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Atomic Formula
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Yuefei Sui – One of the best experts on this subject based on the ideXlab platform.

The Correspondence between Propositional Modal Logic with Axiom $\Box\varphi \leftrightarrow \Diamond \varphi $ and the Propositional Logic
, 2014CoAuthors: Meiying Sun, Shaobo Deng, Yuefei SuiAbstract:The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each Formula in the propositional logic is equivalent to a Formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom $\Box\varphi \leftrightarrow\Diamond\varphi $ to K and get a new system K + . Each Formula in such a logic is equivalent to a Formula in the disjunctive normal form, where □k(k ≥ 0) only occurs before an Atomic Formula p, and $\lnot$ only occurs before a pseudoAtomic Formula of form □k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudoatomcomplete Formulas uniquely determines a maximally consistent set. When a pseudoAtomic Formula □k pi (k,i ≥ 0) is corresponding to a propositional variable qki, each Formula in K + then can be corresponding to a Formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid Formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

The Correspondence between Propositional Modal Logic with Axiom ϕ ↔ ♦ϕ and the Propositional Logic
, 2014CoAuthors: Meiying Sun, Shaobo Deng, Yuefei SuiAbstract:The propositional modal logic is obtained by adding the ne cessity operatorto the propositional logic. Each Formula in the propo sitional logic is equivalent to a Formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom � ϕ ↔ ♦ϕ to K and get a new system K + . Each Formula in such a logic is equivalent to a for mula in the disjunctive normal form, wherek (k ≥ 0) only occurs before an Atomic Formula p ,a nd¬ only occurs before a pseudoAtomic Formula of formk p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudoatomcomplete Formulas uniquely determines a maximally consistent set. When a pseudoAtomic Formula � k pi(k, i ≥ 0) is corresponding to a propositional variable qki, each for mula in K + then can be corresponding to a Formula in the propositional

Intelligent Information Processing – The Correspondence between Propositional Modal Logic with Axiom \Box\varphi \leftrightarrow \Diamond \varphi and the Propositional Logic
Progress in Pattern Recognition Image Analysis Computer Vision and Applications, 2014CoAuthors: Meiying Sun, Shaobo Deng, Yuefei SuiAbstract:The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each Formula in the propositional logic is equivalent to a Formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each Formula in such a logic is equivalent to a Formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an Atomic Formula p, and \(\lnot\) only occurs before a pseudoAtomic Formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudoatomcomplete Formulas uniquely determines a maximally consistent set. When a pseudoAtomic Formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each Formula in K + then can be corresponding to a Formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid Formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .
Meiying Sun – One of the best experts on this subject based on the ideXlab platform.

The Correspondence between Propositional Modal Logic with Axiom $\Box\varphi \leftrightarrow \Diamond \varphi $ and the Propositional Logic
, 2014CoAuthors: Meiying Sun, Shaobo Deng, Yuefei SuiAbstract:The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each Formula in the propositional logic is equivalent to a Formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom $\Box\varphi \leftrightarrow\Diamond\varphi $ to K and get a new system K + . Each Formula in such a logic is equivalent to a Formula in the disjunctive normal form, where □k(k ≥ 0) only occurs before an Atomic Formula p, and $\lnot$ only occurs before a pseudoAtomic Formula of form □k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudoatomcomplete Formulas uniquely determines a maximally consistent set. When a pseudoAtomic Formula □k pi (k,i ≥ 0) is corresponding to a propositional variable qki, each Formula in K + then can be corresponding to a Formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid Formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

The Correspondence between Propositional Modal Logic with Axiom ϕ ↔ ♦ϕ and the Propositional Logic
, 2014CoAuthors: Meiying Sun, Shaobo Deng, Yuefei SuiAbstract:The propositional modal logic is obtained by adding the ne cessity operatorto the propositional logic. Each Formula in the propo sitional logic is equivalent to a Formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom � ϕ ↔ ♦ϕ to K and get a new system K + . Each Formula in such a logic is equivalent to a for mula in the disjunctive normal form, wherek (k ≥ 0) only occurs before an Atomic Formula p ,a nd¬ only occurs before a pseudoAtomic Formula of formk p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudoatomcomplete Formulas uniquely determines a maximally consistent set. When a pseudoAtomic Formula � k pi(k, i ≥ 0) is corresponding to a propositional variable qki, each for mula in K + then can be corresponding to a Formula in the propositional

Intelligent Information Processing – The Correspondence between Propositional Modal Logic with Axiom \Box\varphi \leftrightarrow \Diamond \varphi and the Propositional Logic
Progress in Pattern Recognition Image Analysis Computer Vision and Applications, 2014CoAuthors: Meiying Sun, Shaobo Deng, Yuefei SuiAbstract:The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each Formula in the propositional logic is equivalent to a Formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each Formula in such a logic is equivalent to a Formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an Atomic Formula p, and \(\lnot\) only occurs before a pseudoAtomic Formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudoatomcomplete Formulas uniquely determines a maximally consistent set. When a pseudoAtomic Formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each Formula in K + then can be corresponding to a Formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid Formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .
Ekawit Nantajeewarawat – One of the best experts on this subject based on the ideXlab platform.

ACIIDS (1) – Equivalent Transformation in an Extended Space for Solving QueryAnswering Problems
Intelligent Information and Database Systems, 2014CoAuthors: Kiyoshi Akama, Ekawit NantajeewarawatAbstract:A queryanswering problem QA problem is concerned with finding all ground instances of a query Atomic Formula that are logical consequences of a given logical Formula describing the background knowledge of the problem. Based on the equivalent transformation ET principle, we propose a general framework for solving QA problems on firstorder logic. To solve such a QA problem, the firstorder Formula representing its background knowledge is converted by meaningpreserving Skolemization into a set of clauses typically containing global existential quantifications of function variables. The obtained clause set is then transformed successively using ET rules until the answer set of the original problem can be readily derived. Many ET rules are demonstrated, including rules for unfolding clauses, for resolution, for dealing with function variables, and for erasing independent satisfiable Atomic Formulas. Application of the proposed framework is illustrated.

ACIIDS (1) – Correctness of solving queryanswering problems using satisfiability solvers
Intelligent Information and Database Systems, 2013CoAuthors: Kiyoshi Akama, Ekawit NantajeewarawatAbstract:A queryanswering (QA) problem is concerned with finding the set of all ground instances of a given Atomic Formula that are logical consequences of a specified logical Formula. Recently, many kinds of problems have been solved efficiently by using satisfiability (SAT) solvers, motivating us to use SAT solvers to speed up solving a class of QA problems. Given a finite ground clause set as input, a SAT solver used in this paper generates all models of the input set that contain only Atomic Formulas appearing in it. A method for solving QA problems using SAT solvers is developed, based on the use of a support set to restrict the generation of ground instances of given clauses possibly with constraint Atomic Formulas. The correctness of the proposed method is proved.