4,063 research outputs found
Tuples of disjoint NP-sets
Disjoint NP-pairs are a well studied complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k ≥ 2. We define subclasses of the class of all disjoint k-tuples of NP-sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint NP-pairs exist if and only if complete disjoint k-tuples of NP-sets exist for all k ≥ 2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems
Tuples of disjoint NP-sets
Disjoint NPUnknown control sequence '\mathsf' -pairs are a well studied complexity-theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NPUnknown control sequence '\mathsf' -pairs to disjoint k-tuples of NPUnknown control sequence '\mathsf' -sets for k≥2. We define subclasses of the class of all disjoint k-tuples of NPUnknown control sequence '\mathsf' -sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint NPUnknown control sequence '\mathsf' -pairs exist if and only if complete disjoint k-tuples of NPUnknown control sequence '\mathsf' -sets exist for all k≥2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
On the existence of complete disjoint NP-pairs
Disjoint NP-pairs are an interesting model of computation with important applications in cryptography and proof complexity. The question whether there exists a complete disjoint NP-pair was posed by Razborov in 1994 and is one of the most important problems in the field. In this paper we prove that there exists a many-one hard disjoint NP-pair which is computed with access to a very weak oracle (a tally NP-oracle). In addition, we exhibit candidates for complete NP-pairs and apply our results to a recent line of research on the construction of hard tautologies from pseudorandom generators
Consequences of a Goedel's misjudgment
The fundamental aim of the paper is to correct an harmful way to interpret a
Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the
Goedel's fault is rather venial, its misreading has produced and continues to
produce dangerous fruits, as to apply the incompleteness Theorems to the full
second-order Arithmetic and to deduce the semantic incompleteness of its
language by these same Theorems. The first three paragraphs are introductory
and serve to define the languages inherently semantic and its properties, to
discuss the consequences of the expression order used in a language and some
question about the semantic completeness: in particular is highlighted the fact
that a non-formal theory may be semantically complete despite using a language
semantically incomplete. Finally, an alternative interpretation of the Goedel's
unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic
incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
Hilbert's "Verunglueckter Beweis," the first epsilon theorem, and consistency proofs
In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's Programme,
were working on consistency proofs for arithmetical systems. One proposed
method of giving such proofs is Hilbert's epsilon-substitution method. There
was, however, a second approach which was not reflected in the publications of
the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's
first epsilon theorem and a certain 'general consistency result' due to
Bernays. An analysis of the form of this so-called 'failed proof' sheds further
light on an interpretation of Hilbert's Programme as an instrumentalist
enterprise with the aim of showing that whenever a `real' proposition can be
proved by 'ideal' means, it can also be proved by 'real', finitary means.Comment: 18 pages, final versio
Nondeterministic functions and the existence of optimal proof systems
We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system
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