Sample PTL Axioms What is TL? Temporal Logic ⊢ Axioms ⊢ (start ⇒ Normal Form [Esoterica] (A ⇒ B ) ⇒ ( A⇒ B) Characterising TL A) ⇒ A ¨ Buchi Automata ⊢ Fixpoints Quantification ⊢ ¬A ⇔ ¬ ♦ A ❤(A ∧ B ) ⇔ ❤A ∧ ❤B ⊢ AU(B ∧ AUC ) ⇒ AUC Michael Fisher Department of Computer Science, University of Liverpool, UK Key axiom describing how this form of temporal logic works is the induction axiom: [MFisher@liverpool.ac.uk] ⊢ (ϕ ⇒ ❤ϕ) ⇒ (ϕ ⇒ ϕ) An Introduction to Practical Formal Methods Using Temporal Logic c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 1 / 20 c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic What is Temporal Logic? What is TL? Axioms Characterising TL Normal Form Now that we have seen both syntax and semantics for PTL, together with a variety of examples expressed in its language, we can reconsider the more philosophical question Fixpoints “what is temporal logic?” What is TL? Axioms Characterising TL Normal Form Fixpoints Quantification Quantification While we have given a formal description of the PTL logic, there are a number of alternative formalisations providing different, and interesting, ways to view PTL. An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 2 / 20 – 3 / 20 “PTL corresponds to a specific, decidable (PSPACE-complete) fragment of classical first-order logic (with arithmetic operations).” As above, i |= start is represented by i =0 i |= ❤p is represented by p(i + 1) i |= ♦p is represented by ∃j . ( j ≥ i ) ∧ p ( j ) is represented by ∀j . ( j ≥ i ) ⇒ p ( j ) i |= In the following we will briefly examine a few of these, as this is useful in shedding light on the precise nature of (propositional) temporal logic. c Michael Fisher LOGIC : ESOTERICA ] 1: PTL as First-Order Logic ¨ Buchi Automata ¨ Buchi Automata [ TEMPORAL c Michael Fisher p An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 4 / 20 Example What is TL? Axioms What is TL? Using the above translation, Characterising TL Characterising TL ♦q Normal Form ¨ Buchi Automata Fixpoints ¨ Buchi Automata now becomes Recall that the key axiom describing how PTL works is the (ϕ ⇒ ❤ϕ) ⇒ (ϕ ⇒ ϕ) induction axiom: ⊢ Fixpoints Quantification Quantification ∀i . “PTL captures a simple form of arithmetical induction.” Axioms q ⇒ Normal Form 2: PTL Characterises Induction (∀j . (j ≥ i ) ⇒ q (j )) ⇒ (∃k . (k ≥ i ) ∧ q (k )) As we saw above, this can be described as [ϕ(0) ∧ ∀i . ϕ(i ) ⇒ ϕ(i + 1)] ⇒ ∀j . ϕ(j ) This should now be familiar as the arithmetical induction principle, i.e. if we can Through some (first-order) logical manipulation this can be shown to be true as long as quantification occurs over a non-empty domain. 1. show ϕ(0), and 2. show that, for any i , if we already know ϕ(i ) then we can establish ϕ(i + 1) Since the domain here corresponds to the set of moments in time, then the domain should actually be infinite. c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 5 / 20 then ϕ(i ) is true for all elements, i , of the domain. c Michael Fisher Induction Axiom Again What is TL? Axioms Characterising TL ¨ Buchi Automata Fixpoints Quantification What is TL? So, re-using the translations above: Characterising TL Axioms LOGIC : ESOTERICA ] – 7 / 20 “PTL can be seen as a multi-modal logic, comprising two modalities, [1] and [∗], which interact closely.” Normal Form i |= ❤p i |= ♦ p p i |= → → → p(i + 1) ∃j . ( j ≥ i ) ∧ p ( j ) ∀j . ( j ≥ i ) ⇒ p ( j ) ¨ Buchi Automata Fixpoints Quantification In our syntax, ‘[1]’ is usually represented as ‘ ❤’, while ‘[∗]’ is usually represented as ‘ ’. So, the induction axiom in PTL ⊢ The Induction Axiom can be translated as ϕ) ⊢ [∗](ϕ ⇒ [1]ϕ) ⇒ (ϕ ⇒ [∗]ϕ) which easily transforms into in a modal logic with two modalities. [ϕ(0) ∧ ∀i . ϕ(i ) ⇒ ϕ(i + 1)] ⇒ ∀j . ϕ(j ) An Introduction to Practical Formal Methods Using Temporal Logic (ϕ ⇒ ❤ϕ) ⇒ (ϕ ⇒ can now be viewed as the interaction axiom [∀i . ϕ(i ) ⇒ ϕ(i + 1)] ⇒ [ϕ(0) ⇒ ∀j . ϕ(j )] c Michael Fisher [ TEMPORAL 3: PTL as a Multi-Modal