Sizechange termination is a widely used means of proving termination where source programs are first abstracted to
sizechange graphs which are then analyzed to determine if they satisfy the sizechange termination property.
Here, the choice of the abstraction is crucial to the success of the method, and it is an open problem how to
choose an abstraction such that no critical loss of precision occurs.
In the paper "Lazy Abstraction for SizeChange Termination" we show how to couple
the search for a suitable abstraction and the test for sizechange termination via an encoding to a single SAT
instance. In this way, the problem of choosing the right abstraction is solved en passant by a SAT solver. We show
that for the setting of term rewriting, the integration of this approach into the dependency pair framework works
smoothly and gives rise to a new class of sizechange reduction pairs.
In our experiments, we want to assess the power of sizechange reduction pairs by extensive experiments on a large
collection of examples from the literature and from applications.
Moreover, we compare our new approach of sizechange termination in the term rewriting context to the previous work of Thiemann and Giesl.
Implementation in AProVE
We have implemented our contributions
in the automated termination prover AProVE.
To experiment with this new version of
AProVE, please use the special
web interface.
Tools
To assess the impact of our contributions, in our experiments we compare three configurations of AProVE.
 AProVE SCNP
In the first, we use SCNP reduction pairs in the reduction pair processor of the DP framework.
This configuration is parameterized by the choice whether we allow just max comparisons of
multisets or all four multiset extensions from Def. 7 of our paper. Moreover, the configuration is also
parameterized by the choice whether we use classical sizechange graphs or extended sizechange
graphs. In an extended sizechange graph, to compare
s = F(s_{1},...,s_{n})
with t = G(t_{1},...,t_{m}),
the source and target vertices {s_{1}, ..., s_{n}} and
{t_{1}, ..., t_{m}} are extended by additional
vertices s and t, respectively. Now an edge from s to t_{j}
indicates that the whole term s is greater (or equal) to t_{j}, etc.
So these additional vertices also allow us to compare the whole terms s and t.
By adding these vertices, sizechange termination incorporates the standard comparison of terms as well.
 AProVE reduction pairs:
In the second configuration, we use the base orders directly in the reduction pair processor
(i.e., here we disregard SCNP reduction pairs).
 AProVE SCT:
In the third configuration, we use the implementation of the SCT method as described in previous work of Thiemann and Giesl.
For a fair comparison, we updated that old implementation from the DP approach to the modular DP framework
and used SAT encodings for the base orders. While this approach only uses the embedding order and
argument filters as the base order for the construction of sizechange graphs, it uses more complex orders
(containing the base order) to weakly orient the rules from the TRS.
Examples and Settings
In our experiments, we considered all 1381 examples from the standard TRS category of the
Termination Problem Data Base (TPDB version 7.0.2)
as used in the International Termination Competition 2009.
The experiments were run on a 2.66 GHz Intel Core 2 Quad and as in the termination competition, we used a time
limit of 60 seconds per example. We applied SAT4J to transform propositional
formulas to conjunctive normal form and the SAT solver MiniSAT2
to check the satisfiability of the resulting formulas.
Experiments
The experiments in the table below compare the power and runtimes of the three configurations depending
on the choice of the base order. The column "order" indicates the base order: embedding order with argument
filters (EMB), lexicographic path order with arbitrary permutations and argument filters (LPO),
recursive path order with argument filters (RPO), and linear polynomial interpretations with coefficients
from {0,1} (POLO). For the first configuration, we used three different settings: full SCNP reduction pairs
with extended sizechange graphs ("SCNP all"), SCNP reduction pairs restricted to max comparisons with
extended sizechange graphs ("SCNP max"), and SCNP reduction pairs restricted to max comparisons and
nonextended sizechange graphs ("SCNP fast"). The second and third configuration are called "reduction pairs"
and "SCT", respectively. For each experiment, we give the number of TRSs which could be proved terminating
("proved") and the analysis time in seconds for running AProVE on all 1381 TRSs ("runtime"). The "best"
numbers are always printed in bold. Click on the column headers to view a detailed table with the individual
results of the column. Click on "[Details]" to the right of a row to view a detailed table with the individual
results of the row.
The table allows for the following main observations:

Our SCNP reduction pairs are much more powerful and significantly faster than the implementation of Thiemann and Giesl's approach (column "SCT"). By integrating the search for the base order with SCNP, our new implementation
can use a much larger class of base orders and thus, SCNP reduction pairs can prove significantly more examples.
The reason for the relatively low speed of "SCT" is that this approach iterates through argument filters and
then generates and analyzes sizechange graphs for each of these argument filters. (So the low speed is not due
to the repeated composition of sizechange graphs in the SCT criterion.)

Our new implementation of SCNP reduction pairs is more powerful than using the reduction pairs directly.
Note that when using extended sizechange graphs, every reduction pair can be simulated by an SCNP
reduction pair.

SCNP reduction pairs add significant power when used for simple orders like EMB and LPO.
The difference is less dramatic for RPO and POLO. Intuitively, the reason is that SCNP allows for
multiset comparisons which are lacking in EMB and LPO, while RPO contains multiset comparisons
and POLO can often simulate them. Nevertheless, SCNP also adds some power to RPO and POLO, e.g.,
by extending them by a concept like "maximum". This even holds for more powerful base orders like
matrix orders. See Appendix B of the paper for a TRS where all existing
termination tools fail, but where termination can easily be proved automatically by an SCNP reduction
pair with a matrix base order.
To summarize, the implementation in AProVE confirms the usefulness of our contribution. Our experiments
indicate that the automation of our technique is more powerful than both the direct use of reduction pairs
and the SCT adaptation from previous work.