Tacro is a browser game wherein you solve puzzles to advance, and your solutions provide computer-verifiable formal proofs for mathematical theorems.
Tacro comprises a few different parts:
- An engine which determines the possible moves and their results;
- Some data which defines a progression of levels, each containing axioms and some goal theorems which must be be proved from them to advance to the next level;
- An interface for manipulating the engine into proving the goal theorems; and
- A schema for data interchange (currently hosted on a Firebase backend), allowing each player to publish solutions and new levels.
Yes, at least in Chrome.
Not yet. There are no instructions or tutorial, and the UI controls are highly non-self-evident.
Not yet. If you can somehow figure out the UI controls, and if you enjoy formal theorem proving for its own sake, then you might have fun. The levels have not yet been optimized for Flow.
I think so. Modulo any bugs I'm not aware of, it should not be possible to create unsound proofs with the Tacro engine. Currently this is only guaranteed by the engine's JS, which is probably buggy. In the future, all generated proofs will be double-checked offline by the Ghilbert verifier.
Yes! The current set of levels progresses through propositional calculus, predicate calculus, and Peano arithmetic. I have used the engine to prove induction theorems like the associativity of addition. I believe that anything provable from the Peano axioms could be added as a goal theorem and proved from these levels.
Yes! The engine, interface, and schema are totally separate from the particular data set currently available. It should be possible to use Tacro to work in any axiom system, as long as it has a two-hypothesis inference that "looks like modus ponens", and all of its other axioms have one or zero hypotheses. (I also believe, but am not certain, that any axiom system with more two-hypothesis inferences besides modus ponens can be easily converted to an equipotent axiom system compatible with these requirements.)
Yes. Your work will be published to the server when you come back online.
Not yet. But I'm planning to add a scoreboard, a P2P hint/chat system, a level creator, and systems to publish, browse, rate, and reward new levels and new solutions.
The previous version was called "Orcat" which was short for "Order Categorical". In Orcat, you work forwards from your axioms until your theorem matches a given goal. In Tacro, you work "backwards" from a given goal until your hypothesis matches one of your axioms.