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An important goal of computational linguistics has been to use linguistic theory to guide the construction of computationally efficient real-world natural language processing systems. At first glance, generalized phrase structure grammar (GPSG) appears to be a blessing on two counts. First, the precise formalisms of GPSG might be a direct and transparent guide for parser design and implementation. Second, since GPSG has weak context-free generative power and context-free languages can be parsed in by a wide range of algorithms, GPSG parsers would appear to run in polynomial time. This widely-assumed GPSG efficient parsability result is misleading: here we prove that the universal recognition problem of current GPSG theory is exponential-polynomial time hard, and assuredly intractable. The paper pinpoints sources of complexity (e.g. metarules and the theory of syntactic features) in the current GPSG theory and concludes with some linguistically and computationally motivated restrictions on GPSG.

An important goal of computational linguistics has been to use linguistic theory to guide the construction of computationally efficient real-world natural language processing systems. At first glance, generalized phrase structure grammar (GPSG) appears to be a blessing on two counts. First, the precise formalisms of GPSG might be a direct and transparent guide for parser design and implementation. Second, since GPSG has weak context-free generative power and context-free languages can be parsed in by a wide range of algorithms, GPSG parsers would appear to run in polynomial time. This widely-assumed GPSG efficient parsability result is misleading: here we prove that the universal recognition problem of current GPSG theory is exponential-polynomial time hard, and assuredly intractable. The paper pinpoints sources of complexity (e.g. metarules and the theory of syntactic features) in the current GPSG theory and concludes with some linguistically and computationally motivated restrictions on GPSG.

This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.

One of the central problems in quantum mechanics is to determine the ground state properties of a system of electrons interacting via the Coulomb potential. Since its introduction by Hohenberg, Kohn, and Sham, Density Functional Theory (DFT) has become the most widely used and successful method for simulating systems of interacting electrons, making their original work one of the most cited in physics. In this letter, we show that the field of computational complexity imposes fundamental limitations on DFT, as an efficient description of the associated universal functional would allow to solve any problem in the class QMA (the quantum version of NP) and thus particularly any problem in NP in polynomial time. This follows from the fact that finding the ground state energy of the Hubbard model in an external magnetic field is a hard problem even for a quantum computer, while given the universal functional it can be computed efficiently using DFT. This provides a clear illustration how the field of quantum computing is useful even if quantum computers would never be built.

Blum's machine-independent treatment of the complexity of partial recursive functions is extended to relative algorithms (as represented by Turing machines with oracles). The author proves relativeizations of several results of Blum complexity theory. A recursive relatedness theorem is proved, showing that any two relative complexity measures are related by a fixed recursive function. This theorem allows one to obtain proofs of results for all measures from proofs for a particular measure. The author studies complexity-determined reducibilities, the parallel notion to complexity classes for the relativized case. Truth-table and primitive recursive reducibilities are reducibilities of this type. The concept of a set helping the computation of a function is formalized. Basic properties of the helping relation are proved, including non- transitivity and bounds on the amount of help certain sets can provide.

Blum's machine-independent treatment of the complexity of partial recursive functions is extended to relative algorithms (as represented by Turing machines with oracles). The author proves relativeizations of several results of Blum complexity theory. A recursive relatedness theorem is proved, showing that any two relative complexity measures are related by a fixed recursive function. This theorem allows one to obtain proofs of results for all measures from proofs for a particular measure. The author studies complexity-determined reducibilities, the parallel notion to complexity classes for the relativized case. Truth-table and primitive recursive reducibilities are reducibilities of this type. The concept of a set helping the computation of a function is formalized. Basic properties of the helping relation are proved, including non- transitivity and bounds on the amount of help certain sets can provide.

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs.