BIT predicate

In mathematics and computer science, the BIT predicate or Ackermann coding, sometimes written BIT(i, j), is a predicate which tests whether the jth bit of the number i is 1, when i is written in binary.

History

The BIT predicate was first introduced as the encoding of hereditarily finite sets as natural numbers by Wilhelm Ackermann in his 1937 paper^[1][2] (The Consistency of General Set Theory).

Each natural number encodes a finite set and each finite set is represented by a natural number. This mapping uses the binary numeral system. If the number n encodes a finite set A and the ith binary digit of n is 1 then the set encoded by i is an element of A. The Ackermann coding is a primitive recursive function.^[3]

Implementation

In programming languages such as C, C++, Java, or Python that provide a right shift operator >> and a bitwise Boolean and operator &, the BIT predicate BIT(i, j) can be implemented by the expression (i>>j)&1. Here the bits of i are numbered from the low order bits to high order bits in the binary representation of i, with the ones bit being numbered as bit 0.^[4]

Private information retrieval

In the mathematical study of computer security, the private information retrieval problem can be modeled as one in which a client, communicating with a collection of servers that store a binary number i, wishes to determine the result of a BIT predicate BIT(i, j) without divulging the value of j to the servers. Chor et al. (1998) describe a method for replicating i across two servers in such a way that the client can solve the private information retrieval problem using a substantially smaller amount of communication than would be necessary to recover the complete value of i.^[5]

Mathematical logic

The BIT predicate is often examined in the context of first-order logic, where we can examine the system resulting from adding the BIT predicate to first-order logic. In descriptive complexity, the complexity class FO + BIT resulting from adding the BIT predicate to FO results in a more robust complexity class.^[6] The class FO + BIT, of first-order logic with the BIT predicate, is the same as the class FO + PLUS + TIMES, of first-order logic with addition and multiplication predicates.^[7]

Construction of the Rado graph

The Rado graph: for instance there is an edge from 0 to 3 because the Oth bit of 3 is non zero.

Ackermann in 1937 and Richard Rado in 1964 used this predicate to construct the infinite Rado graph. In their construction, the vertices of this graph correspond to the non-negative integers, written in binary, and there is an undirected edge from vertex i to vertex j, for i < j, when BIT(j,i) is nonzero.^[8]

References

1. Ackermann, Wilhelm (1937). "Die Widerspruchsfreiheit der allgemeinen Mengenlehre". Mathematische Annalen. 114: 305–315. doi:10.1007/bf01594179. Retrieved 2012-01-09.
2. Kirby, Laurence (2009). "Finitary Set Theory". Notre Dame Journal of Formal Logic. 50 (3): 227–244. doi:10.1215/00294527-2009-009. Retrieved 31 May 2011.
3. Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). New York: Springer Science+Business Media. p. 261. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1220-6.
4. Venugopal, K. R. (1997). Mastering C++. Muhammadali Shaduli. p. 123. ISBN 9780074634547..
5. Chor, Benny; Kushilevitz, Eyal; Goldreich, Oded; Sudan, Madhu (1998). "Private information retrieval". Journal of the ACM. 45 (6): 965–981. doi:10.1145/293347.293350..
6. Immerman, Neil (1999). Descriptive Complexity. New York: Springer-Verlag. ISBN 0-387-98600-6.
7. Immerman, Neil (1999). Descriptive Complexity. New York: Springer-Verlag. pp. 14–16. ISBN 0-387-98600-6.
8. Rado, Richard (1964). "Universal graphs and universal functions" (PDF). Acta Arith. 9: 331–340..