[英]What is the running time of the following algorithm (in terms of 𝑂-notation)?
[英]What is an algorithm to enumerate lambda terms?
什么算法將按照長度的順序枚舉lambda演算的表達式? 例如, (λx.x), (λx.(xx)), (λx.(λy.x))
等等?
作為長度,我將在(無類型)lambda表達式的BNF中選擇T
-expansions(“depth”)的數量:
V ::= x | y
T ::= V |
λV.T |
(T T)
在python中,你可以按照上面給定變量的生成規則定義一個生成器,給定的深度如下:
def lBNF(vars, depth):
if depth == 1:
for var in vars:
yield var
elif depth > 1:
for var in vars:
for lTerm in lBNF(vars,depth-1):
yield 'l%s.%s' % (var,lTerm)
for i in range(1,depth):
for lTerm1 in lBNF(vars,i):
for lTerm2 in lBNF(vars,depth-i):
yield '(%s %s)' % (lTerm1,lTerm2)
現在,您可以枚舉/達到給定深度的lambda術語:
vars = ['x','y']
for i in range(1,5):
for lTerm in lBNF(vars,i):
print lTerm
請參閱https://arxiv.org/abs/1210.2610 ,第5頁。以下是一些示例代碼:
from itertools import chain, count
from functools import lru_cache
@lru_cache(maxsize=None)
def terms(size, level=0):
if size == 0:
return tuple(range(level))
else:
abstractions = (
('abs', term)
for term in terms(size - 1, level + 1)
)
applications = (
('app', term1, term2)
for i in range(size)
for term1 in terms(i, level)
for term2 in terms(size - 1 - i, level)
)
return tuple(chain(abstractions, applications))
def string(term):
if isinstance(term, tuple):
if term[0] == 'abs':
return '(λ {})'.format(string(term[1]))
elif term[0] == 'app':
return '({} {})'.format(string(term[1]), string(term[2]))
else:
return term
for size in count():
print('{} terms of size {}'.format(len(terms(size)), size))
for term in terms(size):
pass # input(string(term))
這輸出
0 terms of size 0
1 terms of size 1
3 terms of size 2
14 terms of size 3
82 terms of size 4
579 terms of size 5
4741 terms of size 6
43977 terms of size 7
454283 terms of size 8
等等(即這個序列 )。
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