Mathematical Logic at the Institute of Science Tokyo
I'm a postdoctoral researcher at the Institute of Science Tokyo working with Ryo Kashima. My research is focused on non-classical modal logics, with focus on fixed-point and provability logics.
We give a short overview of recent results on non-classical modal logics, including both semantical and proof-theoretical results.
We study \(\mu\mathsf{CK}\), a variant of the modal μ-calculus based on the constructive modal logic \(\mathsf{CK}\). We define game semantics for the constructive μ-calculus and use it to prove the soundness and completeness of a non-wellfounded proof system. We also describe how to adapt the game semantics and proof system to the \(\mu\)-calculus over other non-classical modal logics.
We develop bi-relational and real valued semantics for a Gödel--Dummett--Gödel--Löb Logic \(\mathsf{GGL}\). We study a fully labeled proof system for it in order to obtain its finite model property and decidability. We use the finite model property to show that both semantics have the same validities.
Artemov and Protopopescu introduced IEL, a modal extension of intuitionistic propositional logic with a modal operator K, standing for knowledge. The K operator is governed by two characteristic axioms: P -> KP and KP -> ~~P. We define and study an alternative semantics for IEL where K is evaluated as a constructive diamond. We show completeness and FMP for our semantics. For applications, we have the following: we separate intuitionistic knowledge and intuitionistic belief; we discuss the relation between Glivenko's Theorem and ignorance; we show that in some sense IEL is indeed intuitionistic, but in another sense it is not. (This is joint work with Igor Sedlár, Czech Academy of Science)
\(\mathsf{IGL}\) is an intuitionistic version of the provability logic \(\mathsf{GL}\) using both box and diamond modalities. It was first studied by Das, van der Giessen, and Marin, who provided two ill-founded proof systems and two semantics for this logic. Later, Aguilera and P. defined a cyclic proof system \(\mathsf{cm\ell IGL}\) for \(\mathsf{IGL}\); they also proved that the set of theorems of \(\mathsf{IGL}\) is recursively enumerable.
We define a well-founded labeled proof system \(\mathsf{\omega m\ell IGL}\) for \(\mathsf{IGL}\) characterized by the following \(\omega\)-rule: \[ \frac{x:\Box^n\bot, \mathbf{R}, \Gamma \vdash \Delta \; (\forall n\in\omega)}{\mathbf{R},\Gamma \vdash \Delta}, \] where \(x\) is a label variable. To prove that all valid formulas of \(\mathsf{IGL}\) can be proved using the \(\omega\)-rule, we use a proof search game argument. To show the soundness of the \(\omega\)-rule are valid, we closely analyze the completeness of \(\mathsf{cm\ell IGL}\). Note that an \(\omega\)-rule for classical \(\mathsf{GL}\) was studied by Tanaka.
Das, van der Giessen and Marin recently defined an intuitionistic version of the provability logic \(\mathsf{GL}\). They define birelational and predicate semantics and two non-wellfounded proof systems \(\ell\mathsf{IGL}\) and \(m\ell\mathsf{IGL}\). They prove the completeness and soundness of the two proof systems with respect to both semantics. In the proof of the completeness of \(m\ell\mathsf{IGL}\) with respect to the predicate semantics, they use \(\Sigma^1_1\)-determinacy; a statement not provable in \(\mathsf{ZFC}\). We define a cyclic proof system \(c\ell\mathsf{IGL}\) for IGL and prove its completeness with respect to predicate semantics using open determinacy. In particular, this implies that the completeness of \(m\ell\mathsf{IGL}\) does not need \(\Sigma^1_1\)-determinacy.
Feedback Turing machines are Turing machines which can query a halting oracle which has information on the convergence or divergence of feedback computations. To avoid a contradiction by diagonalization, feedback Turing machines have two ways of not converging: they can diverge as standard Turing machines, or they can freeze. A natural question to ask is: what about feedback Turing machines which can ask if computations of the same type converge, diverge, or freeze? We define \(\alpha\)th order feedback Turing machines for each computable ordinal \(\alpha\). We also describe feedback computable and semi-computable sets using inductive definitions and Gale–Stewart games.
The modal \(\mu\)-calculus is obtained by adding least and greatest fixed-point operators to modal logic. It's alternation hierarchy classifies the \(\mu\)-formulas by their alternation depth: a measure of the codependence of their least and greatest fixed-point operators. The \(\mu\)-calculus' alternation hierarchy is strict over the class of all Kripke frames: for all \(n\), there is a \(\mu\)-formula with alternation depth \(n+1\) which is not equivalent to any formula with alternation depth \(n\). This does not always happen if we restrict the semantics. For example, every \(\mu\)-formula is equivalent to a formula without fixed-point operators over \(\mathsf{S5}\) frames. We show that the multimodal \(\mu\)-calculus' alternation hierarchy is strict over non-trivial fusions of modal logics. We also comment on two examples of multimodal logics where the \(\mu\)-calculus collapses to modal logic.
