This is the first volume of a systematic two-volume presentation of the various areas of research on structural complexity. The theory of algorithmic complexity, a part of the mathematical theory of computation, can be approached from several points of view, one of which is the structural one. This volume is written for undergraduate students who have taken a first course in Formal Language Theory. It presents the basic concepts of structural complexity, thus providing the background necessary for the understanding of complexity theory. The second corrected edition has been extended by an appendix with recent results on nondeterministic space classes and updated with regard to the bibliographical remarks and the references.

The LBA Problem and its Importance in the Theory of Computing

Structural Complexity I

Turing machine space complexity is related to circuit depth complexity. The relationship complements the known connection between Turing machine time and circuit size, thus enabling us to expose the related nature of some important open problems concerning Turing machine and circuit complexity. We are also able to show some connection between Turing machine complexity and arithmetic complexity.

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On Relating Time and Space to Size and Depth

The chapter consists of two quite different parts. The first part is a survey (including some new results) on finite model theory. One particular point deserves a special attention. In computer science, the standard computation model is the Turing machine whose inputs are strings; other algorithm inputs are supposed to be encoded with strings. However, in combinatorics, database theory, etc., one usually does not distinguish between isomorphic structures (graphs, databases, etc.). For example, a database query should provide information about the database rather than its implementation. In such cases, there is a problem with string presentation of input objects: there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? The question is intimately related to a question by Chandra and Harel in “Structure and complexity of relational queries”, J. Comput. and System Sciences 25 (1982), 99-128. We formalize the question as the question whether there exists a logic that captures polynomial time (without presuming the presence of a linear order) and conjecture the negative answer. The first part is based on lectures given at the 1984 Udine Summer School on Computation Theory and summarized in the technical report “Logic and the Challenge of Computer Science”, CRL-TR-10-85, Sep. 1985, Computing Research Lab, University of Michigan, Ann Arbor, Michigan. In the second part, we introduce a new computation model: evolving algebras (later renamed abstract state machines). This new approach to semantics of computations and in particulur to semantics of programming languages emphasizes dynamic and resource-bounded aspects of computation. It is illustrated on the example of Pascal. The technical report mentioned above contained an earlier version of part 2. The final version was written in 1986.

Logic and the Challenge of Computer Science

@ PTAPE. We show that all NP complete sets known (in the literature) are indeed p-isomorphic and so are the known PTAPE complete sets. Thus showing that, inspite of the radically different origins and attempted simplification of these sets, all the known NP complete sets are identical but for polynomially time bounded permutations.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 also relate the existence of sparse complete sets to the existence of simple combinatorial circuits for the corresponding truncated recognition problem of these languages.

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On isomorphisms and density of NP and other complete sets

A very hard log-space counting class

/pdf/the-lba-problem-and-its-importance-in-the-theory-of-4h2duthba4.pdf

The LBA Problem and its Importance in the Theory of Computing

Citations

Structural Complexity I

On Relating Time and Space to Size and Depth

Logic and the Challenge of Computer Science

On isomorphisms and density of NP and other complete sets

A very hard log-space counting class

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