Linear Temporal Logic and Büchi Automata

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1 Linear Temporal Logic and Büchi Automata Yih-Kuen Tsay Department of Information Management National Taiwan University FLOLAC 2009 Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

2 Outline Introduction Propositional Temporal Logic (PTL) Quantified Propositional Temporal Logic (QPTL) Basic Properties From Temporal Formulae to Automata On-the-fly Translation Tableau Construction Concluding Remarks References Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

3 Introduction We have seen how automata, in particular Büchi automata, may be used to describe the behaviors of a concurrent system. Büchi automata localize temporal dependency between occurrences of events (represented by propositions) to relations between states and tend to be of lower level. We will study an alternative formalism, namely linear temporal logic. Temporal logic formulae describe temporal dependency without explicit references to time points and are in general more abstract. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

4 Introduction (cont.) ~p q ~q s0 p ~q q s1 The above Büchi automaton says that, whenever p holds at some point in time, q must hold at the same time or will hold at a later time. It may not be easy to see that this indeed is the case. In linear temporal logic, this can easily be expressed as (p q), which reads always p implies eventually q. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

5 PTL: The Future Only We first look at the future fragment of Propositional Temporal Logic (PTL). Future operators include (next), (eventually), (always), U (until), and W (wait-for). With W replaced by R (release), this fragment is often referred to as LTL (linear temporal logic) in the model checking community. Let V be a set of boolean variables. The future PTL formulae are defined inductively as follows: Every variable p V is a PTL formula. If f and g are PTL formulae, then so are f, f g, f g, f, f, f, f U g, and f W g. ( f g is also written as f g and (f g) (g f ) as f g.) Examples: ( C 0 C 1 ), (T 1 C 1 ). Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

6 PTL: The Future Only (cont.) A PTL formula is interpreted over an infinite sequence of states σ = s 0 s 1 s 2, relative to a position in that sequence. A state is a subset of V, containing exactly those variables that evaluate to true in that state. If each possible subset of V is treated as a symbol, then a sequence of states can also be viewed as an infinite word over 2 V. The semantics of PTL in terms of (σ, i) = f (f holds at the i-th position of σ) is given below. We say that a sequence σ satisfies a PTL formula f or σ is a model of f, denoted σ = f, if (σ, 0) = f. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

7 PTL: The Future Only (cont.) For a boolean variable p, (σ, i) = p p s i For boolean operators, (σ, i) = f (σ, i) = f does not hold (σ, i) = f g (σ, i) = f or (σ, i) = g (σ, i) = f g (σ, i) = f and (σ, i) = g Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

8 PTL: The Future Only (cont.) For future temporal operators, (σ, i) = f (σ, i + 1) = f 0 f i f i + 1 (σ, i) = f for some j i, (σ, j) = f 0 f i f j (σ, i) = f for all j i, (σ, j) = f 0 f f i f i + 1 f f Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

9 PTL: The Future Only (cont.) For future temporal operators (cont.), (σ, i) = f U g for some k i, (σ, k) = g and for all j, i j < k, (σ, j) = f 0 f U g f i f k 1 g k (σ, i) = f W g (for some k i, (σ, k) = g and for all j, i j < k, (σ, j) = f ) or (for all j i, (σ, j) = f ) f W g holds at position i if and only if f U g or f holds at position i. When R is preferred over W, (σ, i) = p R q for all j 0, (σ, i) = p for every i < j implies (σ, j) = q. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

10 PTL: Adding the Past We now add the past fragment. Past operators include (before), (previous), (once), (so-far), S (since), and B (back-to). The full PTL formulae are defined inductively as follows: Every variable p V is a PTL formula. If f and g are PTL formulae, then so are f, f g, f g, f, f, f, f U g, f W g, f, f, f, f, f S g, and f B g. ( f g is also written as f g and (f g) (g f ) as f g.) Examples: (p q) says every p is preceded by a q. ( p q) is another way of saying p W q! Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

11 PTL: Adding the Past (cont.) For past temporal operators, (σ, i) = f i = 0 or (σ, i 1) = f (σ, i) = f i > 0 and (σ, i 1) = f f 0 f i 1 f f i The difference between f and f occurs at position 0. (σ, i) = f for some j, 0 j i, (σ, j) = f 0 f j f i (σ, i) = f 0 for all j, 0 j i, (σ, j) = f f f f f i 1 f f i Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

12 PTL: Adding the Past (cont.) For past temporal operators (cont.), (σ, i) = f S g for some k, 0 k i, (σ, k) = g and for all j, k < j i, (σ, j) = f 0 g k f k + 1 f S g f i (σ, i) = f B g (for some k, 0 k i, (σ, k) = g and for all j, k < j i, (σ, j) = f ) or (for all j, 0 j i, (σ, j) = f ) f B g holds at position i if and only if f S g or f holds at position i. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

13 QPTL Quantified Propositional Temporal Logic (QPTL) is PTL extended with quantification over boolean variables (so, every PTL formula is also a QPTL formula): If f is a QPTL formula and x V, then x : f and x : f are QPTL formulae. Let σ = s 0 s 1 and σ = s 0 s 1 be two sequences of states. We say that σ is a x-variant of σ if, for every i 0, s i differs from s i at most in the valuation of x, i.e., the symmetric set difference of s i and s i is either {x} or empty. The semantics of QPTL is defined by extending that of PTL with additional semantic definitions for the quantifiers: (σ, i) = x : f (σ, i) = f for some x-variant σ of σ (σ, i) = x : f (σ, i) = f for all x-variant σ of σ Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

14 Equivalence and Congruence A formula p is valid, denoted = p, if σ = p for every σ. Two formulae p and q are equivalent if = p q, i.e., σ = p if and only if σ = q for every σ. Two formulae p and q are congruent, denoted p = q, if = (p q). Congruence is a stronger relation than equivalence: p p and (p p) are equivalent, as they are both true at position 0 of every model. However, they are not congruent; p p holds at all positions of every model, while (p p) holds only at position 0. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

15 Congruences A minimal set of operators:,,, W,, B Other operators could be encoded: p = p p = p W False p = p B False p = p p = p p U q = (p W q q) p S q = (p B q q) Weak vs. strong operators: p = ( p True) p = ( p False) p U q = (p W q q) p W q = (p U q p) p S q = (p B q q) p B q = (p S q p) Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

16 Congruences (cont.) Duality: p = p p = p p = p p = p p = p p = p p = p (p U q) = ( q) W ( p q) (p S q) = ( q) B ( p q) (p U q) = ( p) R ( q) (p W q) = ( q) U ( p q) (p B q) = ( q) S ( p q) (p R q) = ( p) U ( q) x : p = x : p x : p = x : p A formula is in the negation normal form if negation only occurs in front of an atomic proposition. Every PTL/QPTL formula can be converted into an equivalent formula in the negation normal form. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

17 Congruences (cont.) Expansion formulae: p = p p p = p p p U q = q (p (p U q)) p W q = q (p (p W q)) p R q = (q p) (q (p R q)) p = p p p = p p p S q = q (p (p S q)) p B q = q (p (p B q)) These expansion formulae are essential in translation of a temporal formula into an equivalent Büchi automaton. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

18 Congruences (cont.) Idempotence: p = p p = p p = p p = p p U (p U q) = p U q p S (p S q) = p S q p W (p W q) = p W q p B (p B q) = p B q (p U q) U q = p U q (p S q) S q = p S q (p W q) W q = p W q (p B q) B q = p B q Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

19 Expressiveness Theorem PTL is strictly less expressive than Büchi automata. Proof. 1 Every PTL formula can be translated into an equivalent Büchi automaton. 2 p holds at every even position is recognizable by a Büchi automaton, but cannot be expressed in PTL. Theorem QPTL is expressively equivalent to Büchi automata (and hence ω-regular expressions and S1S). Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

20 Simple On-the-fly Translation This is a tableau-based algorithm for obtaining a Büchi automaton from an LTL (future PTL) formula. The algorithm is geared towards being used in model checking in an on-the-fly fashion: It is possible to detect that a property does not hold by only constructing part of the model and of the automaton. The algorithm can also be used to check the validity of a temporal logic assertion. To apply the translation algorithm, we first convert the formula ϕ into the negation normal form. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

21 Data Structure of an Automaton Node ID: A string that identifies the node. Incoming: The incoming edges represented by the IDs of the nodes with an outgoing edge leading to the current node. New: A set of subformulae that must hold at the current state and have not yet been processed. Old: The subformulae that must hold in the node and have already been processed. Next: The subformulae that must hold in all states that are immediate successors of states satisfying the properties in Old. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

22 The Algorithm The algorithm starts with a single node, which has a single incoming edge labeled init (i.e., from an initial node) and expands the nodes in an DFS manner. This starting node has initially one new obligation in New, namely ϕ, and Old and Next are initially empty. With the current node N, the algorithm checks if there are unprocessed obligations left in New. If not, the current node is fully processed and ready to be added to Nodes. If there already is a node in Nodes with the same obligations in both its Old and Next fields, the incoming edges of N are incorporated into those of the existing node. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

23 The Algorithm (cont.) If no such node exists in Nodes, then the current node N is added to this list, and a new current node is formed for its successor as follows: 1 There is initially one edge from N to the new node. 2 New is set initially to the Next field of N. 3 Old and Next of the new node are initially empty. When processing the current node, a formula η in New is removed from this list. In the case that η is a literal (a proposition or the negation of a proposition), then if η is in Old, the current node is discarded; otherwise, η is added to Old. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

