Default Conceptual Graph Rules: Preliminary Results for an Agronomy Application

International audienceIn this paper, we extend Simple Conceptual Graphs with Reiter's default rules. The motivation for this extension came from the type of reasonings involved in an agronomy application, namely the simulation of food processing. Our contribution is many fold: rst, the expressivity of this new language corresponds to our modeling purposes. Second, we provide an effective characterization of sound and complete reasonings in this language. Third, we identify a decidable subclass of Reiter's default logics. Last we identify our language as a superset of SREC-, and provide the lacking semantics for the latter language

Properties of ABA+ for Non-Monotonic Reasoning

We investigate properties of ABA+, a formalism that extends the well studied structured argumentation formalism Assumption-Based Argumentation (ABA) with a preference handling mechanism. In particular, we establish desirable properties that ABA+ semantics exhibit. These pave way to the satisfaction by ABA+ of some (arguably) desirable principles of preference handling in argumentation and nonmonotonic reasoning, as well as non-monotonic inference properties of ABA+ under various semantics.Comment: This is a revised version of the paper presented at the worksho

PyZX: Large Scale Automated Diagrammatic Reasoning

The ZX-calculus is a graphical language for reasoning about ZX-diagrams, a type of tensor networks that can represent arbitrary linear maps between qubits. Using the ZX-calculus, we can intuitively reason about quantum theory, and optimise and validate quantum circuits. In this paper we introduce PyZX, an open source library for automated reasoning with large ZX-diagrams. We give a brief introduction to the ZX-calculus, then show how PyZX implements methods for circuit optimisation, equality validation, and visualisation and how it can be used in tandem with other software. We end with a set of challenges that when solved would enhance the utility of automated diagrammatic reasoning.Comment: In Proceedings QPL 2019, arXiv:2004.1475

Learning to Prove Theorems via Interacting with Proof Assistants

Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as high-level tactics. However, human experts have to construct proofs manually by entering tactics into the proof assistant. In this paper, we study the problem of using machine learning to automate the interaction with proof assistants. We construct CoqGym, a large-scale dataset and learning environment containing 71K human-written proofs from 123 projects developed with the Coq proof assistant. We develop ASTactic, a deep learning-based model that generates tactics as programs in the form of abstract syntax trees (ASTs). Experiments show that ASTactic trained on CoqGym can generate effective tactics and can be used to prove new theorems not previously provable by automated methods. Code is available at https://github.com/princeton-vl/CoqGym.Comment: Accepted to ICML 201

Goal driven theorem proving using conceptual graphs and Peirce logic

The thesis describes a rational reconstruction of Sowa's theory of Conceptual Graphs. The reconstruction produces a theory with a firmer logical foundation than was previously the case and which is suitable for computation whilst retaining the expressiveness of the original theory. Also, several areas of incompleteness are addressed. These mainly concern the scope of operations on conceptual graphs of different types but include extensions for logics of higher orders than first order. An important innovation is the placing of negation onto a sound representational basis. A comparison of theorem proving techniques is made from which the principles of theorem proving in Peirce logic are identified. As a result, a set of derived inference rules, suitable for a goal driven approach to theorem proving, is developed from Peirce's beta rules. These derived rules, the first of their kind for Peirce logic and conceptual graphs, allow the development of a novel theorem proving approach which has some similarities to a combined semantic tableau and resolution methodology. With this methodology it is shown that a logically complete yet tractable system is possible. An important result is the identification of domain independent heuristics which follow directly from the methodology. In addition to the theorem prover, an efficient system for the detection of selectional constraint violations is developed. The proof techniques are used to build a working knowledge base system in Prolog which can accept arbitrary statements represented by conceptual graphs and test their semantic and logical consistency against a dynamic knowledge base. The same proof techniques are used to find solutions to arbitrary queries. Since the system is logically complete it can maintain the integrity of its knowledge base and answer queries in a fully automated manner. Thus the system is completely declarative and does not require any programming whatever by a user with the result that all interaction with a user is conversational. Finally, the system is compared with other theorem proving systems which are based upon Conceptual Graphs and conclusions about the effectiveness of the methodology are drawn