Reasoning with Sets and Sums of Sets Markus Bender - - PowerPoint PPT Presentation

Reasoning with Sets and Sums of Sets

Markus Bender mbender@uni-koblenz.de

Universität Koblenz-Landau

Motivation

Ways for Reasoning with Complex Systems

◮ Introduce calculus/reasoner specifically tailored to the complex system ◮ Reduce complex system to an established problem (abstraction)

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Motivation

Ways for Reasoning with Complex Systems

◮ Introduce calculus/reasoner specifically tailored to the complex system ◮ Reduce complex system to an established problem (abstraction)

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Motivation

Boolean Algebra with Presburger Arithmetic

◮ Introduced by Kuncak et al. ◮ Transform to equisatisfiable problem in pure arithmetic ◮ Proven useful for many reasoning task ◮ Good tool support

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Contributions

Boolean Algebra with Presburger Arithmetic and Sums

◮ Add sum as new bridging function ◮ Arbitrary mixture of quantified and free set variables

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Syntax

F ::= A | F ∧ F | F ∨ F | ¬F | ∃ x .F | ∀ x .F | ∃ k .F | ∀ k .F A ::= B ≈ B | B ⊆ B | T ≈ T | T < T | B ::= x | ∅ | U | B ∪ B | B ∩ B | B T ::= k | K | MAXC | T + T | K · T | card(B) K ::= . . . | −2 | −1 | 0 | 1 | 2 | . . .

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Syntax

F ::= A | F ∧ F | F ∨ F | ¬F | ∃ x .F | ∀ x .F | ∃ k .F | ∀ k .F A ::= B ≈ B | B ⊆ B | T ≈ T | T < T | B ::= x | ∅ | U | B ∪ B | B ∩ B | B T ::= k | K | MAXC | T + T | K · T | card(B) | MAXS | sum(B) K ::= a number

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Syntax

F ::= A | F ∧ F | F ∨ F | ¬F | ∃ x .F | ∀ x .F | ∃ k .F | ∀ k .F A ::= B ≈ B | B ⊆ B | T ≈ T | T < T | B ::= x | ∅ | U | B ∪ B | B ∩ B | B T ::= k | K | MAXC | T + T | K · T | card(B) | MAXS | sum(B) K ::= a number Sets are finite.

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Extension

BAPA

card with sort: α set → N, where α: an arbitrary sort Specific members of the involved sets do not matter.

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Extension

BAPA

card with sort: α set → N, where α: an arbitrary sort Specific members of the involved sets do not matter.

BAPAS

sum with sort: β set → β, where β: a numerical sort Specific members of the involved sets do matter.

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Semantics

Structures

A := (N, F, ΩA, ΠA) N: only codomain for card F: element support; elements of sets (F ⊆ R) ΩA: (more or less) common semantics of functions ΠA: common semantics of predicates

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Overview

Original Formula (OF) Transformed Formula (TF) Model for TF Model for OF Transformation

• Arith. Solver

Transformation:

• 1. Eliminate ≈set
• 2. Eliminate ⊆
• 3. Introduce atomic

decompositions

• 4. Distribute cardinality
• 5. Purify cardinality
• 6. Distribute sum
• 7. Purify sum
• 8. Eliminate quantifiers

and add axioms

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Example

Given Formula

∃ x1 (x0 ⊆ x1 → sum(x0) ≈ sum(x0 ∩ x1))

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Example

Eliminate ≈set , and eliminate ⊆

∃ x1 (x0 ⊆ x1 → sum(x0) ≈ sum(x0 ∩ x1))

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Example

Eliminate ≈set , and eliminate ⊆

∃ x1 (x0 ⊆ x1 → sum(x0) ≈ sum(x0 ∩ x1))

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Example

Eliminate ≈set , and eliminate ⊆

∃ x1 (x0 ⊆ x1 → sum(x0) ≈ sum(x0 ∩ x1))

• ∃ x1 (card(x0 ∩ x1) ≈ 0 → sum(x0) ≈ sum(x0 ∩ x1))

