The logic of explicitly presentation-invariant circuits
We give a novel and simple characterization of first-order logic and its infinitary variants with bounded numbers of variables in terms of circuit computation. The main feature of the proposed circuit model of explicitly symmetric circuits is, that genericity in computations over structures is guaranteed through full combinatorial symmetry of circuits with respect to permutations of the input representation.
Definability in the bounded variable fragment of infinitary logic is equivalent with recognizability in such explicitly symmetric circuits in which the orbits of nodes under symmetry operations are polynomially bounded. First-order definability is equivalent with recognizability in finite depth circuits of that kind.
KeywordsCircuit complexity generic computation finite model theory
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- [AV91]S. Abiteboul and V. Vianu. Generic computation and its complexity. Proc. 23rd ACM Symp. on Theory of Computing (1991), 209–219.Google Scholar
- [BIS86]D.A.M. Barrington, N. Immerman and H. Straubing. On uniformity within NC1. Journal of Computer and System Sciences 41 (1990), 274–306.Google Scholar
- [CH80]A. Chandra and D. Harel. Computable queries for relational databases. Journal of Computer and System Sciences 21 (1980), 156–178.Google Scholar
- [CL86]K. Compton and C. Laflamme. An algebra and a logic for NC1. Information and Computation 87 (1990), 241–263.Google Scholar
- [DGS86]L. Denenberg, Y. Gurevich and S. Shelah. Definability by constant-depth polynomial-size circuits. Information and Control 70 (1986), 216–240.Google Scholar
- [EF95]H.-D. Ebbinghaus, J. Flum. Finite Model Theory. Perspectives in Mathematical logic. Springer 1995.Google Scholar
- [GL81]P. Gacs and L.A. Levin. Causal nets or what is a deterministic computation. Information and Control 51 (1981), 1–19.Google Scholar
- [GL84]Y. Gurevich and H.R. Lewis. A logic for constant-depth circuits. Information and Control 61 (1984), 65–74.Google Scholar
- [I89]N. Immerman. Expressibility and parallel complexity. SIAM Journal of Computation 18 (1989), 625–638.Google Scholar
- [KV92]Ph. G. Kolaitis and M. Y. Vardi. Fixpoint logic vs. in finitary logic infinite-model theory. Proc. 7th IEEE Symp. on Logic and Computer Science (1992), 46–57.Google Scholar