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Kripke Logical Relations and PCF
 Information and Computation
, 1995
"... Sieber has described a model of PCF consisting of continuous functions that are invariant under certain (finitary) logical relations, and shown that it is fully abstract for closed terms of up to thirdorder types. We show that one may achieve full abstraction at all types using a form of "Kripke lo ..."
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Cited by 31 (3 self)
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Sieber has described a model of PCF consisting of continuous functions that are invariant under certain (finitary) logical relations, and shown that it is fully abstract for closed terms of up to thirdorder types. We show that one may achieve full abstraction at all types using a form of "Kripke logical relations" introduced by Jung and Tiuryn to characterize definability. To appear in Information and Computation. (Accepted, October 1994) Supported by NSF grant CCR92110829. 1 Introduction The nature of sequential functional computation has fascinated computer scientists ever since Scott remarked on a curious incompleteness phenomenon when he introduced LCF (Logic for Computable Functions) and its continuous function model in 1969 (Scott, 1993). Scott noted that although the functionals definable by terms in PCFthe term language of LCFadmitted a sequential evaluation strategy, there were functions in the model that seemed to require a parallel evaluation strategy. "Sequen...
An Improved Lower Bound for the Elementary Theories of Trees
, 1996
"... . The firstorder theories of finite and rational, constructor and feature trees possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson [6], based on repr ..."
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Cited by 28 (3 self)
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. The firstorder theories of finite and rational, constructor and feature trees possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson [6], based on representing large binary relations by means of short formulas manipulating with high trees, we prove that all the above theories, as well as all their subtheories, are nonelementary in the sense of Kalmar, i.e., cannot be decided within time bounded by a k story exponential function 1 exp k (n) for any fixed k. Moreover, for some constant d ? 0 these decision problems require nondeterministic time exceeding exp 1 (bdnc) infinitely often. 1 Introduction Trees are fundamental in Computer Science. Different tree structures are used as underlying domains in automated theorem proving, term rewriting, functional and logic programming, constraint solving, symbolic computation, knowledge re...
Using Autoreducibility to Separate Complexity Classes
 In Proceedings of the 36th IEEE Symposium on Foundations of Computer Science
, 1995
"... A language is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate exponential space from doubly exponential space by showing that all Turingcomplete sets for exponential space are autoreducible but there ..."
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Cited by 24 (11 self)
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A language is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate exponential space from doubly exponential space by showing that all Turingcomplete sets for exponential space are autoreducible but there exists some Turingcomplete set for doubly exponential space that is not. We immediately also get a separation of logarithmic space from polynomial space. Although we already know how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's Program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular if we could settle the question as to whether all complete sets for doubly exponential time were autoreducible we would separate polynomial time from either logarithmic space or polynomial space. We also show several other theorems about autore...
A lower cone in the wtt degrees of nonintegral effective dimension
 In Proceedings of IMS workshop on Computational Prospects of Infinity
, 2006
"... ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the e ..."
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Cited by 24 (2 self)
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ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the existence of a wttlower cone of effective dimension r. 1.
Decidability and undecidability results for NelsonOppen and rewritebased decision procedures
 In Proc. IJCAR3, U. Furbach and
, 2006
"... Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but s ..."
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Cited by 22 (15 self)
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Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T1 ∪ T2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable. In the second part of the paper we strengthen the NelsonOppen decidability transfer result, by showing that it applies to theories over disjoint signatures, whose satisfiability problem, in either arbitrary or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, complementing recent work on combination of theories in the rewritebased approach to satisfiability. 1
An Excursion to the Kolmogorov Random Strings
 In Proceedings of the 10th IEEE Structure in Complexity Theory Conference
, 1995
"... We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1) . We show that the class of sets that Turing reduce to R t has measure 0 in EXP with respect to the resourcebounded measure ..."
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Cited by 18 (8 self)
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We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1) . We show that the class of sets that Turing reduce to R t has measure 0 in EXP with respect to the resourcebounded measure introduced by [17]. From this we conclude that R t is not Turingcomplete for EXP . This contrasts the resource unbounded setting. There R is Turingcomplete for coRE . We show that the class of sets to which R t bounded truthtable reduces, has p 2 measure 0 (therefore, measure 0 in EXP ). This answers an open question of Lutz, giving a natural example of a language that is not weaklycomplete for EXP and that reduces to a measure 0 class in EXP . It follows that the sets that are p btt hard for EXP have p 2 measure 0. 1 Introduction One of the main questions in complexity theory is the relation between complexity classes, such as for example P ; NP , and EXP . It is well known that ...
Constructive dimension and weak truthtable degrees
 In Computation and Logic in the Real World  Third Conference of Computability in Europe. SpringerVerlag Lecture Notes in Computer Science #4497
, 2007
"... Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with ..."
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Cited by 13 (3 self)
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Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with dimH(R) ≥ dimH(S)/dimP(S) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1−ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of wtt degrees (and, by extension, Turing degrees). A lower bound of dimH(S)/dimP(S) is shown to hold for the wtt degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of wtt degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the wtt degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor.
Observations on Grammar and Language Families
, 1994
"... In this report, we emphasize the differences of grammar families and their properties versus language families and their properties. To this end, we investigate grammar families from an abstract standpoint, developping a new framework of reasoning. In particular when considering decidability questio ..."
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Cited by 12 (11 self)
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In this report, we emphasize the differences of grammar families and their properties versus language families and their properties. To this end, we investigate grammar families from an abstract standpoint, developping a new framework of reasoning. In particular when considering decidability questions, special care must be taken when trying to use decidability results (which are, in the first place, properties of grammar families) in order to establish results (e.g. hierarchy results) on language families. We illustrate this by inspecting some theorems and their proofs in the field of regulated rewriting. In this way, we also correct the formulation of an important theorem of Hinz and Dassow. As an exercise, we show that there is no `effective' grammatical characterization of the family of recursive languages. Moreover, we show how to prove the strictness of the Chomsky hierarchy using decidability properties only.
Separating Complexity Classes using Autoreducibility
, 1998
"... A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomialtime hierarchy from exponential space by showing that all Turingcomplete sets for certain levels of the exponentialtime hie ..."
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Cited by 8 (1 self)
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A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomialtime hierarchy from exponential space by showing that all Turingcomplete sets for certain levels of the exponentialtime hierarchy are autoreducible but there exists some Turingcomplete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have dierent structural properties, thus applying Post's Program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turingcomplete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial...