Existential second-order logic and modal logic with quantified accessibility relations

Lauri Hella, Antti Kuusisto

Abstract: This article investigates the role of arity of second-order quantifiers in existential second-order logic, also known as Σ11<math><msubsup is="true"><mrow is="true"><mi mathvariant="normal" is="true">Σ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup></math>. We identify fragments L of Σ11<math><msubsup is="true"><mrow is="true"><mi mathvariant="normal" is="true">Σ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup></math> where second-order quantification of relations of arity k>1<math><mi is="true">k</mi><mo is="true">&gt;</mo><mn is="true">1</mn></math> is (nontrivially) vacuous in the sense that each formula of L can be translated to a formula of (a fragment of) monadic Σ11<math><msubsup is="true"><mrow is="true"><mi mathvariant="normal" is="true">Σ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup></math>. Let polyadic Boolean modal logic with identity (PBML=<math><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">PBML</mi></mrow><mrow is="true"><mo is="true">=</mo></mrow></msup></math>) be the logic obtained by extending standard polyadic multimodal logic with built-in identity modalities and with constructors that allow for the Boolean combination of accessibility relations. Let Σ11(PBML=)<math><msubsup is="true"><mrow is="true"><mi mathvariant="normal" is="true">Σ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">PBML</mi></mrow><mrow is="true"><mo is="true">=</mo></mrow></msup><mo stretchy="false" is="true">)</mo></math> be the extension of PBML=<math><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">PBML</mi></mrow><mrow is="true"><mo is="true">=</mo></mrow></msup></math> with existential prenex quantification of accessibility relations and proposition symbols. The principal result of the article is that Σ11(PBML=)<math><msubsup is="true"><mrow is="true"><mi mathvariant="normal" is="true">Σ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">PBML</mi></mrow><mrow is="true"><mo is="true">=</mo></mrow></msup><mo stretchy="false" is="true">)</mo></math> translates into monadic Σ11<math><msubsup is="true"><mrow is="true"><mi mathvariant="normal" is="true">Σ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup></math>. As a corollary, we obtain a variety of decidability results for multimodal logic. The translation can also be seen as a step towards establishing whether every property of finite directed graphs expressible in Σ11(FO2)<math><msubsup is="true"><mrow is="true"><mi mathvariant="normal" is="true">Σ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">FO</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> is also expressible in monadic Σ11<math><msubsup is="true"><mrow is="true"><mi mathvariant="normal" is="true">Σ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">1</mn></mrow></msubsup></math>. This question was left open in the 1999 paper of Grädel and Rosen in the 14th Annual IEEE Symposium on Logic in Computer Science.