529 research outputs found
On a question of Abraham Robinson's
In this note we give a negative answer to Abraham Robinson's question whether
a finitely generated extension of an undecidable field is always undecidable.
We construct 'natural' undecidable fields of transcendence degree 1 over Q all
of whose proper finite extensions are decidable. We also construct undecidable
algebraic extensions of Q that allow decidable finite extensions
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
Decidability vs. undecidability. Logico-philosophico-historical remarks
The aim of the paper is to present the decidability problems from a philosophical and historical perspective as well as to indicate basic mathematical and logical results concerning (un)decidability of particular theories and problems
Categoricity, Open-Ended Schemas and Peano Arithmetic
One of the philosophical uses of Dedekind’s categoricity theorem for Peano Arithmetic is to provide support for semantic realism. To this end, the logical framework in which the proof of the theorem is conducted becomes highly significant. I examine different proposals regarding these logical frameworks and focus on the philosophical benefits of adopting open-ended schemas in contrast to second order logic as the logical medium of the proof. I investigate Pederson and Rossberg’s critique of the ontological advantages of open-ended arithmetic when it comes to establishing the categoricity of Peano Arithmetic and show that the critique is highly problematic. I argue that Pederson and Rossberg’s ontological criterion deliver the bizarre result that certain first order subsystems of Peano Arithmetic have a second order ontology. As a consequence, the application of the ontological criterion proposed by Pederson and Rossberg assigns a certain type of ontology to a theory, and a different, richer, ontology to one of its subtheories
Bounded Quantifier Instantiation for Checking Inductive Invariants
We consider the problem of checking whether a proposed invariant
expressed in first-order logic with quantifier alternation is inductive, i.e.
preserved by a piece of code. While the problem is undecidable, modern SMT
solvers can sometimes solve it automatically. However, they employ powerful
quantifier instantiation methods that may diverge, especially when is
not preserved. A notable difficulty arises due to counterexamples of infinite
size.
This paper studies Bounded-Horizon instantiation, a natural method for
guaranteeing the termination of SMT solvers. The method bounds the depth of
terms used in the quantifier instantiation process. We show that this method is
surprisingly powerful for checking quantified invariants in uninterpreted
domains. Furthermore, by producing partial models it can help the user diagnose
the case when is not inductive, especially when the underlying reason
is the existence of infinite counterexamples.
Our main technical result is that Bounded-Horizon is at least as powerful as
instrumentation, which is a manual method to guarantee convergence of the
solver by modifying the program so that it admits a purely universal invariant.
We show that with a bound of 1 we can simulate a natural class of
instrumentations, without the need to modify the code and in a fully automatic
way. We also report on a prototype implementation on top of Z3, which we used
to verify several examples by Bounded-Horizon of bound 1
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