The Diproche system is an automated proof checker for texts written in a controlled fragment of German, designed for didactical applications in classes introducing students to proofs for the first time. The first version of the system used a controlled natural language for which a Prolog formalization routine was written. In this paper, we explore the possibility of prompting large language models for autoformalization in the context of Diproche, with encouraging first results.

The minimization of propositional formulae is a classical problem in logic, whose first algorithms date back at least to the 1950s with the works of Quine and Karnaugh. Most previous work in the area has focused on obtaining minimal, or quasi-minimal, formulae in conjunctive normal form (CNF) or disjunctive normal form (DNF), with applications in hardware design. In this paper, we are interested in the problem of obtaining an equivalent formula in any format, also allowing connectives that are not present in the original formula. We are primarily motivated in applying minimization algorithms to generate natural language translations of the original formula, where using shorter equivalents as input may result in better translations. Recently, Buchfuhrer and Umans have proved that the (decisional version of the) problem is $\Sigma_2^p$-complete. We analyze three possible (practical) approaches to solving the problem. First, using brute force, generating all possible formulae in increasing size and checking if they are equivalent to the original formula by testing all possible variable assignments. Second, generating the Tseitin coding of all the formulae and checking equivalence with the original using a SAT solver. Third, encoding the problem as a Quantified Boolean Formula (QBF), and using a QBF solver. Our results show that the QBF approach largely outperforms the other two.

Recent developments in support for constructing knowledge graphs have led to a rapid rise in their creation both on the Web and within organisations. Added to existing sources of data, including relational databases, APIs, etc., there is a strong demand for techniques to query these diverse sources of knowledge. While formal query languages, such as SPARQL, exist for querying some knowledge graphs, users are required to know which knowledge graphs they need to query and the unique resource identifiers of the resources they need. Although alternative techniques in neural information retrieval embed the content of knowledge graphs in vector spaces, they fail to provide the representation and query expressivity needed (e.g. inability to handle non-trivial aggregation functions such as regression). We believe that a lingua franca, i.e. a formalism, that enables such representational flexibility will increase the ability of intelligent automated agents to combine diverse data sources by inference. Our work proposes a flexible representation (alists) to support intelligent federated querying of diverse knowledge sources. Our contribution includes (1) a formalism that abstracts the representation of queries from the specific query language of a knowledge graph; (2) a representation to dynamically curate data and functions (operations) to perform non-trivial inference over diverse knowledge sources; (3) a demonstration of the expressiveness of alists to represent the diversity of representational formalisms, including SPARQL queries, and more generally first-order logic expressions.

As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically proves about 60 % of the Mizar theorems in the hammer setting. We also automatically prove 75 % of the Mizar theorems when the automated provers are helped by using only the premises used in the human-written Mizar proofs. We describe the methods and large-scale experiments leading to these results. This includes in particular the E and Vampire provers, their ENIGMA and Deepire learning modifications, a number of learning-based premise selection methods, and the incremental loop that interleaves growing a corpus of millions of ATP proofs with training increasingly strong AI/TP systems on them. We also present a selection of Mizar problems that were proved automatically.

The introduction of automated deduction systems in secondary schools faces several bottlenecks. Beyond the problems related with the curricula and the teachers, the dissonance between the outcomes of the geometry automated theorem provers and the normal practice of conjecturing and proving in schools is a major barrier to a wider use of such tools in an educational environment. Since the early implementations of geometry automated theorem provers, applications of artificial intelligence methods, synthetic provers based on inference rules and using forward chaining reasoning are considered to be best suited for education proposes. Choosing an appropriate set of rules and an automated method that can use those rules is a major challenge. We discuss one such rule set and its implementation using the geometry deductive databases method (GDDM). The approach is tested using some chosen geometric conjectures that could be the goal of a 7th year class ( 12-year-old students). A lesson plan is presented, its goal is the introduction of formal demonstration of proving geometric theorems, trying to motivate students to that goal.

