# Asta (formerly Andreas) Halkjær From

I am a PhD student at DTU Compute since February 2020 working on Formally Correct Deduction Methods for Computational Logic, especially using the proof assistant Isabelle/HOL. My supervisor is Jørgen Villadsen and my co-supervisor is Nina Gierasimczuk.

I was a MSc student on the Honours Programme in Computer Science and Engineering at DTU from 2018 to 2020. My MSc thesis on Hybrid Logic was supervised by Jørgen Villadsen, Alexander Birch Jensen and Patrick Blackburn.

• Co-chair of the Logic and Computation track for the Student Session @ 32nd European Summer School in Logic, Language and Information (ESSLLI 2021).

### Awards

• Travel scholarship awarded by DTU (February 2022).

### Articles

#### Published formalizations

• A Naive Prover for First-Order Logic in Archive of Formal Proofs (March 2022)
Asta Halkjær From

The AFP entry Abstract Completeness by Blanchette, Popescu and Traytel formalizes the core of Beth/Hintikka-style completeness proofs for first-order logic and can be used to formalize executable sequent calculus provers. In the Journal of Automated Reasoning, the authors instantiate the framework with a sequent calculus for first-order logic and prove its completeness. Their use of an infinite set of proof rules indexed by formulas yields very direct arguments. A fair stream of these rules controls the prover, making its definition remarkably simple. The AFP entry, however, only contains a toy example for propositional logic. The AFP entry A Sequent Calculus Prover for First-Order Logic with Functions by From and Jacobsen also uses the framework, but uses a finite set of generic rules resulting in a more sophisticated prover with more complicated proofs.

This entry contains an executable sequent calculus prover for first-order logic with functions in the style presented by Blanchette et al. The prover can be exported to Haskell and this entry includes formalized proofs of its soundness and completeness. The proofs are simpler than those for the prover by From and Jacobsen but the performance of the prover is significantly worse.

The included theory Fair-Stream first proves that the sequence of natural numbers 0, 0, 1, 0, 1, 2, etc. is fair. It then proves that mapping any surjective function across the sequence preserves fairness. This method of obtaining a fair stream of rules is similar to the one given by Blanchette et al. The concrete functions from natural numbers to terms, formulas and rules are defined using the Nat-Bijection theory in the HOL-Library.

• A Sequent Calculus Prover for First-Order Logic with Functions in Archive of Formal Proofs (February 2022)
Asta Halkjær From, Frederik Krogsdal Jacobsen

We formalize an automated theorem prover for first-order logic with functions. The proof search procedure is based on sequent calculus and we verify its soundness and completeness using the Abstract Soundness and Abstract Completeness theories. Our analytic completeness proof covers both open and closed formulas. Since our deterministic prover considers only the subset of terms relevant to proving a given sequent, we do so as well when building a countermodel from a failed proof. We formally connect our prover with the proof system and semantics of the existing SeCaV system. In particular, the prover's output can be post-processed in Haskell to generate human-readable SeCaV proofs which are also machine-verifiable proof certificates.

• Soundness and Completeness of an Axiomatic System for First-Order Logic in Archive of Formal Proofs (October 2021)
Asta Halkjær From

This work is a formalization of the soundness and completeness of an axiomatic system for first-order logic. The proof system is based on System Q1 by Smullyan and the completeness proof follows his textbook "First-Order Logic" (Springer-Verlag 1968). The completeness proof is in the Henkin style where a consistent set is extended to a maximal consistent set using Lindenbaum's construction and Henkin witnesses are added during the construction to ensure saturation as well. The resulting set is a Hintikka set which, by the model existence theorem, is satisfiable in the Herbrand universe.

• Public Announcement Logic in Archive of Formal Proofs (June 2021)
Asta Halkjær From

This work is a formalization of public announcement logic with countably many agents. It includes proofs of soundness and completeness for a variant of the axiom system PA + DIST! + NEC!. The completeness proof builds on the Epistemic Logic theory.

