Tautology testing with a generalized matrix reduction method

AbstractA formalization of the tautology problem in terms of matrices is given. From that a generalized matrix reduction method is derived. Its application to a couple of selected examples indicates a relatively efficient behaviour in testing the validity of a given formula in propositional logic—not only for machines but also for humans. A further result from that formalization is a reduction of the tautology problem to a part of Presburger arithmetic which involves formulas of the ∀∃∃⋯∃-type where all quantifiers have finite range

A Tool for Producing Verified, Explainable Proofs

Mathematicians are reluctant to use interactive theorem provers. In this thesis I argue that this is because proof assistants don't emphasise explanations of proofs; and that in order to produce good explanations, the system must create proofs in a manner that mimics how humans would create proofs. My research goals are to determine what constitutes a human-like proof and to represent human-like reasoning within an interactive theorem prover to create formalised, understandable proofs. Another goal is to produce a framework to visualise the goal states of this system. To demonstrate this, I present HumanProof: a piece of software built for the Lean 3 theorem prover. It is used for interactively creating proofs that resemble how human mathematicians reason. The system provides a visual, hierarchical representation of the goal and a system for suggesting available inference rules. The system produces output in the form of both natural language and formal proof terms which are checked by Lean's kernel. This is made possible with the use of a structured goal state system which interfaces with Lean's tactic system which is detailed in Chapter 3. In Chapter 4, I present the subtasks automation planning subsystem, which is used to produce equality proofs in a human-like fashion. The basic strategy of the subtasks system is break a given equality problem in to a hierarchy of tasks and then maintain a stack of these tasks in order to determine the order in which to apply equational rewriting moves. This process produces equality chains for simple problems without having to resort to brute force or specialised procedures such as normalisation. This makes proofs more human-like by breaking the problem into a hierarchical set of tasks in the same way that a human would. To produce the interface for this software, I also created the ProofWidgets system for Lean 3. This system is detailed in Chapter 5. The ProofWidgets system uses Lean's metaprogramming framework to allow users to write their own interactive, web-based user interfaces to display within the VSCode editor and in an online web-editor. The entire tactic state is available to the rendering engine, and hence expression structure and types of subexpressions can be explored interactively. The ProofWidgets system also allows the user interface to interactively edit the proof document, enabling a truly interactive modality for creating proofs; human-like or not. In Chapter 6, the system is evaluated by asking real mathematicians about the output of the system, and what it means for a proof to be understandable to them. The user group study asks participants to rank and comment on proofs created by HumanProof alongside natural language and pure Lean proofs. The study finds that participants generally prefer the HumanProof format over the Lean format. The verbal responses collected during the study indicate that providing intuition and signposting are the most important properties of a proof that aid understanding.EPSR

A Human Oriented Logic for Automatic Theorem Proving

The automation of first order logic has received comparatively little attention from researcher intent upon synthesizing the theorem proving mechanism used by humans. The dominant point of view [15], [18] has been that theorem proving on the computer should be oriented to the capabilities of the computer rather than to the human mind and therefore one should not be afraid to provide the computer with a logic that humans might find strange and uncomfortable. The preeminence of this point of view is not hard to explain since until now the most successful theorem proving programs have been machine oriented. Nevertheless, there are at least two reasons for being dissatisfied with the machine oriented approach. First, a mathematician often is interested more in understanding the proof of a proposition than in being told that the propositions true, for the insight gained from an understanding of the proof can lead to the proof of additional propositions and the development of new mathematical concepts. However, machine oriented proofs can appear very unnatural to a human mathematician thereby providing him with little if any insight. Second, the machine oriented approach has failed to produce a computer program which even comes close to equaling a good human mathematician in theorem proving ability; this leads one to suspect that perhaps the logic being supplied to the machine is not as efficient as the logic used by humans. The approach taken in this paper has been to develop a theorem proving program as a vehicle for gaining a better understanding of how humans actually prove theorems. The computer program which has emerged from this study is based upon a logic which appears more "natural" to a human (i.e., more human oriented). While the program is not yet the equal of a top flight human mathematician, it already has given indication (evidence of which is presented in section 9) that it can outperform the best machine oriented theorem provers