24 The Algorithm (cont.) When η is not a literal, the current node can be split into two or not split, and new formulae can be added to the fields New and Next. The exact actions depend on the form of η: η = p q, then both p and q are added to New. η = p q, then the node is split, adding p to New of one copy, and q to the other. η = p U q ( = q (p (p R q))), then the node is split. For the first copy, p is added to New and p U q to Next. For the other copy, q is added to New. η = p R q ( = (q p) (q (p R q))), similar to U. η = p, then p is added to Next. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

25 Nodes to GBA The list of nodes in Nodes can now be converted into a generalized Büchi automaton B = (Σ, Q, q 0,, F ): 1 Σ consists of sets of propositions from AP. 2 The set of states Q includes the nodes in Nodes and the additional initial state q 0. 3 (r, α, r ) iff r Incoming(r ) and α satisfies the conjunction of the negated and nonnegated propositions in Old(r ) 4 q 0 is the initial state, playing the role of init. 5 F contains a separate set F i of states for each subformula of the form p U q; F i contains all the states r such that either q Old(r) or p U q Old(r). Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

26 Tableau Construction We next study the Tableau Construction as described in [Manna and Pnueli 1995], which handles both future and past temporal operators. More efficient constructions exist, but this construction is relatively easy to understand. A tableau is a graphical representation of all models/sequences that satisfy the given temporal logic formula. The construction results in essentially a GBA, but leaving propositions on the states (rather than moving them to the incoming edges of a state). Our presentation will be slightly different, to make the resulting GBA more apparent. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

27 Expansion Formulae The requirement that a temporal formula holds at a position j of a model can often be decomposed into requirements that a simpler formula holds at the same position and some other formula holds either at j + 1 or j 1. For this decomposition, we have the following expansion formulae: p = p p p = p p p U q = q (p (p U q)) p W q = q (p (p W q)) Note: this construction does not deal with R. p = p p p = p p p S q = q (p (p S q)) p B q = q (p (p B q)) Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

28 Closure We define the closure of a formula ϕ, denoted by Φ ϕ, as the smallest set of formulae satisfying the following requirements: ϕ Φ ϕ. For every p Φ ϕ, if q a subformula of p then q Φ ϕ. For every p Φ ϕ, p Φ ϕ. For every ψ { p, p, p U q, p W q}, if ψ Φ ϕ then ψ Φ ϕ. For every ψ { p, p S q}, if ψ Φ ϕ then ψ Φ ϕ. For every ψ { p, p B q}, if ψ Φ ϕ then ψ Φ ϕ. So, the closure Φ ϕ of a formula ϕ includes all formulae that are relevant to the truth of ϕ. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

29 Classification of Formulae α p q p p K(α) p, q p, p p, p β K 1 (β) K 2 (β) p q p q p p p p p p p U q q p, (p U q) p W q q p, (p W q) p S q q p, (p S q) p B q q p, (p B q) An α-formula ϕ holds at position j iff all the K(ϕ)-formulae hold at j. A β-formula ψ holds at position j iff either K 1 (ψ) or all the K 2 (ψ)-formulae (or both) hold at j. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

30 Atoms We define an atom over ϕ to be a subset A Φ ϕ satisfying the following requirements: R sat : the conjunction of all state formulae in A is satisfiable. R : for every p Φ ϕ, p A iff p A. R α : for every α-formula p Φ ϕ, p A iff K(p) A. R β : for every β-formula p Φ ϕ, p A iff either K 1 (p) A or K 2 (p) A (or both). For example, if atom A contains the formula p, it must also contain the formulae p and p. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

31 Mutually Satisfiable Formulae A set of formulae S Φ ϕ is called mutually satisfiable if there exists a model σ and a position j 0, such that every formula p S holds at position j of σ. The intended meaning of an atom is that it represents a maximal mutually satisfiable set of formulae. Claim (atoms represent necessary conditions) Let S Φ ϕ be a mutually satisfiable set of formulae. Then there exists a ϕ-atom A such that S A. It is important to realize that inclusion in an atom is only a necessary condition for mutual satisfiability (e.g., { p p, p, p, p} is an atom for the formula p p). Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

32 Basic Formulae A formula is called basic if it is either a proposition or has the form p, p, or p. Basic formulae are important because their presence or absence in an atom uniquely determines all other closure formulae in the same atom. Let Φ + ϕ denote the set of formulae in Φ ϕ that are not of the form ψ. Algorithm (atom construction) 1 Find all basic formulae p 1,, p b Φ + ϕ. 2 Construct all 2 b combinations. 3 Complete each combination into a full atom. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

