# Sorting, linear time and the satisfiability problem

- 113 Downloads
- 17 Citations

## Abstract

Let DLIN denote the class of problems that are computable within*O*(*n*/log*n*) uniform time on a RAM which can read its input (of length*n*) in blocks and which only uses integers*O*(*n*/log*n*). We prove that the sorting problem belongs to DLIN and formulate the Linear Time Thesis: “Each problem computable by a linear time bounded algorithm (in an intuitive sense) belongs to DLIN”. We also study the subclass DTIME_{SORT}(*n*) of problems computable within linear time on a Turing machine using in addition a fixed number of sortings, and we show how the reductions of this class, so-called sort-lin reductions, are useful to classify NP problems: e.g. there is a sort-lin reduction from SAT to 3-SAT (using no sorting) and a sort-lin reduction from the NP-complete graph problem KERNEL to SAT (but we do not know of any similar reduction in DTIME(*n*)). Similarly, problem 2-SAT is linearly equivalent to the problem of Horn renaming (via sort-lin reductions). Finally, SAT is compared with many other combinatorial problems. A problem*A* is SAT-easy (resp. SAT-hard) if there is a sort-lin reduction from*A* to SAT (resp. SAT to*A*). A SAT-equivalent problem is a problem both SAT-easy and SAT-hard. It is shown that the class of SAT-easy (resp. SAT-equivalent) problems is very large and that its generalization to so-called special clauses or, more generally, regular clauses, does not enlarge it. Moreover, we justify our opinion that problem SAT is, in some sense, a minimal NP-complete problem.