Logic One way to view PTL logic is as a fragment of classical logic. Normal Form An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] There are here two distinct accessibility relations, but [∗] represents the reflexive transitive closure of [1]. – 6 / 20 c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 8 / 20 4: PTL Describes Sequences What is TL? Normal Form for PTL What is TL? “PTL can be thought of as a logic over sequences.” Axioms Axioms Characterising TL Characterising TL Normal Form ¨ Buchi Automata As mentioned earlier, the models for PTL are infinite sequences. Normal Form ¨ Buchi Automata Fixpoints Quantification Fixpoints So, a sequence-based semantics can be given for PTL: si , si +1 , . . . |= ❤p si , si +1 , . . . |= si , si +1 , . . . |= ♦p p Quantification It is often useful to replace one complex formula by several simpler formulae. We will use a particular normal form where PTL formulae are represented by if, and only if, si +1 , . . . |= p n ^ if, and only if, there exists a j ≥ i such that sj , . . . |= p where each of the Ri , termed a rule, is an implication in the style if, and only if, for all j ≥ i then sj , . . . |= p An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 9 / 20 formula about current behaviour ⇒ formula about current and future behaviour c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic 5: PTL Describes ω-automata What is TL? Axioms Characterising TL Axioms Characterising TL Fixpoints Quantification ¨ Buchi Automata Fixpoints [ TEMPORAL LOGIC : ESOTERICA ] – 11 / 20 r _ (an initial rule) lb b=1 g ^ ka ⇒ ❤ a=1 formulae such as (p ⇒ ❤q ) give constraints on possible transitions between automaton states; formulae such as ♦ r give constraints on accepting states within an automaton, i.e. states that must be visited infinitely often; and formulae such as s ⇒ t describe global invariants within an automaton. An Introduction to Practical Formal Methods Using Temporal Logic start ⇒ Quantification Correspondingly, temporal formulae might be used to describe certain automata. Intuitively: c Michael Fisher LOGIC : ESOTERICA ] Separated Normal Form (SNF) additionally restricts each Ri to be of one of the following forms Normal Form Models of PTL can be seen as strings accepted by a class of ¨ finite automata — Buchi Automata. [ TEMPORAL Separated Normal Form What is TL? “PTL can be seen as a syntactic characterisation of certain finite-state automata over infinite words.” Normal Form ¨ Buchi Automata Ri i =1 In this way, PTL can be seen as a logic for describing such infinite sequences. c Michael Fisher PTL formulae can become quite complex and difficult to understand. g ^ r _ lb (a step rule) b=1 ka ⇒ ♦l (a sometime rule) a=1 Note, here, that each ka , lb , or l is simply a literal. – 10 / 20 c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 12 / 20 Examples What is TL? Axioms Characterising TL What is TL? The example formulae below correspond to the three different types of SNF rules. Axioms Characterising TL Normal Form ¨ Buchi Automata Fixpoints Quantification Example Automaton (1) Consider the formula ‘ ❤(a ∨ ❤b)’. We can generate a corresponding automaton: Normal Form I NITIAL : S TEP : S OMETIME : start ⇒ losing ∨ hopeful (losing ∧ ¬hopeful ) ⇒ ❤losing hopeful ⇒ ♦ ¬losing ¨ Buchi Automata Fixpoints Quantification i s0 a s1 b Any PTL formula can be transformed into a set of SNF rules that have (essentially) equivalent behaviours at the expense of only a polynomial increase in both the size of the representation and the number of atomic propositions used within the representation. s2 Notation: ‘i ’ represents an initial state and a double circle represents an accepting state (one of which must be visited infinitely often). An edge with no label accepts any value. c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 13 / 20 c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic ¨ Buchi Automata and PTL What is TL? Axioms Characterising TL Normal Form ¨ Buchi Automata Fixpoints LOGIC : ESOTERICA ] – 15 / 20 Example Automaton (2) Models for PTL are essentially infinite, discrete, linear sequences, with an identified start state. What is TL? Thus, each temporal formula corresponds to a set of models on which that formula is satisfied. Normal Form Axioms Characterising TL Quantification [ TEMPORAL a’ has more The automaton translation of the formula ‘ interesting infinite behaviour: ¨ Buchi Automata Fixpoints Quantification s0 Now, look again at these models. See them as strings. a a i Viewing models as strings allows us to utilise the large amount of previous work on finite automata. In particular, we can define a finite automaton that accepts exactly the strings we are interested in and so we can use finite automata to represent temporal models. Thus, now, the “infinite tail” has to always accept an ‘a’. The automata needed are a specific form of automata over ¨ infinite strings, often termed ‘ω-automata’, or Buchi automata. (We will discuss them in detail later.) c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 14 / 20 c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 16 / 20 What are Fixpoints? What is TL? Axioms Characterising TL Normal Form In general, fixpoints are solutions to recursive equations, such as x = f (x ); in basic algebra, we know that a = 0 is a solution of a = (a ∗ 5). Temporal Fixpoint Examples What is TL? Axioms Characterising TL Normal Form ¨ Buchi Automata ¨ Buchi Automata Fixpoints Fixpoints Quantification There can often be several solutions to fixpoint equations; a = 0 and a = 1 are both solutions of a = (a ∗ a). Quantification As we often wish to distinguish solutions, we select using ϕ ♦ϕ ≡ ≡ νξ. (ϕ ∧ ❤ξ) µξ. (ϕ ∨ ❤ξ) ϕ is defined as the maximal fixpoint (ξ) of the Here, formula ξ ⇔ (ϕ ∧ ❤ξ). Thus, the maximal fixpoint above effectively defines ϕ as the ‘infinite’ formula ϕ ∧ ❤ϕ ∧ ❤ ❤ϕ ∧ ❤ ❤ ❤ϕ ∧ . . . ν, representing the maximal (greatest) fixpoint, or µ, representing the minimal (least) fixpoint. Note that the minimal fixpoint of ξ ⇔ (ϕ ∧ ❤ξ) is ‘false’, since putting false in place of ‘ξ’ is legitimate and false is the minimal solution. For example, we write down ν a. (a ∗ a) to denote the maximal fixpoint of a = (a ∗ a); similarly with ‘µ’. N.B: minimality/maximality are defined in relation to ‘⇒’; so ‘false is the minimal element while ‘true’ is the maximal. It is important to note that, in some cases no fixpoint exists. Note: when only one exists, maximal and minimal versions coincide. c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 17 / 20 c Michael Fisher What are Temporal Fixpoints? What is TL? Axioms Characterising TL Axioms Fixpoints Quantification Normal Form A fixpoint solution to the formula ‘ψ ⇔ (ϕ ∧ ❤ψ)’ is some formula ‘ψ’ that makes the statement true. Taking ψ ≡ LOGIC : ESOTERICA ] – 19 / 20 PTL can be extended with quantification. Characterising TL Normal Form ¨ Buchi Automata [ TEMPORAL A Little Quantification What is TL? The µ (least fixpoint) and ν (greatest fixpoint) operators have been transferred to temporal logics. An Introduction to Practical Formal Methods Using Temporal Logic ¨ Buchi Automata Fixpoints Quantification Full first-order quantification is complex, so might allow quantification, but only over Boolean valued variables (specifically, propositions of the language). ϕ does make this formula true, since ϕ ⇔ (ϕ ∧ ❤ Thus, using such a logic, called Quantified Propositional Temporal Logic (QPTL), it is possible to write formulae such as ∃p. p ∧ ❤ ❤p ∧ ♦ ¬p ϕ) is valid, while ψ ≡ ❤ϕ does not, since ❤ϕ 6⇔ (ϕ ∧ ❤ ❤ϕ) . where p is a propositional variable. The fixpoint operators, together with ‘next’, form the basis of a powerful temporal logic called νTL. c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 18 / 20 c Michael Fisher An Introduction to Practical Formal Methods Using Temporal Logic [ TEMPORAL LOGIC : ESOTERICA ] – 20 / 20

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