Kiyoshi Akama

Equivalent Transformation in an Extended Space for Solving Query-Answering Problems

A query-answering 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 first-order logic. To solve such a QA problem, the first-order formula representing its background knowledge is converted by meaning-preserving 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.

Unfolding Existentially Quantified Sets of Extended Clauses

Conventional theories cannot solve many logical problems due to the limitations of the underlying clause space. In conventional clauses, all variables are universally quantified and no existential quantification is allowed. Conventional clauses are therefore not sufficiently expressive for representing first-order formulas. To extend clauses with the expressive power of existential quantification, variables of a new type, called function variables, have been introduced, resulting in a new space of extended clauses, called ECLS_F. This new space is necessary to overcome the limitations of the conventional clause space. To solve problems on ECLS_F, many equivalent transformation rules are used. We formally defined unfolding transformation on ECLS_F, which is applicable not only to definite clauses but also to multi-head clauses. The proposed unfolding transformation preserves the answers to model-intersection problems and is useful for solving many logical problems such as proof problems and query-answering problems on first-order logic with built-in constraint atoms.

Model-Intersection Problems with Existentially Quantified Function Variables: Formalization and a Solution Schema

Built-in constraint atoms play a very important role in knowledge representation and are indispensable for practical applications. It is very natural to use built-in constraint atoms together with user-defined atoms when formalizing logical problems using first-order formulas. In the presence of built-in constraint atoms, however, the conventional Skolemization in general preserves neither the satisfiability nor the logical meaning of a given first-order formula, motivating us to step outside the conventional Skolemization and the usual space of first- order formulas. We propose general solutions for proof problems and query-answering (QA) problems on first-order formulas possibly with built-in constraint atoms. We map, by using new meaning-preserving Skolemization, all proof problems and all QA problems, preserving their answers, into a new class of model-intersection (MI) problems on an extended clause space, where clauses are in a sense ``higher-order n nsince they may contain not only built-in constraint atoms but also function variables. We propose a general schema for solving this class of MI problems by equivalent transformation (ET), where problems are solved by repeated simplification using ET rules. The correctness of this solution schema is shown. Since MI problems in this paper form a very large class of logical problems, this theory is also useful for inventing solutions for many classes of logical problems.

Function-variable Elimination and Its Limitations

The famous proof method by the conventional Skolemization and resolution has a serious limitation. It does n nnot guarantee the correctness of proving theorems in the presence of built-in constraints. In order to understand n nthis difficulty, we use meaning-preserving Skolemization (MPS) and equivalent transformation (ET), which n ntogether provide a general framework for solving query-answering (QA) problems on first-order logic. We n nintroduce a rule for function variable elimination (FVE), by which we regard the conventional Skolemization n nas a kind of the composition of MPS and FVE. We prove that the FVE rule preserves the answers to a class n nof QA problems consisting of only user-defined atoms, while we cannot prove it in the presence of built-in n nconstraints. By avoiding the application of the FVE rule in MPS & ET computation, we obtain a more general n nsolution for proof problems, which guarantees the correctness of computation even in the presence of built-in n nconstraints.

A General Schema for Solving Model-Intersection Problems on a Specialization System by Equivalent Transformation

A model-intersection problem (MI problem) is a pair of a set of clauses and an exit mapping. We define MI n nproblems on specialization systems, which include many useful classes of logical problems, such as proof n nproblems on first-order logic and query-answering (QA) problems in pure Prolog and deductive databases. n nThe theory presented in this paper makes clear the central and fundamental structure of representation and n ncomputation for many classes of logical problems by (i) axiomatization and (ii) equivalent transformation. n nClauses in this theory are constructed based on abstract atoms and abstract operation on them, which can be n nused for representation of many specific subclasses of problems with concrete syntax. Various computation n ncan be realized by repeated application of many equivalent transformation rules, allowing many possible n ncomputation procedures, for instance, computation procedures based on resolution and unfolding. This theory n ncan also be useful for inventing solutions for new classes of logical problems.

Solving Query-answering Problems with If-and-Only-If Formulas

A query-answering 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. A method for solving QA problems on full first-order logic has been invented based on the equivalent n ntransformation (ET) principle, where a given QA problem on first-order logic is converted into a QA problem n non extended clauses and is then further transformed repeatedly and equivalently into simpler forms until its n nanswer set can be readily obtained. In this paper, such a clause-based solution is extended by proposing n na new method for effectively utilizing a universally quantified if-and-only-if statement defining a predicate, n nwhich is called an iff-formula. The background knowledge of a given QA problem is separated into two parts: n n(i) a conjunction of iff-formulas and (ii) other types of knowledge. Special ET rules for manipulating iffformulas n nare introduced. The new solution method deals with both iff-knowledge in first-order logic and a set n nof extended clauses. Application of this solution method is illustrated.

Solving query-answering problems using all-solution satisfiability solvers: Algorithm and correctness

Query-answering (QA) problems have attracted wider interest recently, owing partly to emerging applications involving integration between rules and ontologies in the Semantic Webs layered architecture. Success of using satisfiability (SAT) solvers as low-level solvers for dealing with several kinds of problems motivates us to apply SAT solvers to a class of QA problems. To find all elements of an answer set, a variant of a usual SAT solver, called an all-solution satisfiability (All-SAT) solver, is used for determining all models of a given input set of propositional clauses. Using an All-SAT solver, we propose a procedure for solving QA problems. The procedure generates input for an All-SAT solver from a high-level description of a given QA problem by problem transformation in a clause space that includes set-bounded variables. Based on the equivalent transformation principle, the correctness of the proposed procedure is shown.

An ET-Based Low-Level Solution for Query-Answering Problems

Query-answering QA problems have attracted wider attention in recent years. Methods for solving QA problems based on the equivalent transformation ET principle have been recently developed. Meanwhile efficient satisfiability solvers SAT solvers have been invented and successfully applied to many kinds of problems. In this paper, we propose an ET-based low-level solution for QA problems. By slightly modifying it, we also propose a low-level solution using an all-solution SAT solver. We show that the obtained SAT-solver-based solution can also be seen as another ET-based low-level solution. Our findings clarify that the ET principle supports not only high-level computation but also low-level computation, and it provides a formal basis for correctness verification of computation in both levels.

Solving Query-Answering Problems for the Semantic Web Using Equivalent Transformation

knowledge of the problem are converted into a conjunction of first-order formulas, which is further converted by meaning-preserving Skolemization into a set of extended 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. ET rules for unfolding extended clauses, for removing useless extended clauses, and for dealing with function variables are presented. Application of the framework is illustrated.