2,183 research outputs found
On the Strength of Uniqueness Quantification in Primitive Positive Formulas
Uniqueness quantification (Exists!) is a quantifier in first-order logic where one requires that exactly one element exists satisfying a given property. In this paper we investigate the strength of uniqueness quantification when it is used in place of existential quantification in conjunctive formulas over a given set of relations Gamma, so-called primitive positive definitions (pp-definitions). We fully classify the Boolean sets of relations where uniqueness quantification has the same strength as existential quantification in pp-definitions and give several results valid for arbitrary finite domains. We also consider applications of Exists!-quantified pp-definitions in computer science, which can be used to study the computational complexity of problems where the number of solutions is important. Using our classification we give a new and simplified proof of the trichotomy theorem for the unique satisfiability problem, and prove a general result for the unique constraint satisfaction problem. Studying these problems in a more rigorous framework also turns out to be advantageous in the context of lower bounds, and we relate the complexity of these problems to the exponential-time hypothesis
Global semantic typing for inductive and coinductive computing
Inductive and coinductive types are commonly construed as ontological
(Church-style) types, denoting canonical data-sets such as natural numbers,
lists, and streams. For various purposes, notably the study of programs in the
context of global semantics, it is preferable to think of types as semantical
properties (Curry-style). Intrinsic theories were introduced in the late 1990s
to provide a purely logical framework for reasoning about programs and their
semantic types. We extend them here to data given by any combination of
inductive and coinductive definitions. This approach is of interest because it
fits tightly with syntactic, semantic, and proof theoretic fundamentals of
formal logic, with potential applications in implicit computational complexity
as well as extraction of programs from proofs. We prove a Canonicity Theorem,
showing that the global definition of program typing, via the usual (Tarskian)
semantics of first-order logic, agrees with their operational semantics in the
intended model. Finally, we show that every intrinsic theory is interpretable
in a conservative extension of first-order arithmetic. This means that
quantification over infinite data objects does not lead, on its own, to
proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories
are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were
used to characterize major computational complexity classes Their extensions
described here have similar potential which has already been applied
Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory
In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are establishe
Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory
In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established
Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic
This paper sets out a predicative response to the Russell-Myhill paradox of
propositions within the framework of Church's intensional logic. A predicative
response places restrictions on the full comprehension schema, which asserts
that every formula determines a higher-order entity. In addition to motivating
the restriction on the comprehension schema from intuitions about the stability
of reference, this paper contains a consistency proof for the predicative
response to the Russell-Myhill paradox. The models used to establish this
consistency also model other axioms of Church's intensional logic that have
been criticized by Parsons and Klement: this, it turns out, is due to resources
which also permit an interpretation of a fragment of Gallin's intensional
logic. Finally, the relation between the predicative response to the
Russell-Myhill paradox of propositions and the Russell paradox of sets is
discussed, and it is shown that the predicative conception of set induced by
this predicative intensional logic allows one to respond to the Wehmeier
problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi
Notes on the Riemann Hypothesis
These notes were written from a series of lectures given in March 2010 at the
Universidad Complutense of Madrid and then in Barcelona for the centennial
anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an
introduction to the Riemann Hypothesis and a panoramic view of the world of
zeta and L-functions. We first review Riemann's foundational article and
discuss the mathematical background of the time and his possible motivations
for making his famous conjecture. We discuss some of the most relevant
developments after Riemann that have contributed to a better understanding of
the conjecture.Comment: 2 sections added, 55 pages, 6 figure
Antinomicity and the axiom of choice. A chapter in antinomic mathematics
The present work is an attempt to break ground in mathematics proper, armed with the accepting view just described. Specifically, we shall examine various versions of antinomic set theory, in particular the axiom of choice, keeping the presentation as intuitive as possible, more in the manner of a nineteenth century paper than as a thoroughly formalized system. The reason for such a presentation is the conviction that at this point it should be the mathematics that eventually determines the logic, rather than the other way around
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
Wellordering proofs for metapredicative Mahlo
In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm0 of admissible set theory, transfinite induction along initial segments of the ordinal φω00, for φ being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the paper Jäger and Strahm [11] are indeed shar
- …