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**Unformatted text preview: **CS256/Winter 2007 — Lecture #11 Zohar Manna 11-1 Beyond Temporal Logics Temporal logic expresses properties of infinite sequences of states, but there are interesting properties that cannot be expressed, e.g., “ p is true only (at most) at even positions.” Questions (foundational/practical): • What other languages can we use to express properties of sequences ( ⇒ properties of programs)? • How do their expressive powers compare? • How do their computational complexities (for the decision problems) compare? 11-2 ω-languages Σ : nonempty set (alphabet ) Σ * : set of finite strings of characters in Σ finite word w ∈ Σ * Σ ω : set of all infinite strings of characters in Σ ω-word w ∈ Σ ω (finitary) language: L ⊆ Σ * ω-language: L ⊆ Σ ω 11-3 States Propositional LTL (PLTL) formulas are constructed from the following: • propositions p 1 , p 2 , . . . , p n . • boolean/temporal operators. • a state s ∈ { f, t } n i.e., every state s is a truth-value assignment to all n propositional variables. Example: If n = 3 , then s : h p 1 : t, p 2 : f, p 3 : t i corresponds to state tft . p 1 ↔ p 2 denotes the set of states { fff, fft, t t f, t t t } • alphabet Σ = { f, t } n i.e, 2 n strings, one string for every state. Note : t , f = formulas (syntax) t, f = truth values (semantics) 11-4 Models of PLTL 7→ ω-languages • A model of PLTL for the language with n propositions σ : s , s 1 , s 2 , . . . can be viewed as an infinite string s s 1 s 2 . . . , i.e., σ ∈ ( { f, t } n ) ω • A PLTL formula ϕ denotes an ω-language L = { σ | σ q ϕ } ⊆ ( { f, t } n ) ω Example: If n = 3 , then ϕ : ( p 1 ↔ p 2 ) denotes the ω-language L ( ϕ ) = { fff, fft, t t f, t t t } ω 11-5 Other Languages to Talk about Infinite Sequences • ω-regular expressions • ω-automata • first-order logic of the linear order of N (with relation symbols S , < ) Example: ∀ t [ 0 ≤ t → P ( t ) ] . • (monadic) second-order logic of the linear order of N Example: ∀ Q. Q (0) ∧ ∀ t [ Q ( t ) ↔ ¬ Q ( S t ) ] → ∀ t [ Q ( t ) → P ( t ) ] • right-linear grammars 11-6 Regular Expressions Syntax: r ::= ∅ | ε | a | r 1 r 2 | r 1 + r 2 | r * ( ε = empty word, a ∈ Σ ) Semantics: A regular expression r (on alphabet Σ ) denotes a finitary language L ( r ) ⊆ Σ * : L ( ∅ ) = ∅ L ( ε ) = { ε } L ( a ) = { a } L ( r 1 r 2 ) = L ( r 1 ) · L ( r 2 ) = { xy | x ∈ L ( r 1 ) , y ∈ L ( r 2 ) } L ( r 1 + r 2 ) = L ( r 1 ) ∪ L ( r 2 ) L ( r * ) = L ( r ) * = { x 1 x 2 ··· x n | n ≥ , x 1 , x 2 , . . . , x n ∈ L ( r ) } 11-7 ω-regular expressions Syntax: ωr ::= r 1 ( s 1 ) ω + r 2 ( s 2 ) ω + ··· + r n ( s n ) ω n ≥ 1 , r i , s i = regular expressions Semantics: L ( rs ω ) = { xy 1 y 2 ··· | x ∈ L ( r ) , y 1 , y 2 , . . . ∈ L ( s ) \ { ε }} rs ω denotes all infinite strings with an initial prefix in L ( r ) , followed by a concatenation of infinitely many nonempty words in L (...

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