FL (complexity)

In computational complexity theory, the complexity class FL is the set of function problems which can be solved by a deterministic Turing machine in a logarithmic amount of memory space.{{citation|contribution=Functional oracle queries as a measure of parallel time|first1=Carme|last1=Àlvarez|first2=José L.|last2=Balcázar|first3=Birgit|last3=Jenner|title=STACS 91|series=Lecture Notes in Computer Science|volume=480|year=1991|pages=422–433|publisher=Springer|doi=10.1007/BFb0020817|hdl=2117/327984|isbn=3-540-53709-0 |hdl-access=free}}. As in the definition of L, the machine reads its input from a read-only tape and writes its output to a write-only tape; the logarithmic space restriction applies only to the read/write working tape.

Loosely speaking, a function problem takes a complicated input and produces a (perhaps equally) complicated output. Function problems are distinguished from decision problems, which produce only Yes or No answers and corresponds to the set L of decision problems which can be solved in deterministic logspace. FL is a subset of FP, the set of function problems which can be solved in deterministic polynomial time.

FL is known to contain several natural problems, including arithmetic on numbers. Addition, subtraction and multiplication of two numbers are fairly simple, but achieving division is a far deeper problem which was open for decades.{{citation|first1=A.|last1=Chiu|first2=G.|last2=Davida|first3=B.|last3=Litow|title=Division in logspace-uniform NC1|journal=RAIRO Theoretical Informatics and Applications|volume=35|pages=259–276|year=2001|issue=3 |doi=10.1051/ita:2001119 }}.{{citation|contribution-url=http://ftp.cs.rutgers.edu/pub/allender/division.eatcs.pdf|contribution=The division breakthroughs|first=Eric|last=Allender|authorlink=Eric Allender|title=Current Trends in Theoretical Computer Science|publisher=World Scientific|year=2004|pages=147–164}}.

Similarly one may define FNL, which has the same relation with NL as FNP has with NP.