At the Technical University of Denmark, we currently teach a MSc level course on automated reasoning using the Isabelle proof assistant [11] as our main tool. The course is a 5 ECTS optional course and the homepage is here: https://courses.compute.dtu.dk/02256/ The course is an introduction to automatic and interactive theorem proving, and Isabelle is used to formalize almost all of the concepts we introduce during the course. We have developed a number of external tools to allow us to teach basic proofs in natural deduction and sequent calculus while slowly progressing towards showing students the full power of Isabelle. The learning objectives of the course are as follows: 1. explain the basic concepts introduced in the course 2. express mathematical theorems and properties of IT systems formally 3. master the natural deduction proof system 4. relate first-order logic, higher-order logic and type theory 5. construct formal proofs in the procedural style and in the declarative style 6. use automatic and interactive computer systems for automated reasoning 7. evaluate the trustworthiness of proof assistants and related tools 8. communicate solutions to problems in a clear and precise manner We expect students to already know some logic and to be relatively proficient in functional programming before starting the course. Additionally, we expect students to have some basic knowledge of artificial intelligence algorithms for deduction. Our undergraduate program in computer science and engineering, which many of our students have completed, contains several courses that introduce students to these topics.

Program synthesis aims to automatically construct human-readable programs that satisfy given task specifications, such as input/output pairs or demonstrations. Recent works have demonstrated encouraging results in a variety of domains, such as string transformation, tensor manipulation, and describing behaviors of embodied agents. Most existing program synthesis methods are designed to synthesize programs from scratch, generating a program token by token, line by line. This fundamentally prevents these methods from scaling up to synthesize programs that are longer or more complex. In this work, we present a scalable program synthesis framework that instead synthesizes a program by hierarchically composing programs. Specifically, we first learn a task embedding space and a program decoder that can decode a task embedding into a program. Then, we train a high-level module to comprehend the task specification (e.g., input/output pairs or demonstrations) from long programs and produce a sequence of task embeddings, which are then decoded by the program decoder and composed to yield the synthesized program. We extensively evaluate our proposed framework in a string transformation domain with input/output pairs. The experimental results demonstrate that the proposed framework can synthesize programs that are significantly longer and more complex than the programs considered in prior program synthesis works. Website at https://thoughtp0lice.github.io/hnps_web/

This paper develops a {\em qualitative} and logic-based notion of similarity from the ground up using only elementary concepts of first-order logic centered around the fundamental model-theoretic notion of type.

We formally verify executable algorithms for solving Markov decision processes (MDPs) in the interactive theorem prover Isabelle/HOL. We build on existing formalizations of probability theory to analyze the expected total reward criterion on infinite-horizon problems. Our developments formalize the Bellman equation and give conditions under which optimal policies exist. Based on this analysis, we verify dynamic programming algorithms to solve tabular MDPs. We evaluate the formally verified implementations experimentally on standard problems and show they are practical. Furthermore, we show that, combined with efficient unverified implementations, our system can compete with and even outperform state-of-the-art systems.

Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI), such as belief revision, spatial reasoning, among others. On the other hand, mathematical morphology (MM) is a theory for non-linear analysis of structures, that was widely developed and applied in image analysis. Its mathematical bases rely on algebra, complete lattices, topology. Strong links have been established between MM and mathematical logics, mostly modal logics. In this paper, we propose to further develop and generalize this link between mathematical morphology and modal logic from a topos perspective, i.e. categorial structures generalizing space, and connecting logics, sets and topology. Furthermore, we rely on the internal language and logic of topos. We define structuring elements, dilations and erosions as morphisms. Then we introduce the notion of structuring neighborhoods, and show that the dilations and erosions based on them lead to a constructive modal logic, for which a sound and complete proof system is proposed. We then show that the modal logic thus defined (called morpho-logic here), is well adapted to define concrete and efficient operators for revision, merging, and abduction of new knowledge, or even spatial reasoning.