Weighted linear dynamic logic Manfred Droste 1 and George Rahonis 2 1 - - PowerPoint PPT Presentation

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Weighted linear dynamic logic Manfred Droste 1 and George Rahonis 2 1 Leipzig University, Germany 2 Aristotle University of Thessaloniki, Greece GandALF 2016 Catania, September 15, 2016 George Rahonis (University of Thessaloniki) Weighted linear

SLIDE 1

Weighted linear dynamic logic

Manfred Droste1 and George Rahonis2

1Leipzig University, Germany 2Aristotle University of Thessaloniki, Greece

GandALF 2016 Catania, September 15, 2016

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 1 / 20

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Linear Temporal Logic (LTL), Pnueli 1977

LTL = FO logic. Satis…ability, validity, logical implication of LTL formulas: PSPACE-complete. LTL: reasonable for practical applications. LTL …nite automata. LTL …nite automata over in…nite words.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 2 / 20

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Monadic Second Order (MSO) logic

Büchi 1960, Elgot 1961, Trakhtenbrot 1962: MSO logic = …nite automata. Büchi 1962: MSO logic = …nite automata over in…nite words. MSO logic formulas

non-elementary

…nite automata. MSO logic: not reasonable for practical applications.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 3 / 20

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New logic?

A logic combining the complexity properties of reasoning on LTL and the expressive equivalence to …nite automata was greatly desirable. Vardi and Wolper 1994: ETL a Temporal logic with Automata Connectives. Satis…ability of ETL(=RETL) formulas is PSPACE-complete. Vardi 2000: ForSpec, industrial temporal logic used by Intel: RETL+hardware features (clocks and resets). 2003 PSL an industrial-standard property-speci…cation language: LTL extended with dynamic modalities (borrowed from Dynamic Logic), clocks and resets. Vardi 2011, De Giacomo and Vardi 2013, 2015: Linear dynamic logic (LDL). LDL is a combination of propositional dynamic logic and LTL.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 4 / 20

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Quantitative logics required for modern applications

Droste and Gastin 2005, 2009: Weighted MSO logic over semirings. Weighted automata weighted MSO logic. Restricted weighted MSO logic = weighted automata (Büchi type theorem) but the translation is non-elementary. Kupferman and Lustig 2007: Weighted LTL over De Morgan Algebras. Droste and Vogler 2012: Weighted LTL over arbitrary bounded lattices. Bouyer, Markey and Matteplackel 2014, Almagor, Boker and Kupferman 2014, 2016: Weighted LTL over [0, 1]. Mandrali and Rahonis 2014, 2016: Weighted LTL over semirings. In this paper: Weighted LTL over the naturals is incomparable to weighted automata.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 5 / 20

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Notations

A alphabet w = w(0) . . . w(n 1) 2 A, with w(i) 2 A, 0 i n 1 wi = w(i) . . . w(n 1) for 0 i n 1

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 6 / 20

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LDL - Syntax

Atomic propositions: P = fpa j a 2 Ag.

De…nition

Syntax of LDL formulas ψ over A : ψ ::= true j pa j :ψ j ψ ^ ψ j hθi ψ θ ::= φ j ψ? j θ + θ j θ; θ j θ+ pa 2 P, φ propositional formula over P.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 7 / 20

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LDL - Semantics

ψ LDL formula, w 2 A. De…ne w j = ψ inductively: w j = true, w j = pa i¤ w(0) = a, w j = :ψ i¤ w 6j = ψ, w j = ψ1 ^ ψ2 i¤ w j = ψ1 and w j = ψ2, w j = hφi ψ i¤ w j = φ and w1 j = ψ, w j = hψ1?i ψ2 i¤ w j = ψ1 and w j = ψ2, w j = hθ1 + θ2i ψ i¤ w j = hθ1i ψ or w j = hθ2i ψ, w j = hθ1; θ2i ψ i¤ w = uv, u j = hθ1i true, and v j = hθ2i ψ, w j =

ψ i¤ there exists n with 1 n jwj such that w j = hθni ψ, θn, n 1 is de…ned by θ1 = θ and θn = θn1; θ for n > 1.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 8 / 20

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LDL - Main results (De Giacomo and Vardi 2013)

LDL formulas = rational expressions.

Rational expressions

linear

LDL formulas . LDL formulas

doubly

exponential

rational expressions. LDL formulas

exponential

…nite automata.

Satis…ability, validity, logical implication of LDL formulas: PSPACE-complete.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 9 / 20

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Weighted rational expressions

(K, +, , 0, 1) semiring Weighted rational expressions over A and K : E ::= ka j E + E j E E j E + a 2 A, k 2 K Generalized weighted rational expressions over A and K : E ::= ka j E + E j E E j E + j E E. a 2 A, k 2 K Semantics: kEk : A ! K rational (g-rational)

kkak = ka kE1 + E2k = kE1k + kE2k kE1 E2k = kE1k kE2k (Cauchy product) kE +k = kEk+ (kEk (ε) = 0, proper) kE1 E2k = kE1k kE2k (Hadamard product)

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 10 / 20

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Weighted LDL - Syntax

Atomic propositions: P = fpa j a 2 Ag.

