Tabling as defined
above has a serious limitation. Although the definition of connection/2
from section
section 7.2
can compute the transitive closure of connected cities, it cannot
provide you with a route to travel. The reason is that there are
infinitely many routes if there are cycles in the network and each new
route found will be added to the answer table and cause the tabled
execution's completion algorithm to search for more routes, eventually
running out of memory.

The solution to this problem is called *mode directed tabling*
or
*answer subsumption*.^{185The
term answer subsumption is used by XSB and mode directed
tabling by YAP and B-Prolog. The idea is that some arguments are
considered‘outputs', where multiple values for the same‘input'
are combined. Possibly answer aggregation would have been a
better name.} In this execution model one or more arguments
are *not* added to the table. Instead, we remember a single *aggregated*
value for these arguments. The example below is derived from
section 7.2
and returns the connection as a list of cities. This argument is defined
as a *moded* argument using the
`lattice(PI)`

mode.^{186This
mode is compatible to XSB Prolog.} This causes the tabling
engine each time that it finds an new path to call shortest/3 and keep
the shortest route.

:- table
connection(_,_,lattice(shortest/3)).
shortest(P1, P2, P):-
length(P1, L1),
length(P2, L2),
( L1 < L2
-> P = P1
; P = P2
).
connection(X, Y, [X,Y]) :-
connection(X, Y).
connection(X, Y, P) :-
connection(X, Z, P0),
connection(Z, Y),
append(P0, [Y], P).

The mode declaration scheme is equivalent to XSB with partial
compatibility support for YAP and B-Prolog. The `lattice(PI)`

mode is the most general mode. The YAP `all`

(B-Prolog `@`

)
mode is not yet supported. The list below describes the supported modes
and indicates the portability.

**Var**
`+`

**index**- A variable (XSB), the atom
`index`

(YAP) or a `+`

(B-Prolog, YAP) declare that the argument is tabled normally.
**lattice**(`Pred`)`Pred` denotes a predicate with arity 3. It may be
specified as an predicate indicator (`Name`/3), plain predicate
name (`Name`) or a head term `Name(_,_,_)`

. On each
answer, `PI` is called with three arguments: the current
aggregated answer and the new answer are inputs. The last argument must
be unified with a term that represents the new aggregated answer.
**po**(`PI`)*Partial Ordering*. The new answer is added iff
`call(PI, +Old, +Answer)`

succeeds. For example, `po('<'/2)`

accumulates the smallest result. In SWI-Prolog the arity (2) may be
omitted, resulting in `po(<)`

.
`-`

**first**- The atom
`-`

(B-Prolog, YAP) and `first`

(YAP) declare to keep the first answer for this argument.
**last**- The atom
`last`

(YAP) declares to keep the last answer.
**min**- The atom
`min`

(YAP) declares to keep the smallest answer
according to the standard order of terms (see @</2).
Note that in SWI-Prolog the standard order of terms orders numbers by
value.
**max**- The atom
`max`

(YAP) declares to keep the largest answer
according to the standard order of terms (see @>/2).
Note that in SWI-Prolog the standard order of terms orders numbers by
value.
**sum**- The atom
`sum`

(YAP) declares to sum numeric answers.