MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION
MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION
72 Robert GoldblattThe meaning of µp.α and νp.α for particular α can be hard to fathom, but ithelps to think of them as solutions of the equation “p = α” and repeatedly replacep by α in α itself. It turns out that µp.(α ∨ 〈π〉p) has the same interpretation ina model as the PDL-formula 〈π ∗ 〉α, while νp.(α ∧ [π]p) has the same meaning as[π ∗ π π]α. Also µp.〈π〉p is true at x 0 iff there is an infinite sequence x 0 −→ x1 −→ · · ·in M, which is the condition for truth of the formula repeat(π). Using theseobservations it can be shown that the logic PDL with the repeat construct has asimple translation into the µ-calculus.A CTL-model can be viewed as an Lµ-model with a single transition relationππ π−→, and with a path being a sequence x 0 −→ x1 −→ · · · in the model. CTLtranslates into L µ by translating ∃(α Uβ) as µp.β ∨ (α ∧ 〈π〉p) and ∀(α Uβ) asµp.β ∨ (α ∧ [π]p ∧ 〈π〉⊤). The L µ -formula νp.α ∧ [π][π]p means “along all paths, αis true at every even state”, a property expressible in ECTL* but not CTL*. MadsDam [1994] has constructed algorithms for translating both CTL* and ECTL* intoL µ .Kozen proposed a finite axiomatisation of L µ which, for the binder µ, has theaxiom schemaα(µp.α/p) → µp.αand the inference rule:from α(β/p) → β infer (µp.α) → β if p is not free in β.Validity of the axiom follows from the fact that T = µΘαM is a solution of the“inequality” Θ(T ) ⊆ T , and soundness of the rule is due to µΘαM being the leastsuch solution. Kozen was able to prove the completeness of a limited fragment ofL µ for which he also showed the finite model property and an exponential timedecision procedure. The full L µ was proved decidable by Kozen and Parikh [1984]by reduction to Rabin’s SnS. Streett and Emerson [1984; 1989] used tree automatato improve this to a deterministic triple-exponential time decision algorithm andestablish the full finite model property. Emerson and Jutla [1988; 1999] sharpenedthe complexity result further to a deterministic exponential time algorithm, whichis the best possible result since it is the lower bound for PDL and therefore forthe µ-calculus. Kozen [1988] gave a different proof of the finite model propertyusing techniques from the theory of well-quasi orders, and proved a completenesstheorem for L µ using an infinitary rule of inference.The problem of whether L µ is complete for Kozen’s originally proposed axiomatisationproved challenging, and remained open for some time. It was eventuallysolved in the affirmative by Igor Walukiewicz [1995; 2000].The formalism of the µ-calculus originates in some unpublished notes of Jacode Bakker and Dana Scott from 1969. Kozen’s inference rule derives from theFixpoint Induction rule of [Park, 1969]. Another early independent formulationof a modal program logic with a greatest and least fixpoint operators appears in[Emerson and Clarke, 1980]. For a recent survey of the field of modal µ-calculi,see [Bradfield and Stirling, 2001].
