Space Complexity Vs. Query Complexity - Lachish, Oded; Newman, Ilan; Shapira, Asaf (Computational complexity, 2008)
. Combinatorial property testing deals with the following relaxation of decision problems: Given a fixed property and an input x , one wants to decide whether x satisfies the property or is “far” from satisfying it. The main focus of property testing is in identifying large families of properties that can be tested with a certain number of queries to the input. In this paper we study the relation between the space complexity of a language and its query complexity. Our main result is that for any space complexity s ( n ) ≤ log n there is a language with space complexity O ( s ( n )) and query complexity 2 Ω( s ( n )) . Our result has implications with respect to testing languages accepted by certain restricted machines. Alon et al. [FOCS 1999] have shown that any regular language is testable with a constant number of queries. It is well known that any language in space o (log log n ) is regular, thus implying that such languages can be so tested. It was previously known that there are languages in space O (log n ) that are not testable with a constant number of queries and Newman [FOCS 2000] raised the question of closing the exponential gap between these two results. A special case of our main result resolves this problem as it implies that there is a language in space O (log log n ) that is not testable with a constant number of queries. It was also previously known that the class of testable properties cannot be extended to all context-free languages. We further show that one cannot even extend the family of testable languages to the class of languages accepted by single counter machines.