A formal reduction of the general problem of expressibility of formulas in the Gödel-Löb provability logic
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M.F. Ratsa
Abstract
It is a well-known idea to embed the intuitionistic logic into the modal logic in order to interprete the modality 'provable' as the deducibility in the Peano arithmetics with also well-known difficulties. P. M. Solovay and A. V. Kuznetsov introduced into consideration the Gödel-Löb provability logic whose formulas are constructed from propositional variables with the use of the connectives &, v, ⊃, ¬, and Δ (Gödelised provability). This logic is defined by the classic propositional calculus complemented by the three Δ-axioms
Δ (p ⊃ q) ⊃ (Δp ⊃ Δq), Δ(Δp ⊃ Δp) Δp, Δp ⊃ ΔΔp,
and the extension rule (the Gödel rule). A formula F is called (functionally) expressible in terms of a system of formulas ∑ in logic L if, on the base of ∑ and variables, it is possible to obtain F with the use of the weakened substitution rule and the rule of change by equivalent in L. The general problem of expressibility in a logic L requires to give an algorithm which for any formula F and any finite system of formulas ∑ recognises whether F is expressible in terms of ∑ in L.
In this paper, it is proved that in the Gödel-Löb provability logic and in many of extensions of this logic there is no algorithm which recognises the expressibility. In other words, the general expressibility problem is algorithmically undecidable in these logics.
© 2016 by Walter de Gruyter Berlin/Boston
Artikel in diesem Heft
- Editorial
- On inherited properties of restricted Boolean functions
- Some properties of the inertia groups of Boolean bijunctive functions and an inductive method of generating such functions
- On complexity of representation of multiple-valued logic functions as polarised polynomials
- On the periodicity of the sequence of states of an automaton corresponding to a given initial state and a given periodic input sequence
- On the analysis of fuzzy information graphs
- Lattices of Ω-foliated formations
- A formal reduction of the general problem of expressibility of formulas in the Gödel-Löb provability logic
- On some conditions for absence of a giant component in the generalised allocation scheme
- On 2-regular graphs whose girth is one less than the maximum
- Forthcoming Papers
- Contents
Artikel in diesem Heft
- Editorial
- On inherited properties of restricted Boolean functions
- Some properties of the inertia groups of Boolean bijunctive functions and an inductive method of generating such functions
- On complexity of representation of multiple-valued logic functions as polarised polynomials
- On the periodicity of the sequence of states of an automaton corresponding to a given initial state and a given periodic input sequence
- On the analysis of fuzzy information graphs
- Lattices of Ω-foliated formations
- A formal reduction of the general problem of expressibility of formulas in the Gödel-Löb provability logic
- On some conditions for absence of a giant component in the generalised allocation scheme
- On 2-regular graphs whose girth is one less than the maximum
- Forthcoming Papers
- Contents
