## 7  Performance

### 7.1  Elements of complexity analysis

Given some pattern matching expression of m patterns, algorithm I (Section 5) is called once and U′ (previous section) is called m times with matrix arguments of 0, …, m−1 rows. We can roughly assimilate I with U. As regards U′, after a first, linear phase, U is called once, if they are no or-patterns, or as many times as there are or-pattern alternatives. Disregarding or-patterns and (unreasonably) assuming that U is linear in its input size, we expect a quadratic behavior in the size of the pattern matrices, which we define as the sum of the sizes of its patterns, and will assimilate to source file size in experiments. Namely, for a matrix of size S with m rows, we perform m+1 calls of U on arguments of size at most S, where m is, at worst, of the same order of magnitude as S.

Here, we do not claim to perform a thorough complexity analysis. Rather, we intend to define a reachable target for the running time of a practical implementation. As a matter of fact, even when or-patterns are not considered, the useful clause problem is NP-complete Sekar et al. [1992]. It is thus no surprise that algorithm U can exhibit exponential time behavior in the size of its arguments P and q. This exponential behavior originates from inductive step 2-(a) (Section 3.1): the rows of matrix P whose first pattern is a wildcard get copied into all specialized matrices S(ck, P), and recursive calls on all these matrices may be performed. The most unfavorable situation is as follows: U(P, q ) is false; all patterns in q are wildcards; and the patterns in P contain mostly wildcards and a few different constructor patterns such that constructors collected column-wise are complete signatures. And indeed, one can construct a series of matchings that confirms the exponential time behavior of a straightforward implementation of algorithm U. Besides, randomly generated matrices also confirm that the exponential time behavior can occur. Such problematic random matrices test vectors of booleans and many of their patterns are wildcards.

### 7.2  Safeguards

Input to the Objective Caml compiler a priori is neither nasty nor random, it consists of programs written by human beings attempting to solve specific problems, not to break the compiler. If such reasonable input triggers exponential behavior, then the compilation of useful programs may perhaps become unfeasible.

We now search for means that may prevent U from reacting exponentially to reasonable input. We call such a mean a safeguard. An exhaustive match is a potentially unfavorable situation, since we here compute U(P, (_⋯_)) = False. Hence, we examine how human beings write exhaustive matches. An easy and frequent way to ensure exhaustiveness is to complete a matching with a wildcard pattern. Hence, we could first scan matrices searching for rows of wildcards, and announce exhaustive match as soon as we discover one. However, this process would not be very general. Let us rather consider strict pattern matching and remark that for any defined value we have _ ≼ v. That is, for any pattern p and defined value v, we have pv =⇒ _ ≼ v (or _ ⋠v =⇒ pv). Hence, if some row of some matrix P is made of wildcards, then that row contains all the non-matching potential of the whole matrix P.

Wildcards can also appear inside patterns, as in the following example.

```match v with (1,2) -> 1 | (1,_) -> 2
```

Then, for any defined value v, we have (1,2)v =⇒ (1,_)v (or (1,_)v =⇒ (1,2)v). Thus, before we compute exhaustiveness, we can delete pattern (1,2) from the matching, because its presence does not add non-matching values to those values that do not match the remaining pattern (1,_).

We now consider the general case.

Definition 13   Let p and q be two pattern vectors. By definition, p subsumes q when, for all defined value vectors v, we have: qv =⇒ pv.
Lemma 5   Let P be a pattern matrix. We further assume that there exist two rows of P, p i and p j, such that p i subsumes p j. And we write P′ for P without row p j. Then, for any pattern vector q we have: U(P, q ) = U(P′, q ).
Proof: By definition of ⋠ we have:
P ⋠
 → v
⇐⇒ P′ ⋠
 → v
∧
 → p
j ⋠
 → v
.
Furthermore, by definition of P′ ⋠v and by hypothesis “row p i subsumes p j”, we have:
P′ ⋠v =⇒
 → p
i ⋠
 → v
=⇒
 → p
j ⋠
 → v
Finally, for any defined value v we have: P′ ⋠v  ⇐⇒ Pv. □
Example 4   Let us consider the following matrix P.
P =

 _ (_,_) (_,_) _

Because the constructor of pairs “,” is alone in its signature, pattern (_,_) subsumes pattern _. Namely, any (defined) value v that possesses the type t1 × t2 of a pair can be written v = (v1,v2) and is thus an instance of (_,_)3. As a consequence, erasing any of the two rows of P does not matter and we have:
U(P,
 → q
) = U(
 _ (_,_)

 → q
) = U(
 (_,_) _

 → q
)

It turns out that we can decide the relation “p subsumes q” by using predicate U.

Lemma 6   Let p and q be two pattern vectors. Then p subsumes q, if and only if U(p, q ) does not hold.
Proof: The property is obvious by considering the strict semantics of pattern matching.
U(
 → p
 → q
) = False  ⇐⇒

∀
 → v
 → p
≼
 → v
∨
 → q
 → v

⇐⇒

∀
 → v
 → q
≼
 → v
=⇒
 → p
≼
 → v

Notice that ∀ v here means “for all defined values”. □

By Lemma 5 and Lemma 6, we now have a mean to delete rows from matrix P before we compute U. And this mean is valid for all our semantics of pattern matching. More precisely, we have:

U(
 → p
i
 → p
j) = False =⇒ U(P
 → q
) = U(P′,
 → q
).

