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Theory development with TPTP
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README.mkdn

TPTP theory development emphasizing proof analysis

Overview

In some theory development tasks, a problem is satisfactorily solved once it is shown that a theorem (conjecture) is derivable from the background theory (premises). Depending on one's motivations, the details of the derivation of the conjecture from the premises may or may not be important. In some contexts, though, one wants more from theory development than simply derivability of the target theorems from the background theory. One may want to know which premises of the background theory were used in the course of a proof output by an automated theorem prover (when a proof is available), whether they are all, in suitable senses, necessary (and why), whether alternative proofs can be found, and so forth. The problem, then, is to support proof analysis in theory development; Tipi aims to provide precisely that.

Ready?

Download a release of Tipi by going to

https://github.com/jessealama/tipi/tags

This will give you a list of so-called tags that I've attached to stages of Tipi development. They are ordered by date. I recommend getting the newest release, which is 0.7.

Set?

To run Tipi you will need:

Prover-specific dependencies

Tipi supports various automated reasoning tools, each of which has its own dependencies and further complicates the dependencies of Tipi:

  • Vampire

    Version 0.6 is supported.

  • Prover9/Mace4

    To produce appropriate input for the LADR programs, Tipi requires a recent version of TPTP2X. Moreover, not simply prover9 and mace4 are used by Tipi: interpformat and prooftrans are also required.

    prover9 and mace4 do not directly adhere to the TPTP/SZS philosophy, unfortunately. This complicates things when using them inside Tipi. The behavior at the moment seems fragile to me. If you have suggestions, I'd be happy to discuss. Perhaps one day I'll just write an SZS-friendly wrapper around these venerable programs.

  • E

    We require version 1.4. If you have an earlier version of E that you would like to use, please let me know. Newer versions of E might work, too. I just hope Stephan Schulz doesn't change the interface! From E, Tipi requires not imply eprover, but also epclextract.

Perl modules

Concerning the perl modules: for certain Tipi commands you may be able to get by without all the dependencies, but for safety I recommend you install all of them. If a needed dependency is missing, your perl will quickly complain loudly and die. It's possible that earlier versions of the required modules may work, but that is not tested. To install perl modules, I recommend using either the package system that comes with your operating system (probably all of the packages are available in a recent release of Ubuntu GNU/Linux) or the CPAN module. Get started by doing:

$ perl -MCPAN -e shell

and then installing packages using commands such as

install Algorithm::Combinatorics

and then watch it fly. Any needed but missing dependencies will be downloaded and installed. There is an issue about where CPAN installs the modules. Do you want the modules installed just for you, or for other people on your machine? I can't really help you answer that question.

Concerning the dependency on the TPTP World: I am aware that this is a rather heavy dependency. It is all the more embarrassing when one realizes that in this giant package, Tipi requires only two tools, one of which (GetSymbols) is actually very small. One can find TPTP4X in the leaner TPTP distribution, but if you're not running the exact same architecture as Geoff Sutcliffe you're out of luck. In any case GetSymbols is not in the official TPTP distribution. So I am afraid the TPTP World really is required. It is important for me that eventually the dependency on these tools is removed. I know how to do it, but it will take some time (the problem is simply writing a parser for the TPTP language).

Go!

Tipi is a traditional commandline UNIX-y tool. To get started, if you've got a TPTP theory file problem.p and you know that its conjecture is a consequence of its axioms, try running, e.g., Tipi's reprove command, as follows:

$ tipi reprove problem.p

Tipi comes with many commands for doing all sorts of proof analysis tasks. To get started, try doing

$ tipi --man

to start reading the manual pages. This command will give you the man page for Tipi as a whole. Every command has its own man page. For the reprove command, for example, you can access the man page via:

$ tipi reprove --man

Yes, I know man pages can be pretty bad. But I've really tried to make them attractive and usable for the newcomer and expert alike. If you spot any gaps, infelicities, ambiguities, etc., please let me know.

