# Leonardo Pacheco

Studying Mathematical Logic at Tohoku University

## Recent Presentations

"The alternation hierarchy of the $$\mu$$-calculus over weakly transitive frames" at 28th Workshop on Logic, Language, Information and Computation (with Kazuyuki Tanaka)

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. Slides.

"Determinacy and reflection principles in second-order arithmetic" at Workshop on Reverse Mathematics and its Philosophy (joint presentation with Keita Yokoyama)

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.

"Determinacy and reflection principles in second-order arithmetic" at The second Japan-Russia workshop on effective descriptive set theory, computable analysis and automata

Working in second order arithmetic, we characterize determinacy of finite differences of open and $$\Sigma^0_2$$ sets as reflection principles. In order to do so we study sequences of coded $$\beta$$-models. More precisely, we prove that: over $$\mathsf{ACA}_0$$, $$\Pi^1_2$$-$$\mathsf{Ref}(\mathsf{ACA}_0)$$ is equivalent to $$\forall n.(\Sigma^0_1)_n$$-$$\mathsf{Det}^*_0$$, and $$\Pi^1_3$$-$$\mathsf{Ref}(\Pi^1_1$$-$$\mathsf{CA}_0)$$ is equivalent to $$\forall n.(\Sigma^0_1)_n$$-$$\mathsf{Det}$$; over $$\mathsf{ATR}_0$$, $$\Pi^1_3$$-$$\mathsf{Ref}(\Pi^1_2$$-$$\mathsf{CA}_0)$$ is equivalent to $$\forall n.(\Sigma^0_2)_n$$-$$\mathsf{Det}$$. This is joint work with Keita Yokoyama.

"On the degrees of ignorance: via epistemic logic and μ-Calculus" at SOCREAL 2022

This is joint work with Kazuyuki Tanaka. The extended abstract is available in the booklet here.

"Sequences of $$\beta_k$$-models and reflection in second-order arithmetic" at RIMS共同研究（公開系）「証明と計算の論理と応用」

We study the reflection axioms of the form: if a $$\Pi^1_n$$ sentence $$\varphi$$ is provable in $$T$$, then $$\varphi$$ is true (for a fixed theory $$T$$). We prove the relation between the existence of sequence of $$\beta_k$$ models and reflection axioms for theories of dependent choices. We also comment on the consequences of these results for determinacy in second-order arithmetic. This is a joint work with Keita Yokoyama.

"On the $$\mu$$-calculus between $$\mathsf{S4}$$ and $$\mathsf{S5}$$" at 1o. Enc(ue-o)ntro de Logica Brasil-Col(o-ô)mbia

We investigate the alternation hierarchy of the $$\mu$$-calculus in some modal logics commonly used for epistemic logic.

"Modal semantics for epistemic logic" at 数学基礎論若手の会 2021

Epistemic logic is the logic of knowledge, belief and related notions. In this presentation I define and motivate three different modal semantics for epistemic logic: relational semantics, topological semantics and neighborhood semantics. I also comment on the relation between these semantics and common knowledge. I assume no knowledge of modal logic.

## Publications

PACHECO, Leonardo, YOKOYAMA, Keita, “Determinacy and reflection principles in second-order arithmetic” (arXiv:2209.04082, preprint).

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 $$\mathsf{ACA}_0$$. Preprint.

PACHECO, Leonardo, TANAKA, Kazuyuki, “The alternation hierarchy of the mu-calculus over weakly transitive frames” (To appear, accepted for the proceedings of WoLLIC 2022 – 28th Workshop on Logic, Language, Information and Computation).

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.

PACHECO, Leonardo, TANAKA, Kazuyuki, “On the degrees of ignorance: via epistemic logic and mu-calculus”, Proceedings of SOCREAL 2022: 6th International Workshop on Philosophy and Logic of Social Reality (2022), 74–78.

Abstract: Epistemic Logic normally discourses on knowledge, belief, and related concepts. We here study ignorance instead. With the help of the $$\mu$$-calculus, we analyze the degrees of ignorance in which an agent doesn't know whether or not a given proposition is true. Building on the study by Stalnaker, we argue that logics closer'' to $$\mathsf{S4.2}$$ allow greater degrees of ignorance, compared to logics closer'' to $$\mathsf{S5}$$.

PACHECO, Leonardo, LI, Wenjuan, TANAKA, Kazuyuki, “On one-variable fragments of modal mu-calculus”, Proc. of CTFM 2019, World Scientific Publ. (2022), 17-45.

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.

## Education

• 2020-current: Mathematics PhD candidate at Tohoku University
• 2018-2020: Mathematics Masters at Tohoku University
• Thesis: On the weak hierarchy of $$\mu$$-calculus

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 $$\Sigma^0_2$$.

• 2014-2017: Mathematics Major at Universidade Federal de Goiás
• 2012-2013: Computer Science Major at Universidade Federal de Goiás (Incomplete)

## Contact

Mail: leonardovpacheco [at] gmail [dot] com
Mastodon: @leonardopacheco@mathstodon.xyz