## Keywords

Neural Network Sorting Linear Time Fixed Number Turing Machine## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A.V. Aho, J.E. Hopcroft and J.D. Ullman,
*The Design and Analysis of Computer Algorithms*(Addison-Wesley, Reading, 1974).Google Scholar - [2]A.V. Aho, J.E. Hopcroft and J.D. Ullman,
*Data Structures and Algorithms*(Addison-Wesley, Reading, 1983).Google Scholar - [3]D. Angluin and L. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings, J. Comput. Syst. Sci. 18(1979)155–193.Google Scholar
- [4]B. Apsvall, Recognizing disguised NR(1) instances of the satisfiability problem, J. Algorithms 1(1980)97–103.Google Scholar
- [5]A.M. Ben-Amram and Z. Galil, On pointers versus addresses, J. ACM 39(1992)617–648.Google Scholar
- [6]C. Berge,
*Graphs and Hypergraphs*(North-Holland, Amsterdam, 1973).Google Scholar - [7]K. Berman, J. Franco and J. Schlipf, Unique satisfiability for Horn sets can be solved in nearly linear time, Technical Report CS-TR-92-5, Comput. Sci. Dept., Univ. Cincinnati, OH (1992).Google Scholar
- [8]A. Bertoni, G. Mauri and N. Sabadini, Simulations among classes of random access machines and equivalence among numbers succinctly represented, Ann. Discr. Math. 25(1985)65–90.Google Scholar
- [9]K.J. Compton and C.W. Henson, A uniform method for proving lower bounds on the computational complexity of logical theories, Ann. Pure Appl. Logic 48(1990)1–79.Google Scholar
- [10]S.A. Cook, The complexity of theorem proving procedures,
*3rd STOC*(1971) pp. 151–158.Google Scholar - [11]A. Cook, Short propositional formulas represent nondeterministic computations, Inf. Proc. Lett. (1987/88) 269–270.Google Scholar
- [12]S.A. Cook and R.A. Reckhow, Time-bounded random access machines, J. Comput. Syst. Sci. 7(1973)354–375.Google Scholar
- [13]T.H. Cormen, C.E. Leiserson and R.L. Rivest,
*Introduction to Algorithms*(MIT Press/McGraw-Hill, 1991).Google Scholar - [14]N. Creignou, The class of problems that are linearly equivalent to satisfiability or a uniform method for proving NP-completeness,
*Proc. CSL'92*, Lecture Notes in Computer Science 702 (1993) pp. 115–133; Theor. Comput. Sci. 145(1995)111–145.Google Scholar - [15]N. Creignou, Exact complexity of problems of incompletely specified automata, this volume, Ann. of Math. and AI 16(1996).Google Scholar
- [16]N. Creignou, Temps linéaire et problèmes NP-complet, Thèse de doctorat, Univ. de Caen, France (1993).Google Scholar
- [17]A.K. Dewdney, Linear time transformations between combinatorial problems, Int. J. Comput. Math. 11(1982)91–110.Google Scholar
- [18]W.F. Dowling and J.H. Gallier, Linear-time algorithms for testing the satisfiability of propositional Horn formulas, J. Logic Progr. 3(1984)267–284.Google Scholar
- [19]R. Fagin, Generalized first-order spectra and polynomial-time recognizable sets, in:
*Compl. of Comp., Proc. Symp.*, New York (1973), SIAM-AMS Proc. 7(1974)43–73.Google Scholar - [20]C. Froidevaux, M.C. Gaudel and M. Soria,
*Types de Données et Algorithmes*(McGraw-Hill).Google Scholar - [21]M.R. Garey and D.S. Johnson,
*Computers and Intractability: A Guide to the Theory of NP-Completeness*(Freeman, San Francisco, 1979).Google Scholar - [22]E. Graedel, on the notion of linear time computability,
*Proc. 3rd Italian Conf. on Theor. Comput. Sci.*(World Scientific, 1989) pp. 323–334, also in Int. J. Found. Comput. Sci. 1(1990)295–307.Google Scholar - [23]E. Grandjean, A natural NP-complete problem with a nontrivial lower bound, SIAM J. Comput. 17(1988)786–809.Google Scholar
- [24]E. Grandjean, A nontrivial lower bound for an NP problem on automata, SIAM J. Comput. 19(1990)438–451.Google Scholar
- [25]E. Grandjean, Invariance properties of RAMs and linear time, Comput. Complexity 4(1994)62–106.Google Scholar
- [26]E. Grandjean, Linear time algorithms and NP-complete problems, SIAM J. Comput. 23(1994)573–597.Google Scholar
- [27]E. Grandjean and F. Olive, Monadic logical definability of NP-complete problems, to appear in
*Proc. CSL'94*, Lecture Notes Comput. Sci.Google Scholar - [28]E. Grandjean and J.M. Robson, RAM with compact memory: A realistic and robust model of computations,
*Proc. CSL'90*, Lecture Notes Comput. Sci. 533(1991) pp. 195–233.Google Scholar - [29]Y. Gurevich and S. Shelah, Nearly linear time, Lecture Notes Comput. Sci. 363 (Springer, 1989) pp. 108–118.Google Scholar
- [30]J. Hartmanis and J. Simon, On the power of multiplication in random access machines,
*15th IEEE Symp. on Switching and Automata Theory*(1974) pp. 13–23.Google Scholar - [31]J.J. Hébrard, A linear algorithm for renaming a set of clauses as a Horn set, Theor. Comput. Sci. 124(1994)343–350.Google Scholar
- [32]J.J. Hébrard, Une mise en oeuvre linéaire de la résolution unitaire, Technical Report, Univ. de Caen, Cahiers du LIUC 92-10 (1992).Google Scholar
- [33]J.J. Hébrard, Unique Horn renaming and unique 2-satisfiability, Inf. Proc. Lett. 54(1995)235–239.Google Scholar
- [34]J.E. Hopcroft, W. Paul and L. Valiant, On time versus space and related problems, J. ACM 24(1977)332–337.Google Scholar
- [35]J.E. Hopcroft and J.D. Ullman,
