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 games in logic. I am working in Modal Logic (especially on the modal \(\mu\)-calculus) and Reverse Mathematics (of Gale–Stewart games). I was previously at TU Wien.
Epistemic logics formalize knowledge and related concepts. Artemov and Protopopescu defined an epistemic logic \(\mathsf{IEL}\) to formalize intuitionistic knowledge. The central idea of this logic is that intuitionistic truth implies intuitionistic knowledge. This heavily contrasts with the classical case, where classical knowledge implies classical truth.
The modality \(K\) is interpreted in \(\mathsf{IEL}\) as \[ K\varphi \text{ holds iff it is intuitionistically known that \(\varphi\)}, \] for all formula \(\varphi\). \(\mathsf{IEL}\) satisfies the principles of co-reflection \(\varphi\to K\varphi\) and weak reflection \(K\varphi\to \neg\neg\varphi\). Note that, in a classical setting, these imply that truth and knowledge coincide.
We extend \(\mathsf{IEL}\) with a modality \(\hat K\) for epistemic possibility. We will show that, for all formula \(\varphi\), \(\hat K P\) is equivalent to \(\neg\neg P\). This implies that \(\varphi\) is epistemically possible iff it one can show that it is impossible to prove the negation of \(\varphi\).
\(\mathsf{IGL}\) 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.
We comment on behavior of constructive diamonds in three settings. First, we review a result of Das and Marin showing that the diamond-free fragment constructive and intuitionistic variants of the modal logic K do not coincide. Second, we show that the constructive and intuitionistic variants of S5 coincide. Last, we show that diamonds are equivalent to double negation in Artemov and Protopopescu's intuitionistic epistemic logic.
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.
Both the intuitionistic propositional calculus \(IPC\) and the modal logic \(S4\) are complete with respect to topological spaces. I will sketch how to combine these their topological semantics to obtain bi-topological semantics for the intuitionistic modal logic \(IS4\). Before that, I will briefly describe the history of topological semantics and \(IS4\). I will also comment on bi-topological semantics for other non-classical variations of \(S4\) and other related ongoing work.
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 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.
We prove that the constructive and intuitionistic variants of the modal logic 𝖪𝖡 coincide. This result contrasts with a recent result by Das and Marin, who showed that the constructive and intuitionistic variants of 𝖪 do not prove the same diamond-free formulas.
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 \(\mathsf{IGL}\) 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.
We describe a constructive variation of the provability logic \(\mathsf{GL}\) based on Mendler and de Paiva's constructive modal logic \(\mathsf{CK}\).
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.
Artemov and Protopopescu defined an intuitionistic epistemic logic IEL to reason about intuitionistic knowledge. While classical knowledge implies classical truth, intuitionistic truth implies intuitionistic knowledge. We describe Artemov and Protopopescu's IEL and its BHK interpretation. We characterize epistemic possibility in IEL.
I will talk about the connection between reflection principles and the existence of sequences of beta-models in second-order arithmetic. The reflection principles are of the form: if a formula is provable in a given system, then they are true. Beta-models are coded models which have the same ordinals as the ground model. I comment on how to use this connection to characterize determinacy axioms in reverse mathematics.
We define a constructive version of the \(\mu\)-calculus by adding least and greatest fixed-point operators to constructive modal logic. We define game semantics for the constructive \(\mu\)-calculus and prove its equivalence to bi-relational Kripke semantics. For applications, we study the logic \(\mu\mathsf{CS5}\), a constructive variation of \(\mathsf{S5}\) with fixed-points operators.
The \(\mu\)-calculus is obtained by adding least and greatest fixed-point operators \(\mu\) and \(\nu\) to modal logic. The alternation depth of a formula measures the entanglement of its least and greatest fixed-point operators. Bradfield showed that, for all \(n\in\mathbb{N}\), there is a formula \(W_n\) such that \(W_n\) has alternation depth \(n\) and, over all Kripke frames, \(W_n\) is not equivalent to any formula with alternation depth smaller than \(n\).
The same may not happen over restricted classes of frames: Alberucci and Facchini showed that, the \(\mu\)-calculus collapses to modal logic over \(\mathsf{S5}\) frames. That is, every \(\mu\)-formula is equivalent to a formula without fixed point operators over \(\mathsf{S5}\) frames. Later, Pacheco and Tanaka proved that the \(\mu\)-calculus also collapses to the \(\mu\)-calculus over \(\mathsf{S4.4}\) and \(\mathsf{S4.3.2}\) frames.