Mathematical Modal Logic: A View of its Evolution 737.5 Solovay on Provability in Arithmetic as a ModalityLet PA be the first-order system of Peano Arithmetic that is the subject of Gödel’sincompleteness theorems, and let PA ⊢ σ signify that sentence σ is provable inPA. Gödel showed that this notion can be “arithmetised” and expressed in thelanguage of PA itself. There is a PA-formula Bew(v) with one free variable v suchthat in general PA ⊢ σ iff the sentence Bew(σ) is true (i.e. true of the standardPA-model (ω, +, ·, 0, 1) ). Here σ is the numeral for the Gödel number of σ. Nowall PA-provable sentences are true, so for every σ the sentenceBew(σ) → σis true. But it is not always PA-provable, a fact which is a manifestation ofthe first incompleteness theorem. Gödel gave an example of this in his [1933],observing that if the modality “provable” is taken to mean provable in PA thensome principles of S4 do not hold:For example, B(Bp → p) never holds for that notion, that is it holds for nosystem S that contains arithmetic. For otherwise, for example, B(0 ≠ 0) →0 ≠ 0 and therefore also ¬B(0 ≠ 0) would be provable in S, that is, theconsistency of S would be provable in S.Provability in S of the consistency of S would contradict the second incompletenesstheorem.The question therefore arises as to which modal principles do hold if ✷ is readas “PA-provable”. To make this precise, define a realisation to be a function φassigning to each propositional variable p some PA-sentence p φ . This extendsinductively to all modal formulas by taking ⊤ φ to be (0 = 0), realising the nonmodalconnectives as themselves, and defining(✷α) φ := Bew(α φ ).A modal formula α is PA-valid if PA ⊢ α φ for every realisation φ. The questionbecomes that of determining which modal formulas are PA-valid.The set of all PA-valid formulas is a normal logic, known as G (for Gödel). 62 Toshow that it is normal it is necessary to verify that the following hold in general:PA ⊢ Bew(σ → σ ′ ) → (Bew(σ) → Bew(σ ′ );If PA ⊢ σ, then PA ⊢ Bew(σ).These results were distilled by Martin Löb [1955] from properties of Bew that wereestablished in [Hilbert and Bernays, 1939]. Löb then provedPA ⊢ Bew(σ) → Bew(Bew(σ)),62 Also known as GL for Gödel–Löb.
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Mathematical Modal Logic: A View of its Evolution 737.5 Solovay on Provability in Arithmetic as a ModalityLet PA be the first-order system of Peano Arithmetic that is the subject of Gödel’sincompleteness theorems, and let PA ⊢ σ signify that sentence σ is provable inPA. Gödel showed that this notion can be “arithmetised” and expressed in thelanguage of PA itself. There is a PA-formula Bew(v) with one free variable v suchthat in general PA ⊢ σ iff the sentence Bew(σ) is true (i.e. true of the standardPA-model (ω, +, ·, 0, 1) ). Here σ is the numeral for the Gödel number of σ. Nowall PA-provable sentences are true, so for every σ the sentenceBew(σ) → σis true. But it is not always PA-provable, a fact which is a manifestation ofthe first incompleteness theorem. Gödel gave an example of this in his [1933],observing that if the modality “provable” is taken to mean provable in PA thensome principles of S4 do not hold:For example, B(Bp → p) never holds for that notion, that is it holds for nosystem S that contains arithmetic. For otherwise, for example, B(0 ≠ 0) →0 ≠ 0 and therefore also ¬B(0 ≠ 0) would be provable in S, that is, theconsistency of S would be provable in S.Provability in S of the consistency of S would contradict the second incompletenesstheorem.The question therefore arises as to which modal principles do hold if ✷ is readas “PA-provable”. To make this precise, define a realisation to be a function φassigning to each propositional variable p some PA-sentence p φ . This extendsinductively to all modal formulas by taking ⊤ φ to be (0 = 0), realising the nonmodalconnectives as themselves, and defining(✷α) φ := Bew(α φ ).A modal formula α is PA-valid if PA ⊢ α φ for every realisation φ. The questionbecomes that of determining which modal formulas are PA-valid.The set of all PA-valid formulas is a normal logic, known as G (for Gödel). 62 Toshow that it is normal it is necessary to verify that the following hold in general:PA ⊢ Bew(σ → σ ′ ) → (Bew(σ) → Bew(σ ′ );If PA ⊢ σ, then PA ⊢ Bew(σ).These results were distilled by Martin Löb [1955] from properties of Bew that wereestablished in [Hilbert and Bernays, 1939]. Löb then provedPA ⊢ Bew(σ) → Bew(Bew(σ)),62 Also known as GL for Gödel–Löb.
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