Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Quantitative Stream Reasoning with LARS Rafael Kiesel, Thomas Eiter Vienna University of Technology funded by FWF project W1255-N23 17 th of April 2019 Rafael Kiesel, Thomas Eiter Quantitative LARS 1 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions (Qualitative) Stream Reasoning with LARS ◮ Does a tram arrive at station s within the next 20 minutes? Rafael Kiesel, Thomas Eiter Quantitative LARS 2 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions (Qualitative) Stream Reasoning with LARS ◮ Does a tram arrive at station s within the next 20 minutes? → ⊞ + 20 ♦ Tram ( X , s ) Rafael Kiesel, Thomas Eiter Quantitative LARS 2 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions (Qualitative) Stream Reasoning with LARS ◮ Does a tram arrive at station s within the next 20 minutes? → ⊞ + 20 ♦ Tram ( X , s ) ◮ Can I go from station s to another station s ′ using a tram that arrives within 15 minutes? Rafael Kiesel, Thomas Eiter Quantitative LARS 2 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions (Qualitative) Stream Reasoning with LARS ◮ Does a tram arrive at station s within the next 20 minutes? → ⊞ + 20 ♦ Tram ( X , s ) ◮ Can I go from station s to another station s ′ using a tram that arrives within 15 minutes? → After ( s , s ′ ) ∧ ⊞ + 15 ♦ Tram ( X , s ) ∧ ¬ Full ( X ) Rafael Kiesel, Thomas Eiter Quantitative LARS 2 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions Quantitative? ◮ How many trams will arrive at station s within the next 20 minutes? Rafael Kiesel, Thomas Eiter Quantitative LARS 3 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions Quantitative? ◮ How many trams will arrive at station s within the next 20 minutes? → Expect answer in N Rafael Kiesel, Thomas Eiter Quantitative LARS 3 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions Quantitative? ◮ How many trams will arrive at station s within the next 20 minutes? → Expect answer in N ◮ How likely is it that I can go from station s to another station s ′ using a tram that arrives within 15 minutes? Rafael Kiesel, Thomas Eiter Quantitative LARS 3 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions Quantitative? ◮ How many trams will arrive at station s within the next 20 minutes? → Expect answer in N ◮ How likely is it that I can go from station s to another station s ′ using a tram that arrives within 15 minutes? → Expect answer in [ 0 , 1 ] ◮ ... Rafael Kiesel, Thomas Eiter Quantitative LARS 3 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions Quantitative? Quantitative extensions of LARS ◮ Ad Hoc ◮ Framework Rafael Kiesel, Thomas Eiter Quantitative LARS 4 / 20
Introduction Weighted LARS (Qualitative) Stream Reasoning with LARS Quantitative Stream Reasoning Quantitative? Future/Ongoing work Our Work Questions Our Work ◮ General framework ◮ Semirings as algebraic structure underlying calculations ◮ Introduce weighted LARS formulas (over semirings) ◮ Semantics assigns a numerical value (in the semiring) ◮ Applicability of our framework Rafael Kiesel, Thomas Eiter Quantitative LARS 5 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Preliminaries ◮ Interpretations ( S , t ) , with S = ( v , T ) a stream consisting of an evaluation function v and a set T of time points that are considered, that contains the current time t . ◮ Assign LARS formulas α ::= p | ¬ α | α ∧ α | α ∨ α | ♦ α | � α | @ t α | ⊞ w α a boolean value. ◮ Examples: ◮ ♦ Tram ( x , s ) ◮ ¬ @ T Tram ( x , s ) ∨ ¬ @ T + 1 Tram ( x , s ) Rafael Kiesel, Thomas Eiter Quantitative LARS 6 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Semiring A semiring is an algebraic structure ( R , ⊕ , ⊗ , e ⊕ , e ⊗ ) , s.t. ◮ ( R , ⊕ , e ⊕ ) is a commutative monoid with neutral element e ⊕ ◮ ( R , ⊗ , e ⊗ ) is a monoid with neutral element e ⊗ ◮ multiplication ( e ⊗ ) distributes over addition ( e ⊕ ) ◮ multiplication by e ⊕ annihilates R ( ∀ r ∈ R : e ⊕ ⊗ r = e ⊕ = r ⊗ e ⊕ ) Examples are ◮ ( N , + , · , 0 , 1 ) , the semiring over the natural numbers ◮ ([ 0 , 1 ] , max , · , 0 , 1 ) , a probability semiring ◮ ( {⊥ , ⊤} , ∨ , ∧ , ⊥ , ⊤ ) , a boolean algebra Rafael Kiesel, Thomas Eiter Quantitative LARS 7 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Weighted LARS Syntax We define weighted LARS formulas over a semiring R = ( R , ⊕ , ⊗ , e ⊕ , e ⊗ ) similarly to how weighted MSO formulas are defined in [Droste and Gastin2007] α ::= k | p | ¬ α | α ∧ α | α ∨ α | ♦ α | � α | @ t α | ⊞ w α, where k ∈ R . Rafael Kiesel, Thomas Eiter Quantitative LARS 8 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Weighted LARS Semantics I ◮ Goal: Assign a formula a numerical value ◮ Use e ⊗ and e ⊕ as truth and falsehood respectively ◮ Interpret disjunction as sum and conjunction as product ◮ Formally, for an interpretation ( S , t ) , where S = ( v , T ) : � k � R ( S , t ) = k , for k ∈ R � e ⊗ , if p ∈ v ( t ) � p � R ( S , t ) = e ⊕ , otherwise. � α ∧ β � R ( S , t ) = � α � R ( S , t ) ⊗ � β � R ( S , t ) � α ∨ β � R ( S , t ) = � α � R ( S , t ) ⊕ � β � R ( S , t ) Rafael Kiesel, Thomas Eiter Quantitative LARS 9 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Weighted LARS Semantics II ◮ Negation is close to inversion of the truth value ◮ Interpret existential quantification as sum and universal quantification as product � e ⊗ , iff � α � R ( S , t ) = e ⊕ � ¬ α � R ( S , t ) = otherwise. e ⊕ , t ′ ∈ T � α � R ( S , t ′ ) � ♦ α � R ( S , t ) = � t ′ ∈ T � α � R ( S , t ′ ) � � α � R ( S , t ) = � � @ t ′ α � R ( S , t ) = � α � R ( S , t ′ ) � ⊞ w α � R ( S , t ) = � α � R ( ⊞ w ( S , t ) , t ) Rafael Kiesel, Thomas Eiter Quantitative LARS 10 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Example ◮ How many trams will arrive at station s within the next 20 minutes? Rafael Kiesel, Thomas Eiter Quantitative LARS 11 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Example ◮ How many trams will arrive at station s within the next 20 minutes? → ⊞ + 20 ♦ Tram ( X , s ) over ( N , + , · , 0 , 1 ) Rafael Kiesel, Thomas Eiter Quantitative LARS 11 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Example ◮ How many trams will arrive at station s within the next 20 minutes? → ⊞ + 20 ♦ Tram ( X , s ) over ( N , + , · , 0 , 1 ) ◮ How likely is it that I can go from station s to another station s ′ using a tram that arrives within 15 minutes? Rafael Kiesel, Thomas Eiter Quantitative LARS 11 / 20
Introduction Preliminaries Weighted LARS Syntax Quantitative Stream Reasoning Semantics Future/Ongoing work Example Questions Example ◮ How many trams will arrive at station s within the next 20 minutes? → ⊞ + 20 ♦ Tram ( X , s ) over ( N , + , · , 0 , 1 ) ◮ How likely is it that I can go from station s to another station s ′ using a tram that arrives within 15 minutes? → After ( s , s ′ ) ∧ ⊞ + 15 ♦ Tram ( X , s ) ∧ ¬ Full ( X ) ∨ Tram ( X , s ) ∧ Full ( X ) ∧ 0 . 3 over ([ 0 , 1 ] , max , · , 0 , 1 ) Rafael Kiesel, Thomas Eiter Quantitative LARS 11 / 20
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