It is known that the \(\mu\)-calculus collapses to its alternation-free fragment over transitive frames and to modal logic over equivalence relations. We adapt a proof by D'Agostino and Lenzi to show that the \(\mu\)-calculus collapses to its alternation-free fragment over weakly transitive frames. As a consequence, we show that the \(\mu\)-calculus with derivative topological semantics collapses to its alternation-free fragment. We also study the collapse over frames of \(\mathsf{S4.2}\), \(\mathsf{S4.3}\), \(\mathsf{S4.3.2}\), \(\mathsf{S4.4}\) and \(\mathsf{KD45}\), logics important for Epistemic Logic. At last, we use the \(\mu\)-calculus to define degrees of ignorance on Epistemic Logic and study the implications of \(\mu\)-calculus's collapse over the logics above. This is joint work with Kazuyuki Tanaka. Slides.
See here for a full list of presentations.
The modal \(\mu\)-calculus is obtained by adding least and greatest fixed-point operators to modal logic. It's alternation hierarchy classifies the \(\mu\)-formulas by their alternation depth: a measure of the codependence of their least and greatest fixed-point operators. The \(\mu\)-calculus' alternation hierarchy is strict over the class of all Kripke frames: for all \(n\), there is a \(\mu\)-formula with alternation depth \(n+1\) which is not equivalent to any formula with alternation depth \(n\). This does not always happen if we restrict the semantics. For example, every \(\mu\)-formula is equivalent to a formula without fixed-point operators over \(\mathsf{S5}\) frames. We show that the multimodal \(\mu\)-calculus' alternation hierarchy is strict over non-trivial fusions of modal logics. We also comment on two examples of multimodal logics where the \(\mu\)-calculus collapses to modal logic.
Das, van der Giessen, and Marin recently introduced \(\mathsf{IGL}\), an intuitionistic version of Gödel-Löb logic. Their proof systems involves ill-founded proofs with a progressiveness condition. Their completeness proof uses the principle of \(\Sigma^1_1\)-determinacy; which is not provable in \(\mathsf{ZFC}\). We define a cyclic proof system for \(\mathsf{IGL}\) and give a proof of its completeness theorem avoiding \(\Sigma^1_1\)-determinacy.
Feedback Turing machines are Turing machines which can query a halting oracle which has information on the convergence or divergence of feedback computations. To avoid a contradiction by diagonalization, feedback Turing machines have two ways of not converging: they can diverge as standard Turing machines, or they can freeze. A natural question to ask is: what about feedback Turing machines which can ask if computations of the same type converge, diverge, or freeze? We define \(\alpha\)th order feedback Turing machines for each computable ordinal \(\alpha\). We also describe feedback computable and semi-computable sets using inductive definitions and Gale–Stewart games.
Abstract: It is known that the \(\mu\)-calculus collapses to its alternation-free fragment over transitive frames and to modal logic over equivalence relations. We adapt a proof by D'Agostino and Lenzi to show that the \(\mu\)-calculus collapses to its alternation-free fragment over weakly transitive frames. As a consequence, we show that the \(\mu\)-calculus with derivative topological semantics collapses to its alternation-free fragment. We also study the collapse over frames of \(\mathsf{S4.2}\), \(\mathsf{S4.3}\), \(\mathsf{S4.3.2}\), \(\mathsf{S4.4}\) and \(\mathsf{KD45}\), logics important for Epistemic Logic. At last, we use the \(\mu\)-calculus to define degrees of ignorance on Epistemic Logic and study the implications of \(\mu\)-calculus's collapse over the logics above.
We study a variant of the modal \(\mu\)-calculus based on the constructive modal logic \(\mathsf{CK}\). We define game semantics for the constructive \(\mu\)-calculus and prove its equivalence to the birelational Kripke semantics. We then use the game semantics to prove the soundness and completeness of a fully-labeled non-wellfounded proof system for it. At last, we briefly describe how to adapt the game semantics and proof system to the \(\mu\)-calculus over other non-classical modal logics.
We develop polytopological semantics for various constructive, intuitionistic, and Gödel-Dummett variations of \(\mathsf{K4}\) and \(\mathsf{S4\). In our models, intuitionistic and modal operators are interpreted via various topologies over a single set, equipped with either the closure or derivative operators. We identify regularity conditions to ensure that spaces validate each of our target logics and prove that all the logics considered are sound and strongly complete with respect to their respective semantics.
