Logical Number Theory I

Smoryñski, Craig

doi:10.1007/978-3-642-75462-3

Cited by 109 publications

(42 citation statements)

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“…Addition alone is, in many ways much weaker than addition and multiplication together. For example, this is witnessed by the fact that the first-order theory of the natural numbers with + and × is undecidable, whereas Presburger arithmetic, the first-order theory of the natural numbers with addition only, can be decided using quantifier elimination (cf., e.g., the textbook [30]). It is therefore more than conceivable that addition alone is too weak to make the conjecture fail, and we now show that this is indeed the case.…”

Section: The Cbc With "+" As Numerical Predicatementioning

confidence: 99%

First-order expressibility of languages with neutral letters or: The Crane Beach conjecture

Barrington

Immerman

Lautemann

et al. 2005

Journal of Computer and System Sciences

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A language L over an alphabet A is said to have a neutral letter if there is a letter e ∈ A such that inserting or deleting e's from any word in A * does not change its membership or non-membership in L.The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order, then it is not definable in first-order logic with any set N of numerical predicates. Named after the location of its first, flawed, proof this conjecture is called the Crane Beach conjecture (CBC, for short). The CBC is closely related to uniformity conditions in circuit complexity theory and to collapse results in database theory.We investigate the CBC in detail, showing that it fails for N = {+, ×}, or, possibly stronger, for any set N that allows counting up to the m times iterated logarithm, for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for the addition predicate +, for the fragment BC( 1 ) of first-order logic, for regular languages, and for languages over a binary alphabet. We explain the precise relation between the CBC and so-called natural-generic collapse results in database theory. Furthermore, we introduce a framework that gives better understanding of what exactly may cause a failure of the conjecture.

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Section: The Cbc With "+" As Numerical Predicatementioning

confidence: 99%

First-order expressibility of languages with neutral letters or: The Crane Beach conjecture

Barrington

Immerman

Lautemann

et al. 2005

Journal of Computer and System Sciences

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“…Indeed it can be shown that exponentiation can be defined from addition and multiplication (see e. g. [HP93,p. 301] and [Smo91,p. 192]), and from this it is not so hard to define the BIT predicate, as pointed out by [Lin94] (cf., [Imm98]).…”

Section: Capturing Logcfl Without Bit Theorem 42 There Is a Fixed Gmentioning

confidence: 99%

The Descriptive Complexity Approach to LOGCFL

et al. 1999

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Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC 1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's "hardest context-free language" is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.

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“…A theorem of Visser [14,III.7] states that any class of unary partial recursive functions that (1) contains an upper bound of every partial recursive function and (2) is closed under right composition with all total recursive functions consists of all unary partial recursive functions.…”

Section: Partial Recursive Functions In Monoidal Categoriesmentioning

confidence: 99%

Partial Recursive Functions and Finality

Plotkin

2013

Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky

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Abstract. We seek universal categorical conditions ensuring the representability of all partial recursive functions. In the category Pfn of sets and partial functions, the natural numbers provide both an initial algebra and a final coalgebra for the functor 1 + −. We recount how finality yields closure of the partial functions on natural numbers under Kleene's µ-recursion scheme. Noting that Pfn is not cartesian, we then build on work of Paré and Román, obtaining weak initiality and finality conditions on natural numbers algebras in monoidal categories that ensure the (weak) representability of all partial recursive functions. We further obtain some positive results on strong representability. All these results adapt to Kleisli categories of cartesian categories with natural numbers algebras. However, in general, not all partial recursive functions need be strongly representable.

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Logical Number Theory I

Cited by 109 publications

References 0 publications

First-order expressibility of languages with neutral letters or: The Crane Beach conjecture

First-order expressibility of languages with neutral letters or: The Crane Beach conjecture

The Descriptive Complexity Approach to LOGCFL

Partial Recursive Functions and Finality

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