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To me, solving this problem is all about finding the right language.

As expressed in the OP, Knuth expressions like floor(sqrt((3!)!)) are clunky. The eye gets lost in ! and () and it hard to see structure.

But inspiration hits: every Knuth expression is just a highly nested function of 3, and functions are evaluated from the inside out. So we could use a reverse Polish notation to simplify things. Set f = factorial, s = sqrt and i = floor(int). Then 26 = 3ffsi and in general, Knuth expressions can be represented by strings of the form 3[isf]*. Much simpler!

This representation immediately suggests a solution to the problem: generate all strings [isf]* up to length n and evaluate to each string to see if we have a small integer. Simple, but inefficient. The problem is that the number of strings generated grows as 3**n, and even for small n this search can take long, long time.

This situation can be helped by thinking about the nature of the functions used. Obviously, ii == i, so any sequence i+ can be reduced to a single i. As japhy points out, any sequence of square roots and floors [si]+i can drop the floors to become s+i without affecting the result. And so we can reduce our language of Knuth expressions to the form 3(s|if)+. The number of strings in this language grow as x**n with x somewhat less than 2. This is much better than 3**n and so I wrote a program that was able solve the problem in about 2 hours on my poky PIII notebook:

use warnings; use strict; use Algorithm::Loops qw( NestedLoops ); use Math::BigInt lib => 'GMP'; my $start_base = shift; my $depth = shift; my @func = qw/i s f/; # alphabet used # my @func = qw/s if/; my @seq; # Stores shortest sequences associated with each positive i +nteger [1,200] $seq[3] = ""; my @todo; # Stores base elements yet to be searched. push @todo, $start_base; # Start things off while (@todo) { my $base = shift @todo; print "expanding $base\n"; expand( $base, $depth); } for my $val (1..200) { print "$val: $seq[$val]\n" if defined $seq[$val]; } ### Subroutines # Create all possible strings up to $depth long and evaluate each stri +ng sub expand { my ($base, $depth) = @_; NestedLoops( [ ( [@func] ) x $depth ], { OnlyWhen => 1 }, sub{ evaluate_bigint( $base, @_); }, ); } # Evaluate the result of computing the expression given by a string # Adds useful strings to @seq and pushes newly found values onto @todo # queue. sub evaluate_bigint { my $base = shift; my $val = Math::BigInt->new( $base); # create a BigInt object my $seq_str = join "", @_; my @sequence = split //, $seq_str; my $prefix = defined $seq[$base] ? $seq[$base] : ""; foreach my $f (@sequence) { if ($f eq 'i') { # int # int is a no-op for a BigInt } elsif ($f eq 's') { # sqrt $val->bsqrt(); } else { # factorial return if $val->length() > 5; # limit max size $val->bfac(); } } return if $val->length() > 3; # only want small integers my $num = $val->numify(); # convert to ordinary scalar $seq_str = $prefix . $seq_str; $seq_str =~ s/^i+//; $seq_str =~ s/i+/i/g; $seq_str =~ s/fif/ff/g; if (! defined($seq[$num])) { # new entry push @todo, $num; $seq[$num] = $seq_str; print "$num: $seq_str\n"; } elsif (length $seq_str < length $seq[$num]) { $seq[$num] = $seq_str; } }
This code evaluates strings using the Math::BigInt GMP library in order to handle large factorials. It also uses a memoization technique to reduce the search space. The idea is that starting with 3 and applying all possible strings up to length n yields several small integers as Knuth expressions, e.g., 5 = 3ffssi. Then one can search starting with the small integers, e.g., 5 to yield new small integers, and so on.

This memoization trick, however, leads to another big insight: each Knuth sequence that we have the ability to compute can be described by a sequence of small integers found as we evaluate the string. Between each pair of numbers is an ascent to larger numbers created by f*, followed a descent into smaller numbers created by s*, followed by an i. This yields a further simplification of our language: to map from small integer to small integer, search strings of the form f*s*i. There are just n length n strings of the form f*s*i, versus the 2**n before, so we get a massive speedup.

Here is the same program as before with our newly optimized language:

use warnings; use strict; use Algorithm::Loops qw( NestedLoops ); use Math::BigInt lib => 'GMP'; my $start_base = shift; my $depth = shift; my @seq; # Stores shortest sequences associated with each positive i +nteger [1,200] $seq[3] = ""; my @todo; # Stores base elements yet to be searched. push @todo, $start_base; # Start things off while (@todo) { my $base = shift @todo; print "expanding $base\n"; expand( $base, $depth); } for my $val (1..200) { print "$val: $seq[$val]\n" if defined $seq[$val]; } ### Subroutines # Create all possible strings up to $depth long and evaluate each stri +ng sub expand { my ($base, $depth) = @_; for my $len (1..$depth) { my $max_f = $len < 3 ? $len : 3; for my $f (0..$max_f) { my $str = 'f'x$f . 's'x($len-$f); evaluate_bigint( $base, $str); } } } # Evaluate the result of computing the expression given by a string # Adds useful strings to @seq and pushes newly found values onto @todo # queue. sub evaluate_bigint { my $base = shift; my $val = Math::BigInt->new( $base); # create a BigInt object my $seq_str = shift; my $prefix = defined $seq[$base] ? $seq[$base] : ""; # my $str = $prefix . $seq_str; # print "base: $base, seq_str: $seq_str, str: $str\n"; my @sequence = split //, $seq_str; foreach my $f (@sequence) { if ($f eq 'i') { # int # int is a no-op for a BigInt } elsif ($f eq 's') { # sqrt $val->bsqrt(); } else { # factorial return if $val->length() > 3; # limit max size # print "length before fac:", $val->length(), "\n"; $val->bfac(); } } return if $val->length() > 3; # only want small integers my $num = $val->numify(); # convert to ordinary scalar $seq_str = $prefix . $seq_str; if (! defined($seq[$num])) { # new entry push @todo, $num; $seq[$num] = $seq_str; print "$num: $seq_str\n"; } elsif (length $seq_str < length $seq[$num]) { $seq[$num] = $seq_str; } }
This runs in 1.4 seconds -- a speedup of 5000!

Update: Corrected a spelling mistake, thanks to gam3.

-Mark


In reply to Re: Challenge: Chasing Knuth's Conjecture by kvale
in thread Challenge: Chasing Knuth's Conjecture by kvale

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