566 research outputs found
Undecidability in some field theories
This thesis is a study of undecidability in some field theories. Specifically, we are interested in geometrically oriented problems and have focused our attention in two directions along these lines. The first direction bases on determining the decidability of certain sets of first-order sentences over positive characteristic function fields. We will draw parallel to the problem of algorithmically determining in some cases the existence of points on varieties in positive characteristic function fields; equivalently the existence of certain maps between varieties over other positive characteristic fields.
The second direction bases on determining the decidability of first-order consequences of nonempty finite collections of L_r-sentences, true in fields with plenty of geometric structure. This is connected to the former direction by the fact that a decidable field has a recursive axiomatisation â what if we study a (nonempty) finite subset of the axiomatisation? Undecidability results.
Motivated by classification-theoretic conjectures, we will examine âwilderâ classes of fields in turn and generalise a result of Ziegler to NIP henselian nontrivially valued fields (and beyond). We move to PAC & PRC fields and prove they are finitely undecidable, resolving two open questions of Shlapentokh & Videla, and describe the difficulties that arise in adapting the proof to PpC fields. We pose the question: is every infinite field finitely undecidable
The First-Order Theory of Sets with Cardinality Constraints is Decidable
We show that the decidability of the first-order theory of the language that
combines Boolean algebras of sets of uninterpreted elements with Presburger
arithmetic operations. We thereby disprove a recent conjecture that this theory
is undecidable. Our language allows relating the cardinalities of sets to the
values of integer variables, and can distinguish finite and infinite sets. We
use quantifier elimination to show the decidability and obtain an elementary
upper bound on the complexity.
Precise program analyses can use our decidability result to verify
representation invariants of data structures that use an integer field to
represent the number of stored elements.Comment: 18 page
Undecidability in number theory
These lecture notes cover classical undecidability results in number theory,
Hilbert's 10th problem and recent developments around it, also for rings other
than the integers. It also contains a sketch of the authors result that the
integers are universally definable in the rationals.Comment: 48 pages. arXiv admin note: text overlap with arXiv:1011.342
Invariant Synthesis for Incomplete Verification Engines
We propose a framework for synthesizing inductive invariants for incomplete
verification engines, which soundly reduce logical problems in undecidable
theories to decidable theories. Our framework is based on the counter-example
guided inductive synthesis principle (CEGIS) and allows verification engines to
communicate non-provability information to guide invariant synthesis. We show
precisely how the verification engine can compute such non-provability
information and how to build effective learning algorithms when invariants are
expressed as Boolean combinations of a fixed set of predicates. Moreover, we
evaluate our framework in two verification settings, one in which verification
engines need to handle quantified formulas and one in which verification
engines have to reason about heap properties expressed in an expressive but
undecidable separation logic. Our experiments show that our invariant synthesis
framework based on non-provability information can both effectively synthesize
inductive invariants and adequately strengthen contracts across a large suite
of programs
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
Combining decision procedures for the reals
We address the general problem of determining the validity of boolean
combinations of equalities and inequalities between real-valued expressions. In
particular, we consider methods of establishing such assertions using only
restricted forms of distributivity. At the same time, we explore ways in which
"local" decision or heuristic procedures for fragments of the theory of the
reals can be amalgamated into global ones. Let Tadd[Q] be the
first-order theory of the real numbers in the language of ordered groups, with
negation, a constant 1, and function symbols for multiplication by
rational constants. Let Tmult[Q] be the analogous theory for the
multiplicative structure, and let T[Q] be the union of the two. We
show that although T[Q] is undecidable, the universal fragment of
T[Q] is decidable. We also show that terms of T[Q]can
fruitfully be put in a normal form. We prove analogous results for theories in
which Q is replaced, more generally, by suitable subfields F
of the reals. Finally, we consider practical methods of establishing
quantifier-free validities that approximate our (impractical) decidability
results.Comment: Will appear in Logical Methods in Computer Scienc
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