• Epistemic Logic: Completeness of Modal Logics in Archive of Formal Proofs (October 2018, updated April 2021)
Asta Halkjær From

This work is a formalization of epistemic logic with countably many agents. It includes proofs of soundness and completeness for the axiom system K. The completeness proof is based on the textbook "Reasoning About Knowledge" by Fagin, Halpern, Moses and Vardi (MIT Press 1995). The extensions of system K (T, KB, K4, S4, S5) and their completeness proofs are based on the textbook "Modal Logic" by Blackburn, de Rijke and Venema (Cambridge University Press 2001).

• Hybrid Logic in Archive of Formal Proofs (December 2019, updated June 2020)
Asta Halkjær From

The formalization accompanying my MSc thesis and TYPES 2020 paper.

This work is a formalization of soundness and completeness proofs for a Seligman-style tableau system for hybrid logic. The completeness result is obtained via a synthetic approach using maximally consistent sets of tableau blocks. The formalization differs from previous work in a few ways. First, to avoid the need to backtrack in the construction of a tableau, the formalized system has no unnamed initial segment, and therefore no Name rule. Second, I show that the full Bridge rule is admissible in the system. Third, I start from rules restricted to only extend the branch with new formulas, including only witnessing diamonds that are not already witnessed, and show that the unrestricted rules are admissible. Similarly, I start from simpler versions of the @-rules and show that these are sufficient. The GoTo rule is restricted using a notion of potential such that each application consumes potential and potential is earned through applications of the remaining rules. I show that if a branch can be closed then it can be closed starting from a single unit. Finally, Nom is restricted by a fixed set of allowed nominals. The resulting system should be terminating.

• A Sequent Calculus for First-Order Logic in Archive of Formal Proofs (July 2019)
Asta Halkjær From

This work formalizes soundness and completeness of a one-sided sequent calculus for first-order logic. The completeness is shown via a translation from a complete semantic tableau calculus, the proof of which is based on the First-Order Logic According to Fitting theory. The calculi and proof techniques are taken from Ben-Ari's Mathematical Logic for Computer Science.

• IsaFoL: Isabelle Formalization of Logic

I have contributed to a number of formalizations in the IsaFoL project:

#### Abstracts

• On the Use of Isabelle/HOL for Formalizing New Concise Axiomatic Systems for Classical Propositional Logic at TYPES 2021
Asta Halkjær From and Jørgen Villadsen
• Formalized Soundness and Completeness of Epistemic Logic at LAMAS & SR 2021
Asta Halkjær From, Alexander Birch Jensen and Jørgen Villadsen
• Formalized soundness and completeness of natural deduction for first-order logic at SLS 2018
Andreas Halkjær From.
• Teaching first-order logic with the natural deduction assistant (NaDeA) at SLS 2018
Andreas Halkjær From, Helge Hatteland, Jørgen Villadsen.

### Theses

• MSc: Hybrid Logic (January 2020)

My MSc thesis on formalizing the soundness and completeness of a Seligman-style tableau system for hybrid logic in Isabelle/HOL.

• BSc: Formalized First-Order Logic (July 2017)

My BSc thesis on formalizing the soundness and completeness of NaDeA in Isabelle/HOL.

Report, Slides (Danish)

### Work Experience

#### Teaching Assistant

I have been a TA in the following courses:
• 02101 Introductory Programming, fall 2015
• 02105 Algorithms and Data Structures 1, spring 2016
• 02101 Introductory Programming, fall 2016
• 02110 Algorithms and Data Structures 2, fall 2016
• 02105 Algorithms and Data Structures 1, spring 2017
• 02180 Introduction to Artificial Intelligence, spring 2018
• 02156 Logical Systems and Logic Programming, fall 2018
• 02156 Logical Systems and Logic Programming, fall 2019
• 02102 Introductory Programming, spring 2020
• 02256 Automated Reasoning, spring 2020
• 02287 Logical Theories for Uncertainty and Learning, fall 2020
I have also helped teach kids aged 9-12 years old game programming in MIT Scratch at Hello World.

#### Research Assistant

I worked as a research assistant at DTU Compute from August 2017 to December 2017, working with Isabelle and NaDeA, especially translating students' proofs in NaDeA to their encodings in Isabelle.