Where P′ is P without row number j.

In practice, before computing U(P, q ), our first safeguard consists in deleting any row of P that is subsumed by another row of P. In the worst case, (when no row is subsumed) this requires computing U(p i, p j) for all pairs of (distinct) rows in P. However, in the absence of or-patterns, computing U(p i, p j) cannot be exponential — since bad case 2-(a) may only occur here for constructor signatures of size one, which, in that particular case, yields exactly one recursive call. We can thus approximate the cost of our first safeguard as a quadratic number of “ordinary” pattern operations whose cost is roughly linear in the size of input patterns (still disregarding or-patterns). As a conclusion, if this first safeguard saves us from exponential computations, it is worth its price.

We now design a second, less expensive, safeguard. Two patterns are said to be incompatible when they have no instance in common. Thus, before computing U(P, q ) we can delete some row p i from P, provided p i and q are incompatible (consider strict matching). It turns out that incompatibility of patterns can be computed as an “ordinary” pattern operation by using the rules for computing the incompatibility of pattern and value (see Section 4.2). Thus, we can attempt to simplify matrix P with m rows for the price of m ordinary patterns operations. Unfortunately, this process does not help while checking exhaustiveness, since (_⋯_) is compatible with any pattern vector. Furthermore, the first phase of computing U′(P, q ) for checking utility in fact gets rid of rows that are incompatible with checked row q (at step 1.). However, the price of this second safeguard is low and it may be a good idea to perform it before performing the first safeguard. As a result, we may reduce the size of input to the first safeguard. Of course, this makes sense because the first safeguard performs a quadratic number of ordinary pattern operations while the second safeguard performs a linear number of ordinary pattern operations. In fact, example I of the following section exhibits S3 behavior without the second safeguard, whereas, with second safeguard enabled, it exhibits quasi-quadratic behavior.

### 7.3  Measures

We performed some measures of the machine time taken by our implementation, which is integrated in the Objective Caml compiler. We measure compilation time both with and without pattern-matching diagnostics enabled. Then, a simple subtraction yields the time taken by the detection of anomalies.

Measures apply to series of pattern-matching expressions of identical structure and increasing size. Three of our series T, S and V, may drive compilation of pattern matching to decision trees into producing code of exponential size. We selected those because algorithm U is similar to compilation of pattern matching to decision trees. The fourth series, I, is the matching of n constant constructors. We selected series I because of a real example that triggered excessive compilation times while we were implementing algorithm U. All series are defined precisely in appendix A. In Figure 3, the X-axis shows the size of matchings squared, (expressed as source file size squared), while the Y-axis shows user machine time on a Linux 1Ghz PC. We perform measures both with and without the safeguards described in the previous section (those are named “opt” and “std”).

Experiments S and V demonstrate that algorithm U is more resistant to exponential behavior than simple compilation to decision trees. Namely, exponential behavior of compilation to decision trees have been reported for those examples Sestoft [1996], Maranget [1992]. Observe that algorithm U does not exhibit exponential running time even without safeguards (written “std” in Figure 3).

However, algorithm U is not immune from exponential behavior, as demonstrated by experiment T. In that particular case, safeguards prove efficient and we decided to retain them in our implementation for that reason. Adopting safeguards incurs some penalty, since safeguards significantly degrade performance in experiment S. However, it should be noticed that it requires a 161 × 320 matrix to reach one minute of analysis time in experiment S (with safeguards), whereas it takes only a 42 × 21 matrix to reach a similar time in experiment T (without safeguards).

Finally the plots of figure 3 suggest that the running time of pattern matching analysis with safeguards is approximatively quadratic in input size, at least for our examples and on the studied domain. However, notice that experiment I suggests a slightly more than quadratic running time.

We also estimate the time taken by our algorithm relatively to total compilation time of ordinary programs. To do so, we measure the running time of the compilation of some programs, with pattern matching analysis disabled (no), with pattern matching analysis enabled but without safeguards (std), and with safeguards (opt). We made some effort not to measure irrelevant operations such as linking and the application of external tools. We made ten measures in each experiment and we computed geometric means. The compiled programs are the Objective Caml compiler itself (ocamlc), the hevea LATEX to html translator and another compiler (zyva). All those programs use pattern matching significantly and are of respectable size.

 no std opt ocamlc 13.63 13.92 14.02 hevea 7.84 8.01 8.05 zyva 4.61 4.70 4.74

Those results show that the price of pattern-matching analysis remains quite low in practice. One might even conclude that safeguards are not very useful when it comes to compiling “actual” programs. Still, we adopt safeguards because they are designed to avoid exponential behavior in some specific situations, which do not show up in the above examples. Our choice follows the general philosophy behind preferring backtracking automata to decision trees: avoid exponential behaviors as much as possible.

3
It is worth noticing that all instances of _ no longer are instances of (_,_) when partial values are considered. Namely, the undefined value Ω is not an instance of (_,_).