De…nition

Syntax of weighted LDL formulas ϕ over A and K : ϕ ::= k j ψ j ϕ ϕ j ϕ ϕ j hρi ϕ ρ ::= φ j ϕ? j ρ ρ j ρ ρ j ρ k 2 K, φ propositional formula over P, ψ LDL formula.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 11 / 20

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Weighted LDL - Semantics

ϕ weighted LDL formula. Semantics kϕk : A ! K, for w 2 A: kkk (w) = k, kψk (w) = 1 if w j = ψ

therwise ,

kϕ1 ϕ2k (w) = kϕ1k (w) + kϕ2k (w), kϕ1 ϕ2k (w) = kϕ1k (w) kϕ2k (w), khφi ϕk (w) = kφk (w) kϕk (w1), khϕ1?i ϕ2k (w) = kϕ1k (w) kϕ2k (w), khρ1 ρ2i ϕk (w) = khρ1i ϕk (w) + khρ2i ϕk (w), khρ1 ρ2i ϕk (w) = ∑

w =uv (khρ1i truek (u) khρ2i ϕk (v)) ,

khρi ϕk (w) = ∑

n1

khρni ϕk (w) (khρi truek proper) ρn, n 1 is de…ned by ρ1 = ρ and ρn = ρn1 ρ for n > 1.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 12 / 20

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Weighted LDL - Example

LDL formula: Last ::= htruei

^

a02A :pa0

w = w(0) . . . w(n 1) 2 A, 0 i n 1 wi j = Last i¤ wi+1 6j = pa0 for every a0 2 A i¤ i = n 1 (N, +, , 0, 1), a 2 A, k 2 N n f0g ϕ = D ((h(k pa)?i Last)? (h(k pa)?i Last)?)E true

^

a02A :pa0

kϕk (w) = k2n if w = a2n, n 0

therwise

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 13 / 20

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Weighted LDL - Main results

LDL-de…nable series = g-rational series (No fragments for LDL!).

Weighted g-rational expressions

linear

weighted LDL formulas.

K commutative: LDL-de…nable series = rational series = recognizable series.

Weighted LDL formulas

doubly

exponential

weighted automata. K idempotent:

Weighted LDL formulas

exponential

weighted automata.

K computable …eld, ϕ, ϕ0 weighted LDL formulas, k 2 K : kϕk = kϕ0k , kϕk = e k (constant series): decidable in doubly exponential time.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 14 / 20

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Weighted LDL - Comparison to other weighted logics

Weighted LDL and weighted LTL over the naturals are incomparable. Weighted LDL and weighted FO logic over the naturals are incomparable. K commutative: restricted weighted MSO logic = weighted LDL, restricted weighted FO logic weighted LDL, restricted weighted LTL weighted LDL.

K dual continuous with the Arden …xed point property: weighted LDL = weighted ^ -free µ-calculus.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 15 / 20

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LDL over in…nite words, Vardi 2011

Modi…ed syntax for interpretation over in…nite words. LDL-ω-de…nable languages = ω-rational languages. Satis…ability of LDL formulas: PSPACE-complete.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 16 / 20

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Weighted LDL over in…nite words - Main results

K totally complete semiring. Modi…ed syntax for interpretation over in…nite words. LDL-ω-de…nable series = g-ω-rational series. No fragments for LDL! K totally commutative complete: LDL-ω-de…nable series = ω-rational series = ω-recognizable series.

K idempotent:

Weighted LDL formulas

exponential

weighted Büchi automata.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 17 / 20

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Weighted LDL over in…nite words - Comparison to other weighted logics

K totally commutative complete: restricted weighted ω-MSO logic = weighted ω-LDL, restricted weighted ω-FO logic weighted ω-LDL, restricted weighted ω-LTL weighted ω-LDL.

K dual continuous semiring with the Arden …xed point property: weighted ω-LDL = weighted ω- ^ -free µ-calculus.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 18 / 20

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Future research

Translation of weighted LDL formulas over in…nite words to weighted automata (not idempotent semirings).

Complexity results.

Weighted LDL over more general structures, reasonable for practical applications, e.g. valuation monoids.

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 19 / 20

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References

M. Y. Vardi, P. Wolper, Reasoning about in…nite computations,

Information and Computation 115(1994) 1–37.

M. Y. Vardi, The rice and fall of LTL, Presentation at GandALF 2011.
G. De Giacomo, M. Y. Vardi, Linear temporal logic and linear dynamic

logic on …nite traces, in: Proceedings of IJCAI 2013, pp. 854–860.

G. De Giacomo, M. Y. Vardi, Synthesis for LTL and LDL on …nite

traces, in: Proceedings of IJCAI 2015, pp. 1558–1564. Thank you Eυχαριστ ´ ω

George Rahonis (University of Thessaloniki) Weighted linear dynamic logic Catania, September 15, 2016 20 / 20