Other modal tableau systems

There are several other programs provers for modal formula, many of which are also based on tableau reasoning.

The Object Oriented Prover for S5n, OOPS is a modal prover written in Java. OOPS is uses a design with a flexible rule engine, the rule are specified as objects and can be changed at runtime. The price of this flexibility is that OOPS can be very slow, taking several minutes for formulas such as those from the three hats example.

ModLeanTAP is another implementation of modal tableaux, written in Prolog. This prover is very compact, measuring just under 10kB. The source code is set up to correspond closely to the accompanying paper, which also gives correctness proofs.

A disdvantage of both of these provers is the lack of a good user interface. OOPS requires formulas to be entered on the command line in a non-standard syntax. There is no option to load input from a file. ModLeanTAP does not include a user interface at all, it is just a decision procedure callable from other Prolog code.

The Logics Workbench also includes support for modal logic, among various other logical systems. It has a more evolved user interface compared to the other two systems mentioned above. Unfortunately this program is no longer maintained.

Labeled tableaux

Both OOPS and ModLeanTAP use labeled tableaux, in this system each formula is labled with the world it is in. For example what we would write as [φ+,ψ+]i is written as

In these systems. 0 is the label of the root world, while 1i is another world reachable by agent i. Multiple formulas are grouped together by these labels. So the above tableau should be read as “In all worlds '1' reachable from world '0' by agent i, either φ holds or ψ holds”

OOPS uses Java objects for the labels, which are essentially unique identifiers. With unique identifiers care must be taken to not create multiple worlds for the same formula. In OOPS that problem is solved by maintaining a list of formulas for which a world has been created.

ModLeanTAP uses a slightly different approach, where the world created for a formula i φ is labled by the formula itself. This also solves the problem of duplicating work.

In labeled tableau proofs it is also possible to create worlds for negatively occurring boxes (i φ-). These formulas are then labeled with a variable, which is later unified with a ground label from a normal positive box. An advantage of this approach is that formulas inside a negative box can also be expanded, allowing them to be inspected only once. In our system a formula like φ above will be copied for each world, and can therefore be inspected multiple times.

These variable labels also come with disadvantages. In non-serial models something like 0.x. φ, 0.x. ¬φ is not an axiom, because there may not exist a world matching x. Another disadvantage is that working with variables requires complex machinery such as unifications. For ModLeanTAP this is not a problem, since it is written in Prolog these come for free. OOPS does not in fact use variable labels, instead it uses a table of necessities, just like our prover.