Contact

I welcome all sorts of feedback: bug reports, feature requests, documentation gaps or errors, discussion of algorithms, efficiency, complexity, etc. Contact me at:

jesse.alama@gmail.com

You are also welcome to submit bug reports or other issues by using the Tipi issue tracking system provided by GitHub.

Introduction

A characteristic feature of theorem proving problems arising in theory development is that we often do not know which premises of our background theory are needed for a proof until we find one. If we are working in a stable background theory in which the axioms are fixed, we naturally include all premises of the background theory because it a safe estimate (perhaps an overestimate) of what is needed to solve the problem. We may add lemmas on top of the background theory to help a theorem prover find a solution or to make our theory more comprehensible. Since computer-assisted theorem proving is beset on all sides by intractability, any path through a formal theory development task is constantly threatened by limitations both practical (time, memory, patience, willpower) and theoretical (undecidability of first-order validity). Finding even one solution (proof, model, etc.) is often no small feat, so declaring victory once the first solution is found is thus quite understandable and may be all that is wanted.

In some theory development tasks, though, we want to learn more about our problem beyond its solvability. This paper announces \tipi, a tool that helps us to go beyond mere solvability of a reasoning problem by providing support for answer such questions as:

  • What premises of the problem were used in the solution?

  • Do other automated reasoning systems derive the conclusion from the same premises?

  • Are my premises consistent? Do they admit unintended models?

  • What premises are truly needed for the conclusion? Can we find multiple sets of such premises? Is there a a ``minimal'' theory that derives the conclusion?

  • Are my axioms independent of one another?

Let us loosely call the investigation of these and related questions proof analysis.

Tipi is useful for theory exploration both in the context of discovery and in the convex of justification. In the context of discovery, one crafts lemmas, adds or deletes axioms, changes existing axioms, modifies the problem statement, etc., with the aim of eventually showing that the theory adequate for one's purposes (it derives a certain conjecture, is satisfiable, is unsatisfiable, etc.). In the context of discovery, the set of one's axioms is in flux, and one needs tools to help ensure that the development is not veering too far off course into the unexpected countersatisfiability, admitting `nonsense' models, being inconsistent, etc. In the context of justification, after the initial work is done and a solution is found, one wants to know more about the relationship between the theory and the conjecture than simply that the latter is derivable from the former. What is the proof like? Are there other proofs? \tipi{} is designed to facilitate answering questions such as these.

The theorem provers and model finders that make up Tipi include E, Vampire, Prover9, Mace4, and Paradox. The system is extensible; adding support for new automated reasoning systems is straightforward because we rely only on the SZS ontology to make judgments about theorem proving problems.

Tipi uses a variety of automated reasoning technology to carry out its analysis. It uses theorem provers and model finders and is based on the TPTP syntax for expressing reasoning problems and the SZS problem status ontology so it can thereby can flexibly use of a variety of automated reasoning tools that support this syntax.

Motivation (you needn't care about this)

The philosophical background of Tipi is a classic problem in the philosophy of logic known as the proof identity problem:

When are two proofs the same?

Standard approaches to the proof identity problem work with natural deduction derivations or category theory. One well-known proposal is to identify 'proof' with a natural deduction derivation, define a class of conversion operations on natural deduction derivations, and declare that two proofs are the same if one can be converted to the other. See this survey paper for a discussion of this approach to the proof identity problem. The inspiration for Tipi is to take on the proof identity problem with the assistance of modern automated reasoning tools. From this perspective, the TPTP library can be seen as a useful resource on which to carry out experiments about 'practical' proof identity. TPTP problems typically don't contain proofs in the usual sense of the term, but they do contain hints of proofs in the sense that they specify axioms and perhaps some intermediate lemmas.

One does not need to share the philosophical background (or even care about it) to start using Tipi, which in any case was designed to facilitate TPTP-based theory development.

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