Tensor-Based Abduction in Horn Propositional Programs Yaniv Aspis, - - PowerPoint PPT Presentation

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September 2018 Tensor-Based Abduction in Horn Propositional Programs Yaniv Aspis, Krysia Broda, Alessandra Russo Department of Computing, Imperial College London {yaniv.aspis17,k.broda,a.russo}@imperial.ac.uk Why Linear Algebra? AI

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Tensor-Based Abduction in Horn Propositional Programs

Yaniv Aspis, Krysia Broda, Alessandra Russo

September 2018

Department of Computing, Imperial College London

{yaniv.aspis17,k.broda,a.russo}@imperial.ac.uk

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Why Linear Algebra?

AI software is moving towards GPU-based solutions
Optimized for matrix/tensor multiplication
Highly-Parallel computations
Develop algorithms for GPU

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Abduction

Logical Inference through Explanations

𝑄, 𝑕, 𝐵𝑐

Assume 𝑄 is a Horn Logic Program
Observation 𝑕
Find Δ ⊆ 𝐵𝑐 such that 𝑕 ∈ 𝑀𝐼𝑁(𝑄 ∪ Δ)
𝑄 ∪ Δ is consistent

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Background: Embedding Atoms[1]

𝑕 ← 𝑞 ∧ 𝑟 𝑞 ← 𝑟 𝑟 ← ← 𝑕

𝑤 𝑕 = 1 𝑤 𝑞 = 1 𝑤 𝑟 = 1 𝑤 ⊥ = 1 𝑤 ⊤ = 1 𝑤 𝑕,𝑟 = 1 1 1

1. Sakama, C., Inoue, K., Sato, T.: Linear Algebraic Characterization of Logic Programs. In: KSEM. Lecture

Notes in Computer Science, vol. 10412, pp. 520-533. Springer (2017)

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Background: Embedding Programs[1]

𝑕 ← 𝑞 ∧ 𝑟 𝑞 ← 𝑟 𝑟 ← ← 𝑕 𝐸𝑄 = 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 ⊥ ⊤ 𝑕 𝑞 𝑟 ⊥ ⊤ 𝑕 𝑞 𝑟 𝐾 = 𝑈𝑄 𝐽 ∪ 𝐽 ⇔ 𝑤 𝐾 = 𝐼1 𝐸𝑄 ⋅ 𝑤 𝐽

𝐼1 𝑦 = 0, 𝑦 < 1 1, 𝑦 ≥ 1 Immediate Consequence via matrix multiplication:

1. Sakama, C., Inoue, K., Sato, T.: Linear Algebraic Characterization of Logic Programs. In: KSEM. Lecture

Notes in Computer Science, vol. 10412, pp. 520-533. Springer (2017)

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Invert implications:

𝑕 ← 𝑞 ∧ 𝑟 𝑕 ← 𝑟 𝑕 ← 𝑢

Abduction - Main Idea

𝑕 𝑞, 𝑟 𝑟 𝑢

𝑞, 𝑟 , 𝑟 , 𝑢 , 𝑕 Potential solutions:

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Tensor Embedding

1 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢 ⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

𝑕 → 𝑞 ∧ 𝑟

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𝐵∷1 = 1 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

𝑕 → 𝑞 ∧ 𝑟

𝐵∷2 = 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

𝑕 → 𝑟

𝐵∷3 = 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

𝑕 → 𝑢

𝐵∷4 = 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

Keep 𝑕

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Frontal Slices

𝐵∷𝑙 Third-Order Tensor 𝐵

1 1 1 1 1 1 1

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Abductive Step

𝐼1 𝐵 ×2 1 1 = 1 1 1 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

𝑕 ⇒ 𝑞, 𝑟 , 𝑟 , 𝑢 , 𝑕

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Inconsistencies

𝑕 ← 𝑞 ∧ 𝑟 𝑕 ← 𝑟 𝑕 ← 𝑢 ← 𝑢

1 1 1 1 1 1 1 1 1

So far:

Idea: Compute 𝑀𝐼𝑁(𝑄 ∪ Δ) for each column

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

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Inconsistencies

1 1 1 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

⇒

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

𝑀𝐼𝑁

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Inconsistencies

1 1 1 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

⇒

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

𝑀𝐼𝑁 Inconsistent!

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Inconsistencies

1 1 1 1 1 1 1 1 1

⊥ ⊤ 𝑕 𝑞 𝑟 𝑢

⇒

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

𝑀𝐼𝑁 Inconsistent! Remove inconsistencies, duplicates, and continue to Abductive Step…

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Abducibles

Δ ⊆ 𝐵𝑐 if and only if:

𝑤 Δ × 𝑤 𝐵𝑐 = 𝑤 Δ

Post-Filtering
If Abducibles are not defined, then Δ ⊆ 𝐵𝑐 implies

𝐼1 𝐵𝑄 ×2 𝑤 Δ = 𝑤 Δ after duplicates are removed (Filter during run)

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Discussion and Future Work

Clauses with negation
First-Order Predicate Logic
Optimisation and Scalability Testing

Future Work:

Proof of correctness has been completed
An unoptimised implementation has been made

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Tensor Multiplication

× = × = =

Flatten

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Multiple Definitions

𝑕 ← 𝑞 ∧ 𝑟 𝑕 ← 𝑞 ∧ 𝑠

𝐼 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 ⋅ 1 1 1 = 1 1

Introduce auxiliary variables:

𝑕1 ← 𝑞 ∧ 𝑟 𝑕2 ← 𝑞 ∧ 𝑠 𝑕 ← 𝑕1 𝑕 ← 𝑕2

⊥ ⊤ 𝑕 𝑞 𝑟 𝑠