33 Example Consider the formula ϕ 1 : p p whose basic formulae are p, p, p. Following is the list of all atoms of ϕ 1 : A 0 : { p, p, p, p, p, ϕ 1 } A 1 : {p, p, p, p, p, ϕ 1 } A 2 : { p, p, p, p, p, ϕ 1 } A 3 : {p, p, p, p, p, ϕ 1 } A 4 : { p, p, p, p, p, ϕ 1 } A 5 : {p, p, p, p, p, ϕ 1 } A 6 : { p, p, p, p, p, ϕ 1 } A 7 : {p, p, p, p, p, ϕ 1 } Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

34 The Tableau Given a formula ϕ, we construct a directed graph T ϕ, called the tableau of ϕ, by the following algorithm. Algorithm (tableau construction) 1 The nodes of T ϕ are the atoms of ϕ. 2 Atom A is connected to atom B by a directed edge if all of the following are satisfied: R : For every p Φ ϕ, p A iff p B. R : For every p Φ ϕ, p A iff p B. R : For every p Φ ϕ, p A iff p B. An atom is called initial if it does not contain a formula of the form p or p ( = p). Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

35 Example Figure: Tableau T ϕ1 for ϕ 1 = p p. Source: [Manna and Pnueli 1995]. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

36 From the Tableau to a GBA Create an initial node and link it to every initial atom that contains ϕ. Label each directed edge with the atomic propositions that are contained in the ending atom. Add a set of atoms to the accepting set for each subformula of the following form: q: atoms with q or q. p U q: atoms with q or (p U q). q ( = q): atoms with q or q. ( q W p) ( = p U (q p)): atoms with q or q W p. q ( = q): atoms with q or q. (q W p) ( = p U ( q p)): atoms with q or q W p. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

37 Correctness: Models vs. Paths For a model σ, the infinite atom path π σ : A 0, A 1, in T ϕ is said to be induced by σ if, for every position j 0 and every closure formula p Φ ϕ, (σ, j) = p iff p A j. Claim (models induce paths) Consider a formula ϕ and its tableau T ϕ. For every model σ : s 0, s 1,, there exists an infinite atom path π σ : A 0, A 1, in T ϕ induced by σ. Furthermore, A 0 is an initial atom, and if σ = ϕ then ϕ A 0. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

38 Correctness: Promising Formulae A formula ψ Φ ϕ is said to promise the formula r if ψ has one of the following forms: r, p U r, r, ( r W p). or if r is the negation q and ψ has one of the forms: q, (q W p). Claim (promise fulfillment by models) Let σ be a model and ψ, a formula promising r. Then, σ contains infinitely many positions j 0 such that (σ, j) = ψ or (σ, j) = r. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

39 Correctness: Fulfilling Paths Atom A fulfills a formula ψ that promises r if ψ A or r A. A path π : A 0, A 1, in the tableau T ϕ is called fulfilling: A 0 is an initial atom. For every promising formula ψ Φ ϕ, π contains infinitely many atoms A j that fulfill ψ. Claim (models induce fulfilling paths) If π σ : A 0, A 1, is a path induced by a model σ, then π σ is fulfilling. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

40 Correctness: Fulfilling Paths (cont.) Claim (fulfilling paths induce models) If π : A 0, A 1, is a fulfilling path in T ϕ, there exists a model σ inducing π, i.e., π = π σ and, for every ψ Φ ϕ and every j 0, (σ, j) = ψ iff ψ A j. Proposition (satisfiability and fulfilling paths) Formula ϕ is satisfiable iff the tableau T ϕ contains a fulfilling path π = A 0, A 1, such that A 0 is an initial ϕ-atom. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

41 Concluding Remarks PTL can be extended in other ways to be as expressive as Büchi automata, i.e., to express all ω-regular properties. For example, the industry standard IEEE 1850 Property Specification Language (PSL) is based on an extension that adds classic regular expressions. Regarding translation of a temporal formula into an equivalent Büchi automaton, there have been quite a few algorithms proposed in the past. How to obtain an automaton as small as possible remains interesting, for both theoretical and practical reasons. Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

42 References E.M. Clarke, O. Grumberg, and D.A. Peled. Model Checking, The MIT Press, E.A. Emerson. Temporal and modal logic, Handbook of Theoretical Computer Science (Vol. B), MIT Press, G.J. Holzmann. The SPIN Model Checker: Primer and Reference Manual, Addison-Wesley, Z. Manna and A. Pnueli. The Temporal Logic of Reactive and Concurrent Systems: Specification, Springer, Z. Manna and A. Pnueli. Temporal Verification of Reactive Systems: Safety, Springer, Yih-Kuen Tsay IM.NTU) Linear Temporal Logic and Büchi Automata FLOLAC / 42

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