𝑛𝑔𝑒𝑖𝑛, π‘π‘–π‘Ÿπ‘‘}

by which
                                         𝐸 = {¬𝑓 𝑙𝑦 ← 𝑝𝑒𝑛𝑔𝑒𝑖𝑛}
                                            𝐹 = {𝑓 𝑙𝑦 ← π‘π‘–π‘Ÿπ‘‘}
such that (𝐸, 𝐹 ) is a minimal explanation of ¬𝑓 𝑙𝑦 with respect to Ξ“2 .

Theorem 13. For the ALP95 program Ξ“ =< 𝑃,  > and a positive observation 𝐺.
    1. If 𝑆 is a minβŠ‚ assumable answer set of πœƒ β€² (Ξ“)βˆͺ{← 𝐺}, then there exists a minimal anti-explanation
       of 𝐺 with respect to Ξ“ is
                                       ({π‘Ÿ|π‘Žπ‘π‘π‘™π‘¦π‘Ÿ ∈ 𝑆}, {π‘Ÿ|π‘π‘™π‘œπ‘π‘˜π‘Ÿ ∈ 𝑆})
    2. If (𝐸, 𝐹 ) is a minimal anti-explanation of 𝐺 with respect to Ξ“, then there exists a minβŠ‚ assumable
       answer set 𝑆 of πœƒ β€² (Ξ“) βˆͺ {← 𝐺} such that

                                          𝐸 = {π‘Ÿ|π‘Žπ‘π‘π‘™π‘¦π‘Ÿ ∈ 𝑆}, 𝐹 = {π‘Ÿ|π‘π‘™π‘œπ‘π‘˜π‘Ÿ ∈ 𝑆}

Example 10. Continue to consider the abductive logic program Ξ“2 used in the Example 9. By the The-
orem 13, the anti-explanation of 𝑓 𝑙𝑦 with respect to Ξ“2 is the minβŠ‚ assumable answer set of the AASP
program
                                           πœƒ β€² (Ξ“2 ) βˆͺ {← 𝑓 𝑙𝑦}
Obviously, a minβŠ‚ assumable answer set of the program is {𝑝𝑒𝑛𝑔𝑒𝑖𝑛, π‘π‘–π‘Ÿπ‘‘} that tells neither (1) nor (2)
is used, and the corresponding minimal anti-explanation (𝐸, 𝐹 ) of 𝑓 𝑙𝑦 is

                                             𝐸=βˆ…         𝐹 = {𝑓 𝑙𝑦 ← π‘π‘–π‘Ÿπ‘‘}

.


5. Conclusion
This paper introduce AASP that extends ASP with an operator : to express the notion of assump-
tion in logic programming. AASP can be viewed as a tool to design the intelligent agent capable of
assumption-based reasoning that is a framework of many intelligent behaviors in the case of incom-
plete information. AASP provides a rational approach to assumption-based reasoning by using the
answer set-based reasoning in both assumption making and belief building, which makes the exit-
ing ASP solvers can be used to facilitate the implementment of the AASP solver: ASP solvers can be
directly used to generate assumption sets and then the assumable answer set of an AASP program.
Several strategies of assumption making are given in the definition of the semantics of AASP. Those
strategies depicts the attitude of agents to assumptions. The preliminary exploration results on the
relationship between AASP and other existing knowledge representations show that AASP provides
a more general way to model the problems with defaults, exceptions, and to solve the explanation
problems. Especially, the close relationship between AASP and constrained default logic implies that
AASP is able to address several limitations of the original ASP in representing defaults 1 .
   Future work includes more properties of the AASP languages, implementation, and applications.
The first next step is to investigate the properties of the assumption set of the AASP programs, and the
algorithm and its complexity in solving AASP programs. Besides, it is still needed to explore the rela-
tion of AASP to ASP and its extensions, and the relation of AASP to other knowledge representation
languages more deeply.


Acknowledgments
We are grateful to the anonymous referees for their useful comments on the earlier version of this
paper. The work was supported by the Pre-research Key Laboratory Fund for Equipment (Grant No.
6142101190304).


    1
      The limitations of Reiter’s default logic are extensively studied in [28]. Because of the close relationship between ASP
and Reiter’s default logic shown in [33] and [22], ASP cannot meet some properties, such as the property of joint consistency
of the justifications of applying default rules
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