In this note, we prove that the constructive and intuitionistic variants of the modal logic \(\mathsf{KB}\) coincide. This result contrasts with a recent result by Das and Marin, who showed that the constructive and intuitionistic variants of \(\mathsf{K}\) do not prove the same diamond-free formulas.
Abstract: It is known that several variations of the axiom of determinacy play important roles in the study of reverse mathematics, and the relation between the hierarchy of determinacy and comprehension are revealed by Tanaka, Nemoto, Montalbán, Shore, and others. We prove variations of a result by Kołodziejczyk and Michalewski relating determinacy of arbitrary boolean combinations of \(\Sigma^0_2\) sets and reflection in second-order arithmetic. Specifically, we prove that: over \(\mathsf{ACA}_0\), \(\Pi^1_2\)-\(\mathsf{Ref}(\mathsf{ACA}_0)\) is equivalent to \(\forall n.(\Sigma^0_1)_n\)-\(\mathsf{Det}^*_0\); \(\Pi^1_3\)-\(\mathsf{Ref}(\Pi^1_1\)-\(\mathsf{CA}_0)\) is equivalent to \(\forall n.(\Sigma^0_1)_n\)-\(\mathsf{Det}\); and \(\Pi^1_3\)-\(\mathsf{Ref}(\Pi^1_2\)-\(\mathsf{CA}_0)\) is equivalent to \(\forall n.(\Sigma^0_2)_n\)-\(\mathsf{Det}\). We also restate results by Montalbán and Shore to show that \(\Pi^1_3\)-\(\mathsf{Ref}(\mathsf{Z}_2)\) is equivalent to \(\forall n.(\Sigma^0_3)_n\)-\(\mathsf{Det}\) over \(\mathsf{ACA}_0\).
See here for a full list of publications.
Available here.
Short abstract: In this thesis, we study two problems related to difference hierarchies. The difference hierarchy for a point class \(\Gamma\) classifies the Boolean combinations of sets in \(\Gamma\) by their complexity. Gale–Stewart games play essential roles in both problems.
In the first part of this thesis, we study the \(\mu\)-calculus' alternation hierarchy over various semantics. The \(\mu\)-calculus' alternation hierarchy classifies its formulas by how many interdependent fixed-point operators appear in a given formula. Bradfield showed that the alternation hierarchy is strict over arbitrary frames. This may not happen if we modify the semantics.
We refine Alberucci and Facchini's proof of the collapse to modal logic over equivalence relations to show that the alternation hierarchy collapses to modal logic in bigger classes of frames. We use this characterization to study degrees of ignorance in various epistemic logics. Afterwards, we show that, on graded semantics, constructive semantics and modal logic with impossible worlds, the alternation hierarchy collapses to modal logic over equivalence relations. On the other hand, the alternation hierarchy is strict on multimodal \(\mu\)-calculus over equivalence relations. We also show that current proofs of the collapse do not work on the non-monotone \(\mu\)-calculus. Furthermore, we show that the alternation hierarchy collapses to its alternation-free fragment over weakly transitive frames.
In the second part of this thesis, we study the connection between Gale–Stewart games and reflection principles in second-order arithmetic. Gale–Stewart games have been studied in reverse mathematics since its beginning and are central to descriptive set theory. Sets definable by the \(\mu\)-calculus are exactly the winning regions of Gale–Stewart games whose payoffs are Boolean combinations of \(\Sigma^0_2\) sets.
Kołodziejczyk and Michalewski proved that the determinacy of Boolean combinations of \(\Sigma^0_2\) sets is equivalent to the reflection principle \(\Pi^1_3\)-\(\mathrm{Ref}(\Pi^1_2\)-\(\mathsf{CA}_0)\). We use finite sequences of coded \(\beta\)-models to prove that the determinacy of Boolean combinations of \(\Sigma^0_1\) sets is equivalent to the reflection principle \(\Pi^1_3\)-\(\mathrm{Ref}(\Pi^1_1\)-\(\mathsf{CA}_0)\). We also use the same method to give a new proof of Kołodziejczyk and Michalewski's result. At last, we use a modified version of the method to prove that the determinacy of Boolean combinations of \(\Sigma^0_1\) sets of Cantor space is equivalent to the reflection principle \(\Pi^1_2\)-\(\mathrm{Ref}(\mathsf{ACA}_0)\).
Available here.
In this thesis we present the weak \(\mu\)-calculus and weak \(\mu\)-arithmetic and their alternations hierarchies. We also prove a refinement of a result in Reverse Mathematics related to the \(\mu\)-arithmetic and the determinacy of the finite levels of the difference hierarchy of \(\Sigma^0_2\).
Mail: leonardovpacheco@gmail.com
CV: available here
Blog: https://leonardopacheco.xyz/blog
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