We show how Alberucci and Facchini's proof generalize to the \(\mu\)-calculus's collapse over \(n\)-pigeonhole frames. Let \(n\in\omega\). A frame \(F =\langle W,R\rangle\) is an \(n\)-pigeonhole frame iff, for all sequence \(w_0 R^+ w_1 R^+ \cdots R^+ w_n\), there is \(i< j\leq n\) such that \(w_{i}R = w_{j}R\). We also comment about ongoing work to prove the converse: if the \(\mu\)-calculus collapses to modal logic over a class of frames \(\mathsf{F}\), then there is \(n\in\omega\) such all frames \(F\in\mathsf{F}\) are \(n\)-pigeonhole.
(This is joint work with Kazuyuki Tanaka.)
I will review some recent results on the reverse mathematics of automata on infinite words. This is still a new subarea of reverse mathematics, but it already touches very weak and very strong subsystems of second order arithmetic. I also comment on some of my recent research on the reverse mathematics of the Wagner hierarchy.
The \(\mu\)-calculus is obtained by adding to modal logic the least and greatest fixed-point operators \(\mu\) and \(\nu\). The alternation depth of a formula measures the entanglement of its least and greatest fixed-point operators. Bradfield showed that, for all \(n\in\mathbb{N}\), there is a formula \(W_n\) such that \(W_n\) has alternation depth \(n\) and, over all Kripke frames, \(W_n\) is not equivalent to any formula with alternation depth smaller than \(n\).
The same may not happen over restricted classes of frames: Alberucci and Facchini showed that, over frames of \(\mathsf{S5}\), every \(\mu\)-formula is equivalent to a formula without fixed point operators. In this case, we say the \(\mu\)-calculus collapses to modal logic over frames of \(\mathsf{S5}\).
We show how Alberucci and Facchini’s proof generalize to the \(\mu\)-calculus’s collapse over frames of intuitionistic \(\mathsf{S5}\). This generalization can also be done for some non-normal logics and for graded modal logics. We also show that, on the other hand, the \(\mu\)-calculus does not collapse over the bimodal logic \(\mathsf{S5}_2\).
We study some consequences of the \(\mu\)-calculus' alternation hierarchy collapse on some variations of epistemic logic with only one agent; and sketch how to prove that the alternation hierarchy is strict if we have more than one agent. Slides.
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.
It is well-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 (especially the level of \(\Sigma^0_2\) and \(\Sigma^0_3\)) and comprehension axioms are revealed by Tanaka, Nemoto, Montalbán, Shore, and others. In this talk, we show variations of a result by Kołodziejczyk and Michalewski relating determinacy and reflection in second-order arithmetic based on a model-theoretic characterization of the reflection principles. Slides. Video.
More presentations here.
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.
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.
Artemov and Protopopescu defined an intuitionistic epistemic logic IEL to reason about intuitionistic knowledge. While classical knowledge implies classical truth, intuitionistic truth implies intuitionistic knowledge. We describe Artemov and Protopopescu's IEL and its BHK interpretation. We characterize epistemic possibility in IEL.
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.
We define game semantics for the constructive \(\mu\)-calculus and prove its correctness. We use these game semantics to prove that the \(\mu\)-calculus collapses to modal logic over \(\mathsf{CS5}\) frames. Finally, we prove the completeness of \(\mathsf{\mu CS5}\) over \(\mathsf{CS5}\) frames.
Abstract: We survey recent results on reflection in second-order arithmetic. The reflection principles we consider can be roughly divided into two categories: semantic reflection and syntactic reflection.
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 \(\mathsf{ACA}_0\).
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.
Abstract: In this paper, we study one-variable fragments of modal \(\mu\)-calculus and their relations to parity games. We first introduce the weak modal \(\mu\)-calculus as an extension of the one-variable modal \(\mu\)-calculus. We apply weak parity games to show the strictness of the one-variable hierarchy as well as its extension. We also consider games with infinitely many priorities and show that their winning positions can be expressed by both \(\Sigma^\mu_2\) and \(\Pi^\mu_2\) formulas with two variables, but requires a transfinite extension of the \(L_\mu\)-formulas to be expressed with only one variable. At last, we define the \(\mu\)-arithmetic and show that a set of natural numbers is definable by both a \(\Sigma^\mu_2\) and a \(\Pi^\mu_2\) formula of \(\mu\)-arithmetic if and only if it is definable by a formula of the one-variable transfinite \(\mu\)-arithmetic.
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 [at] gmail [dot] com
CV: available here
Mastodon: @leonardopacheco@mathstodon.xyz
Blog: https://leonardopacheco.xyz/blog
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