ABSTRACT
For each natural number d we consider a finite structure md whose universe is the set of all 0,1-sequence of length n=2d, each representing a natural number in the set [0,1,...,2n-1] in binary form. The operations included in the structure are the four constants 0,1,2n-1,n, multiplication and addition modulo 2n, the unary function min[2x, 2n-1], the binary functions ⌊ x/y⌋ (with ⌊ x/0 ⌋ =0), max(x,y), min(x,y), and the boolean vector operations ∧,∨,- defined on 0,1 sequences of length n by performing the operations on all components simultaneously. These are essentially the arithmetic operations that can be performed on a RAM, with wordlength n, by a single instruction. We show that there exists a term (that is, an algebraic expression) F(x,y) built up from the mentioned operations, with the only free variables x,y, such that for all terms G(y), which is also built up from the mentioned operations, the following holds. For infinitely many positive integers d, there exists an a ∈ md such that the following two statements are not equivalent: (i) md |= ∃ x, F(x,a), (ii) md models G(a)=0. In other words, the question whether an existential statement, depending on the parameter a ∈ md is true or not, cannot be decided by evaluating an algebraic expression at a.
Another way of formulating the theorem, in a slightly stronger form, is, that over the structures md, quantifier elimination is not possible in the following sense. Let calm be a first-order language with equality, containing function symbols for all of the mentioned arithmetic operations. Then there exists an existential first-order formula φ(y) of calm, containing a single existential quantifier and the only free variable y, such that for each propositional formula P(y) of calm, we have that for infinitely many positive integers d, φ(y) and P(y) are not equivalent on md, that is, md |= - ∀ y, φ(y) >-> P(y).
We also show that the theorem, in both forms, remains true if the binary operation min [xy, 2n-1] is added to the structure md. A general theorem is proved as well, which describes sufficient conditions for a set of operations on a sequence of structures kd, d=1,2,... which guarantees that the analogues of the mentioned theorems holds for the structures kd too.
Supplemental Material
- .V. Aho, J.E. Hopcroft, J.D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974. Google ScholarDigital Library
- . Ajtai, Determinism versus Nondeterminism for Linear Time RAMs with Memory Restrictions, Journal of Computer andSystems Science, 65(1): 2--37, (2002) Google ScholarDigital Library
- M. Ajtai, Oblivious $RAM$s without cryptographic assumptions, Electronic Colloquium on Computational Complexity (ECCC),17:28, 2010.Google Scholar
- . Ajtai. Oblivious $RAM$s without cryptographic assumptions,Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010,Cambridge, Massachusetts, USA, 5--8 June 2010, pages 181--190. ACM,2010. Google ScholarDigital Library
- . Ajtai, Y. Gurevich, Monotone versus positive, Journal of the ACM (JACM), Vol. 34, Issue 4, Oct. 1987, pp. 1004--1015. Google ScholarDigital Library
- E. Artin. Galois Theory, Dover Publications, 1998.(Reprinting of second revised edition of 1944, The University of Notre Dame Press).Google Scholar
- P. Beame, S. A. Cook, and H. J. Hoover, Log Depth Circuits for Division and Related Problems, SIAM Journal on Computing, 15(4):994--1003, November 1986. Google ScholarDigital Library
- . Beame, T. S. Jayram, M. Sacks, Time-space tradeoffs for branching programs, Journal of Computer and Systems Science,63(4):542--572, December 2001. Google ScholarDigital Library
- . Beame, M. Sacks, Xiadong Sun, E. Vee, Time-space trade-off lower bounds for randomized computation of decision problems,Journal of ACM, 50(2):154--195,2003. Google ScholarDigital Library
- M. Davis, H. Putnam and J. Robinson, Thedecision problem for exponential diophantine equations, Ann. of Math. (2) 74 (1961),425--436. 3. J. P. Jones, Three universal representations of r.e. sets,J. Symbolic Logic 43 (1978), 335--351.Google ScholarCross Ref
- W. Hesse, Division is in Uniform $TC^0$, ICALP 2001: 104--114. Google ScholarDigital Library
- A. Magid, Differential Galois theory,Notices of the American Mathematical Society 46 (9): 1999.Google Scholar
- L. Fortnow, Time-space tradoffs for satisfiability, Journal of Computer and System Sciences, 60:337--353, 2000. Google ScholarCross Ref
- Ju. V. Matijasevic, Enumerable sets are diophantine, Dokl. Akad.Nauk SSSR 191 (1970), 279--282. English transi.: Soviet Math. Doklady11 (1970), 354--358.Google Scholar
- . Paul, N. Pippenger, E. Szemerédi, and W. Trotter. On determinism versus nondeterminism and related problems,In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 429--438. IEEE, New York, 1983. Google ScholarDigital Library
- . Takeuti, Proof Theory,North-Holland, Studies in Logic and the Foundations of Mathematics, Vol. 81, Second edition, 1987.Google Scholar
- . Yao Separating the polynomial time hierarchy by oracles, Proc.26th Annu. IEEE Symp. Found. Comp. Sci. 1--10(1985). Google ScholarDigital Library
Index Terms
- Determinism versus nondeterminism with arithmetic tests and computation: extended abstract
Recommendations
Lower bounds for RAMs and quantifier elimination
STOC '13: Proceedings of the forty-fifth annual ACM symposium on Theory of ComputingFor each natural number d we consider a finite structure Md whose universe is the set of all 0,1-sequence of length n=2d, each representing a natural number in the set {0,1,...,2n-1} in binary form. The operations included in the structure are the four ...
Arithmetic, first-order logic, and counting quantifiers
This article gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers. Precisely, this ...
Determinism vs. nondeterminism for two-way automata: representing the meaning of states by logical formulæ
DLT'12: Proceedings of the 16th international conference on Developments in Language TheoryThe question whether nondeterminism is more powerful than determinism for two-way automata is one of the most famous old open problems on the border between formal language theory and automata theory. An exponential gap between the number of states of ...
Comments