Deciding Equations in Residuated Lattices

The algorithm implemented here is due to Okada and Terui, JSL 64(2), 1999. A version of this algorithm for residuated lattices is described in the preprint A Gentzen systems and decidability for residuated lattices (P. Jipsen, Jan 2000) pdf version

The (in)equations are in the language of (bounded) residuated lattices. Specifically, the language used here has

Click here if you want to enter your own (in)equation.

A list of standard equational results about residuated lattices can be viewed on this page.

Test yourself by deciding the following randomly generated RL equations.

The JavaScript code is available for browsing.

Here is the list of residuated lattice equations, followed by their Gentzen system proofs (generated by the embedded JavaScript program)

To display the proofs in a linear fashion, rather than the more usual Gentzen style, the proof trees are displayed from the root downwards, with the depth of nodes indicated by the indentation level. (Only the branching rules increase the indentation depth.) Each sequent is labeled by the name of the rule that produced it. When a terminal node is reached, the indentation level decreases again. With a bit of practice, these proofs are actually easier to read (and write) than the standard tree-like presentation. The symbol !- can be interpreted as not less or equal, in which case each line implies the one below it (if at the same indentation level) or one of the two indented alternatives (in the case of a branching rule). With this interpretation, the terminal nodes represent contradictions, reminiscent of a tableau style proof.