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44
Finite permutation groups and finite simple groups
 Bull. London Math. Soc
, 1981
"... In the past two decades, there have been farreaching developments in the problem of determining all finite nonabelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been ..."
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Cited by 92 (3 self)
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In the past two decades, there have been farreaching developments in the problem of determining all finite nonabelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of
Gödel's Theorem and Information
, 1982
"... Gödel's theorem may be demonstrated using arguments having an informationtheoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the tr ..."
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Cited by 53 (6 self)
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Gödel's theorem may be demonstrated using arguments having an informationtheoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.
The Equality Problem for Rational Series With Multiplicities in the Tropical Semiring is Undecidable
, 1994
"... this paper that the equality problem for Mrational series over an alphabet with at least two letters is undecidable. ..."
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Cited by 52 (2 self)
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this paper that the equality problem for Mrational series over an alphabet with at least two letters is undecidable.
The Convenience of Tilings
 In Complexity, Logic, and Recursion Theory
, 1997
"... Tiling problems provide for a very simple and transparent mechanism for encoding machine computations. This gives rise to rather simple master reductions showing various versions of the tiling problem complete for various complexity classes. We investigate the potential for using these tiling proble ..."
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Cited by 36 (0 self)
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Tiling problems provide for a very simple and transparent mechanism for encoding machine computations. This gives rise to rather simple master reductions showing various versions of the tiling problem complete for various complexity classes. We investigate the potential for using these tiling problems in subsequent reductions showing hardness of the combinatorial problems that really matter. We ilustrate our approach by means of three examples: a short reduction chain to the Knapsack problem followed by a Hilbert 10 reduction using similar ingredients. Finally we reprove the Deterministic Exponential Time lowerbound for satisfiablility in Propositional Dynamic Logic. The resulting reductions are relatively simple; they do however infringe on the principle of orthogonality of reductions since they abuse extra structure in the instances of the problems reduced from which results from the fact that these instances were generated by a master reduction previously. 1 Introduction This paper...
On the Classical Decision Problem
 Perspectives in Mathematical Logic
, 1993
"... this paper. In particular, their comments inspired and gave arguments for the discussion on the value of the classical decision problem after Church's and Turing's results. References ..."
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Cited by 36 (0 self)
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this paper. In particular, their comments inspired and gave arguments for the discussion on the value of the classical decision problem after Church's and Turing's results. References
Propositional Dynamic Logic of Nonregular Programs
 Journal of Computer and System Sciences
, 1983
"... this paper indicate that this line is extremely close to the original regular PDL ..."
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Cited by 27 (2 self)
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this paper indicate that this line is extremely close to the original regular PDL
Incorporating XSL Processing Into Database Engines
, 2002
"... The two observations that 1) many XML documents are stored in a database or generated from data stored in a database and 2) processing these documents with XSL stylesheet processors is an important, often recurring task justify a closer look at the current situation. Typically, the XML documen ..."
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Cited by 23 (1 self)
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The two observations that 1) many XML documents are stored in a database or generated from data stored in a database and 2) processing these documents with XSL stylesheet processors is an important, often recurring task justify a closer look at the current situation. Typically, the XML document is retrieved or constructed from the database, exported, parsed, and then processed by a special XSL processor. This cumbersome process clearly sets the goal to incorporate XSL stylesheet processing into the database engine.
Three counterexamples refuting Kieu’s plan for “quantum adiabatic hypercomputation”; and some uncomputable quantum mechanical tasks
 J.Applied Mathematics and Computation
, 2006
"... Abstract — Tien D. Kieu, in 10 papers posted to the quantph section of the xxx.lanl.gov preprint archive [some of which were also published in printed journals such as Proc. Royal Soc. A 460 (2004) 1535] had claimed to have a scheme showing how, in principle, physical “quantum adiabatic systems ” c ..."
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Cited by 21 (0 self)
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Abstract — Tien D. Kieu, in 10 papers posted to the quantph section of the xxx.lanl.gov preprint archive [some of which were also published in printed journals such as Proc. Royal Soc. A 460 (2004) 1535] had claimed to have a scheme showing how, in principle, physical “quantum adiabatic systems ” could be used to solve the prototypical computationally undecidable problem, Turing’s“halting problem,”in finite time, with success probability> 2/3 (where this 2/3 is independent of the input halting problem). There were several errors in those papers, most which ultimately could be corrected. More seriously, we here exhibit counterexamples to a crucial step in Kieu’s argument. The counterexamples are small quantum adiabatic systems in which “decoy ” nonground states arise with high probability (> 99.999%). Kieu had wrongly claimed no decoy state could ever acquire occupation probability greater than 50%. These counterexamples destroy Kieu’s entire plan and there seems no way to correct the plan to escape them. Nevertheless, there are some important consequences salvageable from Kieu’s idea: we can prove that the “halflife” of Kieu’s quantum systems is uncomputably large and no fully general form of the quantum adiabatic theorem can exist that yields computable upper bounds on adiabatic convergence times (both unless Church’s thesis is false so that finite time adiabatic quantum system evolution is unsimulable); and we can prove that no algorithm exists to find the ground state energy of Kieu’s class of quantum Hamiltonians and hence their longterm thermal behavior is
Undecidability and incompleteness in classical mechanics
 Internat. J. Theoret. Physics
, 1991
"... We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary realvalued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian ..."
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Cited by 15 (4 self)
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We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary realvalued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian mechanics and dynamical systems theory. Our examples deal with the algorithmic impossibility of deciding whether a given Hamiltonian can be integrated by quadratures and related questions; they lead to a version of G6del's incompleteness theorem within Hamiltonian mechanics. A similar application to the unsolvability of the decision problem for chaotic dynamical systems is also obtained. 1.
Topology of Diophantine sets: remarks on Mazur’s conjectures. In Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent
 of Contemp. Math
, 1999
"... Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S ..."
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Cited by 15 (1 self)
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Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S, P on D whose graphs are diophantine in Q 3i (via the inclusion D 3 ⊂ Q 3i), and such that for two specific elements d0, d1 ∈ D the structure (D, S, P, d0, d1) is a model for integer arithmetic (Z,+, ·,0, 1). Using a construction of Pheidas, we give a counterexample to the analogue of Mazur’s conjecture over a global function field, and prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over a finite field. 1.