Anantha Padmanabha
Abstract
First order modal logic (𝖥𝖮𝖬𝖫) is built by extending First Order Logic (𝖥𝖮) with modal operators. A typical formula is of the form \(\forall x \exists y \Box P(x,y)\). Not only is 𝖥𝖮𝖬𝖫 undecidable, even simple fragments like that of restriction to unary predicate symbols, guarded fragment and two variable fragment, which are all decidable for 𝖥𝖮 become undecidable for 𝖥𝖮𝖬𝖫. In this paper we study Term Modal logic (𝖳𝖬𝖫) which allows modal operators to be indexed by terms. A typical formula is of the form \(\forall x \exists y~\Box _x P(x,y)\). There is a close correspondence between 𝖳𝖬𝖫 and 𝖥𝖮𝖬𝖫 and we explore this relationship in detail in the paper.
In contrast to 𝖥𝖮𝖬𝖫, we show that the two variable fragment (without constants, equality) of 𝖳𝖬𝖫 is decidable. Further, we prove that adding a single constant makes the two variable fragment of 𝖳𝖬𝖫 undecidable. On the other hand, when equality is added to the logic, it loses the finite model property.
- [1] . 1998. Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic 27, 3 (1998), 217–274.Google Scholar
Cross Ref
- [2] . 2001. Modal Logic (Cambridge Tracts in Theoretical Computer Science). Cambridge University Press.Google Scholar
- [3] . 1975. Normal forms in modal logic. Notre Dame Journal of Formal Logic 16, 2 (1975), 229–237.Google ScholarCross Ref
- [4] . 2012. First-order Modal Logic. Vol. 277. Springer Science & Business Media.Google Scholar
- [5] . 2001. Term-modal logics. Studia Logica 69, 1 (2001), 133–169.Google ScholarCross Ref
- [6] . 2009. Quantification in Nonclassical Logic. Elsevier.Google Scholar
- [7] . 1997. On the decision problem for two-variable first-order logic. Bulletin of Symbolic Logic 3, 1 (1997), 53–69.Google ScholarCross Ref
- [8] . 1999. On logics with two variables. Theoretical Computer Science 224, 1-2 (1999), 73–113.Google ScholarCross Ref
- [9] . 1993. Naming and identity in epistemic logics part I: The propositional case. Journal of Logic and Computation 3, 4 (1993), 345–378.Google ScholarCross Ref
- [10] . 2012. Small substructures and decidability issues for first-order logic with two variables. The Journal of Symbolic Logic (2012), 729–765.Google ScholarCross Ref
- [11] . 2005. Undecidability of first-order intuitionistic and modal logics with two variables. Bulletin of Symbolic Logic 11, 3 (2005), 428–438. Google ScholarCross Ref
- [12] . 2007. Dynamic term-modal logic. In A Meeting of the Minds. 173–186.Google Scholar
- [13] . 1962. The undecidability of monadic modal quantification theory. Mathematical Logic Quarterly 8, 2 (1962), 113–116.Google ScholarCross Ref
- [14] . 2020. Dynamic term-modal logics for first-order epistemic planning. Artificial Intelligence 286 (2020), 103305.Google ScholarCross Ref
- [15] . 2020. Decidability results in first-order epistemic planning. In IJCAI. 4161–4167.Google Scholar
- [16] . 2022. Generalized bundled fragments for first-order modal logic. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022) (Leibniz International Proceedings in Informatics (LIPIcs)), , , and (Eds.), Vol. 241. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 70:1–70:14. Google ScholarCross Ref
- [17] . 1975. On languages with two variables. Mathematical Logic Quarterly 21, 1 (1975), 135–140.Google ScholarCross Ref
- [18] . 2017. Decidable term-modal logics. In Multi-Agent Systems and Agreement Technologies. Springer, 147–162.Google Scholar
- [19] . 2019. Propositional Term Modal Logic. Ph.D. Dissertation. Institute of Mathematical Sciences, Homi Bhabha National Institute.Google Scholar
- [20] . 2019. The monodic fragment of propositional term modal logic. Studia Logica 107, 3 (2019), 533–557.Google ScholarDigital Library
- [21] . 2019. Propositional modal logic with implicit modal quantification. In Logic and Its Applications - 8th Indian Conference, ICLA 2019, Delhi, India, March 1-5, 2019, Proceedings (Lecture Notes in Computer Science), and (Eds.), Vol. 11600. Springer, 6–17. Google ScholarDigital Library
- [22] . 2019. Two variable fragment of term modal logic. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.Google Scholar
- [23] . 2018. Bundled fragments of first-order modal logic: (Un) decidability. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science.Google Scholar
- [24] . 2010. Epistemic term-modal logic. Proceedings of the 15th Student Session of the European Summer School in Logic, Language and Information (2010), 37–46.Google Scholar
- [25] . 2019. Undecidability of first-order modal and intuitionistic logics with two variables and one monadic predicate letter. Studia Logica 107, 4 (2019), 695–717.Google ScholarDigital Library
- [26] . 2021. A Proof-Theoretic Study of Term-Sequence-Dyadic Deontic Logic and Common Sense Modal Predicate Logic. Ph.D. Dissertation. Hokkaido University.Google Scholar
- [27] . 2019. Term-sequence-modal logics. In International Workshop on Logic, Rationality and Interaction. Springer, 244–258.Google Scholar
- [28] . 2018. Propositional epistemic logics with quantification over agents of knowledge. Studia Logica 106, 2 (2018), 311–344.Google ScholarDigital Library
- [29] . 2000. Term-modal Logic and Quantifier-free Dynamic Assignment Logic. Ph.D. Dissertation. Acta Universitatis Upsaliensis.Google Scholar
- [30] . 1996. Why is modal logic so robustly decidable?. In Descriptive Complexity and Finite Models, Proceedings of a DIMACS Workshop 1996, Princeton, New Jersey, USA, January 14-17, 1996 (DIMACS Series in Discrete Mathematics and Theoretical Computer Science), and (Eds.), Vol. 31. DIMACS/AMS, 149–183. Google ScholarCross Ref
- [31] . 2017. A new modal framework for epistemic logic. In Proceedings Sixteenth Conference on Theoretical Aspects of Rationality and Knowledge, TARK 2017, Liverpool, UK, 24-26 July 2017.515–534. Google ScholarCross Ref
- [32] . 2018. Beyond knowing that: A new generation of epistemic logics. In Jaakko Hintikka on Knowledge and Game Theoretical Semantics. Springer, 499–533.Google Scholar
- [33] . 2022. Quantifier-free epistemic term-modal logic with assignment operator. Annals of Pure and Applied Logic 173, 3, 103071.Google ScholarCross Ref
- [34] . 2001. Decidable fragments of first-order modal logics. The Journal of Symbolic Logic 66, 3 (2001), 1415–1438.Google ScholarCross Ref
- [35] Anantha Padmanabha. 2021. Relative expressive powers of first order modal logic and term modal logic. In Proceedings of the (ICLA’21). Vol. 97. https://scholar.google.com/citations?view_op=view_citation&hl=en&user=10ihAiIAAAAJ&citation_for_view10ihAiIAAAAJ:WF5omc3nYNoC.Google Scholar
Index Terms
- A Decidable Fragment of First Order Modal Logic: Two Variable Term Modal Logic
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