On propositional quantifiers in provability logic.
TL;DR: It is proved that the first order theory of the O-generated subalgebra of DA(PA), the Diagonalizable Algebra of Peano Arith- metic, is decidable but not elementary recursive; the same theory, enriched by a single free variable ranging overDA(PA, is already undecidable.
read more
Abstract: The first order theory of the Diagonalizable Algebra of Peano Arith- metic (DA(PA)) represents a natural fragment of provability logic with proposi- tional quantifiers. We prove that the first order theory of the O-generated subalgebra of DA(PA) is decidable but not elementary recursive; the same theory, enriched by a single free variable ranging over DA(PA), is already undecidable. This gives a negative answer to the question of the decidability of provability logics for recur- sive progressions of theories with quantifiers ranging over their ordinal notations. We also show that the first order theory of the free diagonalizable algebra on n independent generators is undecidable iff n Φ 0.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
Problems in the Logic of Provability
Lev D. Beklemishev,Albert Visser +1 more
- 01 Jan 2006
TL;DR: In the first part of the paper as discussed by the authors, we discuss some conceptual problems related to the notion of proof in Provability Logic, and in the second part we survey five major open problems in provability logic and possible directions for future research.
Undecidability in diagonalizable algebras
Abstract: Abstract If a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.
26
Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard
Bartosz Bednarczyk,Stéphane Demri +1 more
- 24 Jun 2019
TL;DR: Tower-hardness of QCTLt restricted to EF or to EXEF and of the well-known modal logics, with propositional quantification under a semantics based on classes of trees, is proved.
Second-order propositional modal logic and monadic alternation hierarchies
Antti Kuusisto,Antti Kuusisto +1 more
TL;DR: It is established that the quantifier alternation hierarchy of formulae of second-order propositional modal logic (SOPML) induces an infinite corresponding semantic hierarchy over the class of finite directed graphs.
11
Undecidability of the elementary theory of the semilattice of GLP-words
TL;DR: In this paper, the authors prove the undecidability of the elementary theory of the subsemilattice of the Lindenbaum algebra of Peano PA with 0-consistency operators only.
9
References
On a Decision Method in Restricted Second Order Arithmetic
J. Richard Büchi
- 01 Jan 1990
TL;DR: The interpreted formalism of SC as mentioned in this paper is a fraction of the restricted second order theory of natural numbers, or of the first-order theory of real numbers, and it is easy to see that SC is equivalent to the first order theory [Re, +, Pw, Nn], whereby Re, + are the sets of non-negative reals, integral powers of 2, and natural numbers.
1.5K
Provability interpretations of modal logic
TL;DR: In this article, the authors consider interpretations of modal logic in Peano arithmetic determined by an assignment of a sentencev * ofP to each propositional variablev. They show that a modal formula, χ, is valid if ψ* is a theorem ofP in each interpretation.
484
Transfinite recursive progressions of axiomatic theories
TL;DR: In this paper, the authors considered a set of non-logical axioms of the classical functional calculus with the assumption that the set A is recursive, or at least recursively enumerable.
317
•Book
The Unprovability of Consistency: An Essay in Modal Logic
George Boolos
- 30 Apr 1979
TL;DR: The Unprovability of Consistency as mentioned in this paper is concerned with connections between two branches of logic: proof theory and modal logic, and it is the study of the principles that govern the concepts of necessity and possibility that govern provability and consistency.
117
Characters and fixed-points in provability logic.
Zachari Gleit,Warren D. Goldfarb +1 more
TL;DR: Some basic theorems about provability logic ― the system of modal logic that reflects the behavior of formalized provability predicates in theories such as arithmetic ― are given simplified, model-theoretic proofs.