Semantics-Based Translation Methods for Modal Logics

Abstract

A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known 'relational' translation makes the modal logic's possible worlds structure explicit by introducing a distinguished predicate symbol to represent the accessibility relation. In the second approach, the 'functional' translation method, paths in the possible worlds structure are represented by compositions of functions which map worlds to accessible worlds. On the syntactic level this means that every flexible symbol is parametrized with particular terms denoting whole paths from the initial world to the actual world. The 'target logic' for the translation is a first-order many-sorted logic with built in equality. Therefore the 'source logic' may also be first-order many-sorted with built in equality. Furthermore flexible function symbols are allowed. The modal operators may be parametrized with arbitrary terms and particular properties of the accessibility relation may be specified within the logic itself.

A Multi-level Approach to Program Synthesis

“…For the sake of completeness axioms encoding the properties of the accessibility relation R will be added which then must imply the translated formula. This technique is known as the relational translation [26,27,3]. A major difficulty of this approach is the potential undecidability of the resulting classical formula.…”

Problem-oriented applications of automated theorem proving

“…The core of each ATP-system is the inference machine which amounts to sort of a "microprocessor" for theorem proving [Ohl91]. A formula to be proved usually is preprocessed by some input layer in order to transform it into the "machine language" of the inference machine.…”

“…A popular example is the normal form transformation (NFT) which serves to translate arbitrary first order formulae into clause-form which is the preferred language of most inference machines. Another important field of translation is formed by the so-called "logic morphisms" [Ohl91]. They basically aim at encoding semantical meta-knowledge for non-classical logics within the language of classical logic.…”

The user modeling shell system BGP-MS

“…To achieve this, a procedure is needed that translates modal expressions into predicate calculus while preserving their original semantics. The so-called functional translation [Ohlbach, 1991] is a method based on the standard possible-worlds semantics for modal logic that transforms sequences of modal operators into complex logical terms. The transformation uses context access functions that map worlds to accessible worlds, and consecutively applies one function for each modal operator to the initial world.…”