Deciding Equations in Residuated Lattices

The algorithm implemented here is from Okada and Terui, JSL 64(2), 1999, but the Gentzen system and decidability are originally due to Ono and Komori, JSL 50(1), 1985. An algebraic version of this algorithm for residuated lattices is described in the article
P. Jipsen and C. Tsinakis, A Survey of Residuated Lattices, in "Ordered Algebraic Structures" (J. Martinez, editor), Kluwer Academic Publishers, Dordrecht, 2002, 19-56.

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

Talk given at the l-groups and f-rings conference, University of Florida, Feb 28-Mar 03, 2001

Enter your own (in)equation

Standard equational results about residuated lattices can be viewed

Test yourself by deciding 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.