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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

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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

17th of April 2019

Rafael Kiesel, Thomas Eiter Quantitative LARS 1 / 20

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

(Qualitative) Stream Reasoning with LARS

◮ Does a tram arrive at station s within the next 20 minutes?

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

(Qualitative) Stream Reasoning with LARS

◮ Does a tram arrive at station s within the next 20 minutes?

→ ⊞+20♦Tram(X, s)

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

(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?

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

(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)

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

Quantitative?

◮ How many trams will arrive at station s within the next 20

minutes?

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

Quantitative?

◮ How many trams will arrive at station s within the next 20

minutes? → Expect answer in N

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

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

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

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]

◮ ...

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

Quantitative?

Quantitative extensions of LARS

◮ Ad Hoc ◮ Framework

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions (Qualitative) Stream Reasoning with LARS Quantitative? Our Work

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

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

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 | @tα | ⊞wα a boolean value.

◮ Examples:

◮ ♦Tram(x, s) ◮ ¬@TTram(x, s) ∨ ¬@T+1Tram(x, s) Rafael Kiesel, Thomas Eiter Quantitative LARS 6 / 20

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

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

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

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 | @tα | ⊞wα, where k ∈ R.

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

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):

kR(S, t) = k, for k ∈ R pR(S, t) = e⊗, if p ∈ v(t) e⊕,

therwise.

α ∧ βR(S, t) = αR(S, t) ⊗ βR(S, t) α ∨ βR(S, t) = αR(S, t) ⊕ βR(S, t)

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

Weighted LARS Semantics II

◮ Negation is close to inversion of the truth value ◮ Interpret existential quantification as sum and universal

quantification as product ¬αR(S, t) = e⊗, iff αR(S, t) = e⊕ e⊕,

therwise.

♦αR(S, t) =

t′∈TαR(S, t′)

αR(S, t) =

t′∈TαR(S, t′)

@t′αR(S, t) = αR(S, t′) ⊞wαR(S, t) = αR(⊞w(S, t), t)

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

Example

◮ How many trams will arrive at station s within the next 20

minutes?

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

Example

◮ How many trams will arrive at station s within the next 20

minutes? → ⊞+20♦Tram(X, s) over (N, +, ·, 0, 1)

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

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?

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Preliminaries Syntax Semantics Example

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

ver ([0, 1], max, ·, 0, 1)

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions LARS measure Problem definitions Applications Example Relation to other formalisms

LARS measure

◮ A LARS measure µ is defined by a triple Π, α, R, where

◮ Π is a LARS program ◮ α is a weighted LARS formula over R ◮ R is a semiring

◮ We set

µ(S, t) = αR(S, t) iff S is an answer stream of Π at t, e⊕

therwise.

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions LARS measure Problem definitions Applications Example Relation to other formalisms

Problem definitions

◮ Optimisation:

argmax(S,t) µ(S, t)

◮ Probabilistic reasoning:

Pµ(S, t) = µ(S, t)

(S′,t′) µ(S′, t′)

Pµ(φ, t) =

S,(S,t)|

=φ

P(S, t) Eµ[β] =

(S,t)

βR(S, t)P(S, t)

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions LARS measure Problem definitions Applications Example Relation to other formalisms

Applications

◮ P-log [Baral et al.2009]: Probabilistic reasoning. Can be

expressed using the framework.

◮ Problog [De Raedt et al.2007]: Probabilistic reasoning. Can

be expressed using the framework.

◮ LPMLN [Lee and Yang2017]: Probabilistic reasoning. Can be

expressed with µ(S, t) = αR(S, t) iff S is an answer stream of ΠS,t at t, e⊕

therwise.

◮ PrASP [Nickles and Mileo2015]: Probabilistic reasoning. No

bvious relation to our framework.

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions LARS measure Problem definitions Applications Example Relation to other formalisms

Reliability of constraint satisfaction

◮ Using the semiring over the natural numbers, we can evaluate

how many proofs there are for a formula.

◮ We consider answer streams of a program Π more reliable if

there are more proofs for a constraint α

◮ Assume that the probability of an answer stream is

proportional to the number of proofs for the constraint → probability distribution given as Pµ induced by Π, α, N

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions LARS measure Problem definitions Applications Example Relation to other formalisms

Relation to other formalisms

◮ ASP expressible in second order logic ◮ for the propositional case even in monadic second order logic

(MSO)

◮ Fragment of weighted MSO by [Droste and Gastin2007]

equivalent to weighted automata

◮ Similarly a fragment of the problems definable using LARS

measures is equivalent to weighted automata

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions

Future/Ongoing work

◮ Weighted LARS formulas for aggregates, weighted constraints

and more

◮ Implementation ◮ Complexity considerations ◮ General properties of extensions formalised using weighted

formulas?

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Questions

Questions?

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Questions

References I

Chitta Baral, Michael Gelfond, and Nelson Rushton. Probabilistic reasoning with answer sets. Theory and Practice of Logic Programming, 9(1):57–144, 2009. Luc De Raedt, Angelika Kimmig, and Hannu Toivonen. Problog: A probabilistic prolog and its application in link discovery. In IJCAI, volume 7, pages 2462–2467. Hyderabad, 2007. Manfred Droste and Paul Gastin. Weighted automata and weighted logics. Theoretical Computer Science, 380(1):69, 2007.

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Introduction Weighted LARS Quantitative Stream Reasoning Future/Ongoing work Questions Questions

References II

Joohyung Lee and Zhun Yang. Lpmln, weak constraints, and p-log. In Thirty-First AAAI Conference on Artificial Intelligence, 2017. Matthias Nickles and Alessandra Mileo. A system for probabilistic inductive answer set programming. In International Conference on Scalable Uncertainty Management, pages 99–105. Springer, 2015.

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