Answer-set programming (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation>) is a declarative logic programming paradigm that provides an efficient problem-solving approach in logic-based artificial intelligence. In <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation>, problems are represented as logic programs, and solutions are identified through their answer sets. Equilibrium logic (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \textsf{EL} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">EL</mi> </math></EquationSource> </InlineEquation>) is a general-purpose nonmonotonic reasoning formalism based on a monotonic logic called here-and-there logic (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \textsf{HT} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">HT</mi> </math></EquationSource> </InlineEquation>). <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \textsf{HT} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">HT</mi> </math></EquationSource> </InlineEquation> is a three-valued intermediate logic that lies strictly between intuitionistic logic and classical logic. <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \textsf{EL} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">EL</mi> </math></EquationSource> </InlineEquation> was originally proposed as a foundational framework of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation>, where answer sets of an <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation> program are captured by the equilibrium models of the corresponding <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ \textsf{HT} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">HT</mi> </math></EquationSource> </InlineEquation> theory. While <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation> has proven successful as a knowledge-representation formalism, it encounters specific situations where its language falls short of accurately representing and reasoning about incomplete information. Researchers now widely agree that <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation> requires powerful introspective reasoning with the use of epistemic modal operators. Therefore, epistemic specifications (<InlineEquation ID="IEq12"> <EquationSource Format="TEX">$ \textsf{ES} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ES</mi> </math></EquationSource> </InlineEquation>) have been proposed as extensions of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation> programs with subjective literals. These new modal constructs in the <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation> language make it possible to check whether a regular literal of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation> is true in every (or some) answer set of a logic program, which is required to model incomplete information in <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation>. Thus, <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$ \textsf{ES} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ES</mi> </math></EquationSource> </InlineEquation> programs are interpreted by world-view structures, which are essentially collections of answer sets (or equilibrium models). However, despite long-lasting debates on how to capture the intended meaning of <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$ \textsf{ES} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ES</mi> </math></EquationSource> </InlineEquation> programs via world views, researchers have not reached a consensus on fully satisfactory semantics. Recently, Cabalar et al. have argued that such research on <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$ \textsf{ES} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ES</mi> </math></EquationSource> </InlineEquation> semantics should be grounded in formal robustness rather than in test examples. Thus, inspired by <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation>’s foundational properties, they introduced a new structural principle called the epistemic splitting property (<InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\texttt {E}\text {-}\texttt {SP}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="monospace">E</mi> <mtext>-</mtext> <mi mathvariant="monospace">SP</mi> </mrow> </math></EquationSource> </InlineEquation>) and designated it as one of the compulsory criteria for epistemic <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation>. However, this criterion has left several intuitive semantic approaches unsatisfactory. This paper generalises Cabalar et al.’s approach to a more comprehensive, meticulous, and conservative extension of <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$ \textsf{ASP} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ASP</mi> </math></EquationSource> </InlineEquation>’s original splitting property, thereby broadening the applicability and enhancing the efficiency of epistemic splitting property for general epistemic equilibrium logics.

Splitting Property for Epistemic Equilibrium Logics

期刊论文
发表日期：

2025年11月26日

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Ezgi Iraz Su

摘要

Answer-set programming ( $ \textsf{ASP} $ $ASP$ ) is a declarative logic programming paradigm that provides an efficient problem-solving approach in logic-based artificial intelligence. In $ \textsf{ASP} $ $ASP$ , problems are represented as logic programs, and solutions are identified through their answer sets. Equilibrium logic ( $ \textsf{EL} $ $EL$ ) is a general-purpose nonmonotonic reasoning formalism based on a monotonic logic called here-and-there logic ( $ \textsf{HT} $ $HT$ ). $ \textsf{HT} $ $HT$ is a three-valued intermediate logic that lies strictly between intuitionistic logic and classical logic. $ \textsf{EL} $ $EL$ was originally proposed as a foundational framework of $ \textsf{ASP} $ $ASP$ , where answer sets of an $ \textsf{ASP} $ $ASP$ program are captured by the equilibrium models of the corresponding $ \textsf{HT} $ $HT$ theory. While $ \textsf{ASP} $ $ASP$ has proven successful as a knowledge-representation formalism, it encounters specific situations where its language falls short of accurately representing and reasoning about incomplete information. Researchers now widely agree that $ \textsf{ASP} $ $ASP$ requires powerful introspective reasoning with the use of epistemic modal operators. Therefore, epistemic specifications ( $ \textsf{ES} $ $ES$ ) have been proposed as extensions of $ \textsf{ASP} $ $ASP$ programs with subjective literals. These new modal constructs in the $ \textsf{ASP} $ $ASP$ language make it possible to check whether a regular literal of $ \textsf{ASP} $ $ASP$ is true in every (or some) answer set of a logic program, which is required to model incomplete information in $ \textsf{ASP} $ $ASP$ . Thus, $ \textsf{ES} $ $ES$ programs are interpreted by world-view structures, which are essentially collections of answer sets (or equilibrium models). However, despite long-lasting debates on how to capture the intended meaning of $ \textsf{ES} $ $ES$ programs via world views, researchers have not reached a consensus on fully satisfactory semantics. Recently, Cabalar et al. have argued that such research on $ \textsf{ES} $ $ES$ semantics should be grounded in formal robustness rather than in test examples. Thus, inspired by $ \textsf{ASP} $ $ASP$ ’s foundational properties, they introduced a new structural principle called the epistemic splitting property ( $\texttt {E}\text {-}\texttt {SP}$ $E - SP$ ) and designated it as one of the compulsory criteria for epistemic $ \textsf{ASP} $ $ASP$ . However, this criterion has left several intuitive semantic approaches unsatisfactory. This paper generalises Cabalar et al.’s approach to a more comprehensive, meticulous, and conservative extension of $ \textsf{ASP} $ $ASP$ ’s original splitting property, thereby broadening the applicability and enhancing the efficiency of epistemic splitting property for general epistemic equilibrium logics.