Isomorphism is one kind of equivalence test for MRSs. Two MRSs are isomorphic if they have the same structure--that is, they have the same kind and number of predicates, and the variables in the predicates map to the same positions in other predicates. Variable names can differ.

The following two (contrived) MRSs are isomorphic:

[ LTOP: u1 INDEX: u2
  RELS: < [ "rel1" LBL: u3 ARG0: u4 ]
          [ "rel2" ARG0: u5 ]
          [ "rel3" LBL: u6 ARG0: u7 ARG1: u4 ] >
  HCONS: < u4 qeq u5 > ]

[ LTOP: u9 INDEX: u8
  RELS: < [ "rel1" LBL: u7 ARG0: u6 ]
          [ "rel2" ARG0: u5 ]
          [ "rel3" LBL: u4 ARG0: u3 ARG1: u6 ] >
  HCONS: < u6 qeq u5 > ]

The following two (also contrived, minimally different) MRSs are not isomorphic. (See what the ARG0 in rel1 maps to).

[ LTOP: u1 INDEX: u2
  RELS: < [ "rel1" LBL: u3 ARG0: u4 ]
          [ "rel2" ARG0: u5 ]
          [ "rel3" LBL: u6 ARG0: u7 ARG1: u4 ] >
  HCONS: < u4 qeq u5 > ]

[ LTOP: u1 INDEX: u2
  RELS: < [ "rel1" LBL: u3 ARG0: u5 ]
          [ "rel2" ARG0: u5 ]
          [ "rel3" LBL: u6 ARG0: u7 ARG1: u4 ] >
  HCONS: < u4 qeq u5 > ]

Checking for isomorphism mainly involves checking that the variables appear in the same arguments of the predicates and QEQs. One also should check that the variable properties are the same.

One way to accomplish this task is through a recursive procedure: Given two lists of unmapped variables (one from each MRS), assume a mapping between two variables and then repeat on the remaining variables. Variables that can be mapped are those that appear on the same position in the same kind of predicate (e.g. ARG0 on a _def_q_rel), and have the same properties. If no remaining variables can be mapped and the lists are not empty, backtrack and try other mappings until one is found. If one is found, the MRSs are isomorphic.

Because the recursive check for isomorphism can take some time, several optimizations should be implemented. Some possible optimizations are listed below:

• Simple check for equivalence of string/XML representation (if the string or XML representations are equal, then they are the exact same MRS (same structure and same variable names))
• Check the number of relations, HCONS, etc.
• Only map variables between the same type of relation (e.g. don't try to map _def_q_rel with dog_n_rel) and same type of argument (e.g. ARG0s of dog_n_rel)
• * Further, only map variables that appear in the same positions of the same relations across the whole MRS (i.e. don't just look at one relation, but all of them. In the first MRS example above, u4 has the following pattern: ("rel1":ARG0,"rel3":ARG1) (you could also include the HCONS))
• Order the sets of unmapped variables (where a set is all variables in a predicate) by the length of the set, fewest first. Because there are only 3 mappings for predicates with 2 arguments, 6 mappings when 3 arguments, 24 for 4, etc.

One may choose to not compare variable properties as they can differ between MRS that have gone through SEM-I mappings and those that haven't. This may occur when comparing MRS from parsed and generated sentences. This would mean that sentences varying in some way not encoded in predicates and arguments would be considered isomorphic. Examples (depending on the grammar) may be "Kim runs the process." (present, active) and "Kim ran the process" (past, active) and "The process was run by Kim." (past, passive)