*Introduction to Automata Theory, Languages and Computation*(Addison-Wesley, Reading, 1979).Google Scholar - [36]H.B. Hunt and R.E. Stearns, The complexity of very simple Boolean formulas with applications, SIAM J. Comput. 19(1990)44–70.Google Scholar
- [37]N. Immerman, Nondeterministic space is closed under complementation, SIAM J. Comput. 17(1988)935–938.Google Scholar
- [38]A. Itai and J.A. Makowsky, On the complexity of Herbrand's theorem. Technical Report 243, Computer Science Dept., Technion, Haifa, Israel (1982).Google Scholar
- [39]A. Itai and J.A. Makowsky, Unification as a complexity measure for logic programming, Technical Report 301, Computer Science Dept., Technion, Haifa, Israel (1993), in revised form in J. Logic. Progr. 4(1987)105–117.Google Scholar
- [40]N.D. Jones, Space-bounded reducibility among combinatorial problems, J. Comput. Syst. Sci. 11(1975)68–85.Google Scholar
- [41]N.D. Jones and W.T. Laaser, Complete problems for deterministic polynomial time, Theor. Comput. Sci. 3(1977)105–117.Google Scholar
- [42]N.D. Jones, Y.E. Lien and W.T. Laaser, New problems complete for nondeterministic log space, Math. Syst. Th. 10(1976)1–17.Google Scholar
- [43]R.M. Karp, Reducibility among combinatroial problems,
*IBM Symp. 1972, Complexity of Computer Computations*(Plenum Press, New York, 1972).Google Scholar - [44]J. Katajainen, J. van Leuwen and M. Penttonen, Fast simulation of Turing machines by random access machines, SIAM J. Comput. 17(1988)77–88.Google Scholar
- [45]D.E. Knuth,
*The Art of Programming*, Vols. 1, 2, 3 (Addison-Wesley, Reading, 1968, 1969, 1973).Google Scholar - [46]H.R. Lewis, Renaming a set of clauses as a Horn set, J. ACM 25(1978)134–135.Google Scholar
- [47]M.C. Loui and D.R. Luginbuhl, The complexity of on-line simulations between multidimensional Turing machines and random access machines, Math. Syst. Th. 25(1992)293–308.Google Scholar
- [48]K. Melhorn,
*Data Structures and Algorithms*, Vols. 1–3 (Springer, 1984).Google Scholar - [49]M. Minoux, LTUR: A simplified linear-time resolution algorithm for Horn formulae and computer implementation, Inform. Proc. Lett. 29(1988)1–12.Google Scholar
- [50]B. Monien, About the derivation languages of grammars and machines,
*4th ICALP*, Lecture Notes Comput. Sci 52 (Springer, 1977) pp. 337–351.Google Scholar - [51]W. Paul, N. Pippenger, E. Szemeredi and W.T. Trotter, On determism versus non-determinism and related problems,
*24th Symp. Found. of Comput. Sci.*(1983) pp. 429–438.Google Scholar - [52]S. Ranaivoson, Ph.D. Thesis, Université de Caen, France (1991).Google Scholar
- [53]S. Ranaivoson, Nontrivial lower bounds for some NP-complete problems on directed graphs,
*CSL'90*, Lecture Notes Comput. Sci. 533 (1991) pp. 318–339.Google Scholar - [54]J.M. Robson, Random access machines and multi-dimensional memories, Inform. Proc. Lett. 34(1990)265–266.Google Scholar
- [55]J.M. Robson, An
*O*(*T*log*T*) reduction from RAM computations to satisfiability, Theor. Comput. Sci. 82(1991)141–149.Google Scholar - [56]J.M. Robson, Subexponential algorithms for some NP-complete problems, unpublished manuscript (1985).Google Scholar
- [57]T.J. Schaeffer, The complexity of satisfiability problems,
*Proc. 10th ACM Symp. on Theory of Computing*(1978) pp. 216–226.Google Scholar - [58]C.P. Schnorr, Satisfiability is quasilinear complete in NQL, J. ACM 25(1978)136–145.Google Scholar
- [59]A. Schoenhage, Storage modifications machines, SIAM J. Comput. 9(1980)490–508.Google Scholar
- [60]A. Schoenhage, A nonlinear lower bound for random access machines under logarithmic cost, J. ACM 35(1988)748–754.Google Scholar
- [61]R. Sedgewick,
*Algorithms in C++*(Addison-Wesley, Reading, 1992).Google Scholar - [62]S.O. Slissenko, Complexity problems of theory of computation, Russ. Math. Surveys 36(1981)23–125.Google Scholar
- [63]C. Slot and P. van Emde Boas, The problem of space invariance for sequential machines, Inform. Comput. 77(1988)93–122.Google Scholar
- [64]R.E. Stearns and H.B. Hunt III, Power indices and easier hard problems, Math. Syst. Th. 23(1990)209–225.Google Scholar
- [65]R. Szelepcsenyi, The method of forced enumeration for nondeterministic automata, Acta Informatica 26(1988)279–284.Google Scholar
- [66]R.E. Tarjan, Depth-first search and linear time algorithms, SIAM J. Comput. 1(1972)146–160.Google Scholar
- [67]P. van Emde Boas, Machine models and simulations,
*Handbook of Theoretical Computer Science*, Vol. A, ed. J. van Leuwen (Elsevier/MIT Press, 1990) Chap. 1, pp. (1–66).Google Scholar - [68]J. van Leuwen, Graph algorithms,
*Handbook of Theoretical Computer Science*, Vol. A, ed. J. van Leuwen (Elsevier/MIT Press, 1990) Chap. 10, pp. 525–631.Google Scholar - [69]K. Wagner and G. Wechsung,
*Computational Complexity*(Reidel, Berlin, 1986).Google Scholar - [70]J. Wiedermann, Deterministic and nondeterministic simulation of the RAM by the Turing machine,
*Proc. IFIP Congress 83*, ed. R.E.A. Mason (North-Holland, Amsterdam, 1983) pp. 163–168.Google Scholar - [71]J. Wiedermann, Normalizing and accelerating RAM computations and the problem of reasonable space measures, Technical Report OPS-3, Dept. of Programming Systems, Bratislava, Czechoslovakia (1990), and
*Proc. 17th ICALP*, Lecture Notes Comput Sci. 443 (Springer, 1990).Google Scholar