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Posts Tagged ‘binary operator’

Number of possible different one line programs

February 22nd, 2012 No comments

Writing one line programs is a popular activity in some programming languages (e.g., awk and Perl). How many different one line programs is it possible to write?

First we need to get some idea of the maximum number of characters that written on one line. Microsoft Windows XP or later has a maximum command line length of 8191 characters, while Windows 2000 and Windows NT 4.0 have a 2047 limit. POSIX requires that _POSIX2_LINE_MAX have a value of at least 2048.

In 2048 characters it is possible to assign values to and use at least once 100 different variables (e.g., a1=2;a2=2.3;....; print a1+a2*a3...). To get a lower bound lets consider the number of different expressions it is possible to write. How many functionally different expressions containing 100 binary operators are there?

If a language has, say, eight binary operators (e.g., +, -, *, /, %, &, |, ^), then it is possible to write 8^100 right 2.03703598*10^90 visually different expressions containing 100 binary operators. Some of these expressions will be mathematically equivalent (adopting the convention of leaving out the operands), e.g., + * can also be written as * + (the appropriate operands will also have the be switched around).

If we just consider expressions created using the commutative operators (i.e., +, *, &, |, ^), then with these five operators it is possible to write 1170671511684728695563295535920396 mathematically different expressions containing 100 operators (assuming the common case that the five operators have different precedence levels, which means the different expressions have a one to one mapping to a rooted tree of height five); this 1.17067*10^33 is a lot smaller than 5^100 right 7.88860905*10^69.

Had the approximately 10^9 computers/smart phones in the world generated expressions at the rate of 10^6 per second since the start of the Universe, 4.336*10^17 seconds ago, then the 4.336*10^32 created so far would be almost half of the total possible.

Once we start including the non-commutative operators such a minus and divide the number of possible combinations really starts to climb and the calculation of the totals is real complicated. Since the Universe is not yet half way through the commutative operators I will leave working this total out for another day.

Update (later in the day)

To get some idea of the huge jump in number of functionally different expressions that occurs when operator ordering is significant, with just the three operators -, / and % is is possible to create 3^100 right 5.15377521*10^47 mathematically different expressions. This is a factor of 10^14 greater than generated by the five operators considered above.

If we consider expressions containing just one instance of the five commutative operators then the number of expressions jumps by another two orders of magnitude to 5*100*3^99. This count will continue to increase for a while as more commutative operators are added and then start to decline; I have not yet worked things through to find the maxima.

Update (April 2012).
Sequence A140606 in the On-Line Encyclopedia of Integer Sequences lists the number of inequivalent expressions involving n operands; whose first few values are: 1, 6, 68, 1170, 27142, 793002, 27914126, 1150212810, 54326011414, 2894532443154, 171800282010062, 11243812043430330, 804596872359480358, 62506696942427106498, 5239819196582605428254, 471480120474696200252970, 45328694990444455796547766, 4637556923393331549190920306

Relative spacing of operands affects perception of operator precedence

January 22nd, 2012 1 comment

What I found most intriguing about Google Code Search (shutdown Nov 2011) was how quickly searches involving regular expressions returned matches. A few days ago Russ Cox, the implementor of Code Search not only explained how it worked but also released the source and some precompiled binaries. Google’s database of source code did not include the source of R, so I decided to install CodeSearch on my local machine and run some of my previous searches against the latest (v2.14.1) R source.

In 2007 I ran an experiment that showed developers made use of variable names when making binary operator precedence decisions. At about the same time two cognitive psychologists, David Landy and Robert Goldstone, were investigating the impact of spacing on operator precedence decisions (they found that readers showed a tendency to pair together the operands that were visibly closer to each other, e.g., a with b in a+b * c rather than b with c).

As somebody very interested in finding faults in code the psychologists research findings on spacing immediately suggested to me the possibility that ‘incorrectly’ spaced expressions were a sign of failure to write code that had the intended behavior. Feeding some rather complicated regular expressions into Google’s CodeSearch threw up a number of ‘incorrectly’ spaced expressions. However, this finding went no further than an interesting email exchange with Landy and Goldstone.

Time to find out whether there are any ‘incorrectly’ spaced expressions in the R source. cindex (the tool that builds the database used by csearch) took 3 seconds on a not very fast machine to process all of the R source (56M byte) and build the search database (10M byte; the Linux database is a factor of 5.5 smaller than the sources).

The search:

csearch "\w(\+|\-)\w +(\*|\/) +\w"

returned a few interesting matches:

...
modules/internet/nanohttp.c:       used += tv_save.tv_sec + 1e-6 * tv_save.tv_usec;
modules/lapack/dlapack0.f:     $          ( T*( ONE+SQRT( ONE+S / T ) ) ) )
modules/lapack/dlapack2.f:               S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) )
modules/lapack/dlapack4.f:     $          ( T*( ONE+SQRT( ONE+S / T ) ) ) )

There were around 15 matches of code like 1e-6 * var (because the pattern \w is for alphanumeric sequences and that is not a superset of the syntax of floating-point literals).

The subexpression ONE+S / T is just the sort of thing I was looking for. The three instances all involved code that processed tridiagonal matrices in various special cases. Google search combined with my knowledge of numerical analysis was not up to the task of figuring out whether the intended usage was (ONE+S)/T or ONE+(S/T).

Searches based on various other combination of operator pairs failed to match anything that looked suspicious.

There was an order of magnitude performance difference for csearch vs. grep -R -e (real 0m0.167s vs. real 0m2.208s). A very worthwhile improvement when searching much larger code bases with more complicated patterns.