Definable fixed points in modal and temporal logics : A survey

Abstract

The paper presents a survey of author's results on definable fixed points in modal, temporal, and intuitionistic propositional logics. The well-known Fixed Point Theorem considers the modalized case, but here we investigate the positive case. We give a classification of fixed point theorems, describe some classes of models with definable least fixed points of positive operators, special positive operators, and give some examples of undefinable least fixed points. Some other interesting phenomena are discovered -definability by formulas that do not preserve positivity of parameters and definability by finite sets of formulas. We also consider negative operators, graded modalities, construct undefinable inflationary fixed points, and put some problems. . Modal logic in Russia Let us first recall some standard definitions. Modal propositional formulas are constructed from propositional variables p, q, r . . . and the constant ⊥ (falsity) using the binary connectives ∧, ∨, and the unary connectives ¬ and . We introduce the following abbreviations:A modal Kripke frame W, R consists of a non-empty set W with a binary relation R. A Kripke model based on a frame W, R is a triple W, R, v , where v is a valuation function assigning a subset of W (the value) to every propositional variable. This function is extended to formulas in the usual way as follows: the value of the constant ⊥ is always the empty set; the connectives ¬, ∧, ∨ respectively correspond to the complement, the intersection, and the union; and ♦ correspond to the following operations on sets:A = {x | ∀y (xRy ⇒ y ∈ A)}, ♦A = {x | ∃y (xRy ∧ y ∈ A)}.

Fixed-point Elimination in the Intuitionistic Propositional Calculus

“…From now on, our goal shall be to give an upper bound for cl(φ) when φ is a formula such as the one in (either side of) equation (26). Since our proofs actually yield upper bounds for Ruitenburg's numbers ρ(φ) (and a proof of Ruitenburg's theorem for these formulas) we present our results directly as bounds for the numbers ρ(φ).…”

“…We come back now to our original goal, that of estimating upper bounds for formulas φ of the form φ = i∈I φ i as in display (26), where now φ i ∈ Disj(A, B) for each i ∈ I. The next Proposition reduces the problem of giving a closed expression for φ ρ(φ) and estimating an upper bound for the Ruitenburg number of φ as in (26) to that of a conjunction of star formulas, that is, formulas of the form φ = def i φ i , with φ i = def j∈Ji φ wi,j and w i,j ∈ (P (A) × P (B)) * .…”

“…On the proof-theoretic side, this phenomenon is known as the deduction theorem; on the algebraic side it translates to equationally definable principal congruences [5]. In modal logic the deduction theorem is equivalent to having a master modality [24,Theorem 64]; as a matter of fact, having a master modality appears to be a common pattern in several works on alternation-depth hierarchies modal µ-calculi [28,26,1,10,3]. Other ingredients are the following.…”

Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

“…An interesting consequence of this result is that least (and greatest) fixpoints of monotonic formulae are definable in (IPC) (Ghilardi et al 2016(Ghilardi et al , 2019Mardaev 2007Mardaev , 1993: this is because the sequence (1) becomes increasing when evaluated on ⊥/x (if A is monotonic in x), so that the period is decreased to 1. Thus, the index of the sequence becomes a finite upper bound for the fixpoint approximations convergence: in fact, we have IPC A N (⊥/x) → A N+1 (⊥/x) and IPC A N+1 (⊥/x) → A N+2 (⊥/x) by the monotonicity of A, yielding IPC A N (⊥/x) ↔ A N+1 (⊥/x) by (2).…”

Stability of the Blok Theorem

“…For quite a while, the result seemed relatively neglected. One of the few researchers making extensive use of it was the late Sergey Mardaev [26][27][28][29][30]. It also got mentioned in Humberstone's monograph on logical connectives [23].…”