Showing papers by "Juris Hartmanis published in 1977"

On Isomorphisms and Density of $NP$ and Other Complete Sets

Abstract: If all $NP$ complete sets are isomorphic under deterministic polynomial time mappings (p-isomorphic) then $P e NP$ and if all PTAPE complete sets are P-isomorphic then $P e {\text{PTAPE}}$. We show that all $NP$ complete sets known (in the literature) are indeed p-isomorphic and so are the known PTAPE complete sets. This shows that, in spite of the radically different origins and attempted simplification of these sets, all the known $NP$ complete sets are identical but for simple isomorphic codings computable in deterministic polynomial time.Furthermore, if all $NP$ complete sets are p-isomorphic then they all must have similar densities and, for example, no language over a single letter alphabet can be $NP$ complete, nor can any sparse language over an arbitrary alphabet be $NP$ complete. We show that complete sets in EXPTIME and EXPTAPE cannot be sparse and therefore they cannot be over a single letter alphabet. Similarly, we show that the hardest context-sensitive languages cannot be sparse. We als...

Relations between diagonalization, proof systems, and complexity gaps

Abstract: In this paper we study diagonal processes over time-bounded computations of one-tape Turing machines by diagonalizing only over those machines for which there exist formal proofs that they operate in the given time bound. This replaces the traditional “clock” in resource bounded diagonalization by formal proofs about running times and establishes close relations between properties of proof systems and existence of sharp time bounds for one-tape Turing machine complexity classes. Furthermore, these diagonalization methods show that the Gap Theorem for resource bounded computations does not hold for complexity classes consisting only of languages accepted by Turing machines for which it can be formally proven that they run in the required time bound.

Relations Between Diagonalization, Proof Systems, and Complexity Gaps (Preliminary Version)

Abstract: Abstract In this paper we study diagonal processes over time bounded computations of one-tape Turing machines by diagonalizing only over those machines for which there exist formal proofs that they operate in the given time bound. This replaces the traditional “clock” in resource bounded diagonalization by formal proofs about running times and establishes close relations between properties of proof systems and existence of sharp time bounds for one-tape Turing machine complexity classes. These diagonalization methods also show that the Gap Theorem for resource bounded computations can hold only for those complexity classes which differ from the corresponding provable complexity classes. Furthermore, we show that there exist recursive time bounds T ( n ) such that the class of languages for which we can formally prove the existence of Turing machines which accept them in time T ( n ) differs from the class of languages accepted by Turing machines for which we can prove formally that they run in time T ( n ). We also investigate the corresponding problems for tape bound computations and discuss the difference time and tapebounded computations.