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Atomic Sets and Atomic Decompositions1

x0 ∩ x1 x0 ∩ x1 x0 ∩ x1 x0 ∩ x1

1Ohlbach, Kuncak 11 / 30

Atomic Sets and Atomic Decompositions1

x0 ∩ x1 x0 ∩ x1 x0 ∩ x1 x0 ∩ x1 S00 := x0 ∩ x1, S01 := x0 ∩ x1, S10 := x0 ∩ x1, S11 := x0 ∩ x1

1Ohlbach, Kuncak 11 / 30

Atomic Sets and Atomic Decompositions1

x0 ∩ x1 x0 ∩ x1 x0 ∩ x1 x0 ∩ x1 S00 := x0 ∩ x1, S01 := x0 ∩ x1, S10 := x0 ∩ x1, S11 := x0 ∩ x1 x0 := S10 ∪ S11 = (x0 ∩ x1) ∪ (x0 ∩ x1) x1 := S01 ∪ S11 = (x0 ∩ x1) ∪ (x0 ∩ x1).

1Ohlbach, Kuncak 11 / 30

Example

Introduce Atomic Decompositions

∃ x1 (card(x0 ∩ x1) ≈ 0 → sum(x0) ≈ sum(x0 ∩ x1))

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Example

Introduce Atomic Decompositions

∃ x1 (card(x0 ∩ x1) ≈ 0 → sum(x0) ≈ sum(x0 ∩ x1))

• ∃ x1 (card(S10) ≈ 0 → sum(S10 ∪ S11) ≈ sum(S11))

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Example

Distribute card, purify card

∃ x1 (card(S10) ≈ 0 → sum(S10 ∪ S11) ≈ sum(S11))

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Example

Distribute card, purify card

∃ x1 (card(S10) ≈ 0 → sum(S10 ∪ S11) ≈ sum(S11))

• ∃ x1

∃ l00, l01, l10, l11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ (l10 ≈ 0 → sum(S10 ∪ S11) ≈ sum(S11))

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Example

Distribute sum

∃ x1 ∃ l00, l01, l10, l11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ (l10 ≈ 0 → sum(S10 ∪ S11) ≈ sum(S11))

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Example

Distribute sum

∃ x1 ∃ l00, l01, l10, l11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ (l10 ≈ 0 → sum(S10 ∪ S11) ≈ sum(S11))

• ∃ x1

∃ l00, l01, l10, l11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ (l10 ≈ 0 → sum(S10) + sum(S11) ≈ sum(S11))

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Example

Purify sum

∃ x1 ∃ l00, l01, l10, l11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ (l10 ≈ 0 → sum(S10) + sum(S11) ≈ sum(S11))

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Example

Purify sum

∃ x1 ∃ l00, l01, l10, l11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ (l10 ≈ 0 → sum(S10) + sum(S11) ≈ sum(S11))

• ∃ x1

∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ sum(S00) ≈ s00 ∧ · · · ∧ sum(S11) ≈ s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

∃ x1 ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ sum(S00) ≈ s00 ∧ · · · ∧ sum(S11) ≈ s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Quantifier Elimination2

x

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Quantifier Elimination2

x l, k natural numbers card(x) = k + l.

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Quantifier Elimination2

y y x ∩ y x ∩ y l, k natural numbers card(x) = k + l.

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Quantifier Elimination2

y y x ∩ y x ∩ y l, k natural numbers card(x) = k + l. ⇔ ∃ y (card(x ∩ y) = k ∧ card(x ∩ y) = l), with y a finite set.

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Example

Remove Quantifier (Use Equivalence)

∃ x1 ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ sum(S00) ≈ s00 ∧ · · · ∧ sum(S11) ≈ s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Use Equivalence)

∃ x1 ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 card(S00) ≈ l00 ∧ · · · ∧ card(S11) ≈ l11 ∧ sum(S00) ≈ s00 ∧ · · · ∧ sum(S11) ≈ s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

• ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11

card(x0) ≈ l00 + l01 ∧ card(x0) ≈ l10 + l11 ∧ sum(x0) ≈ s00 + s01 ∧ sum(x0) ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Purify)

∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 card(x0) ≈ l00 + l01 ∧ card(x0) ≈ l10 + l11 ∧ sum(x0) ≈ s00 + s01 ∧ sum(x0) ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Purify)

∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 card(x0) ≈ l00 + l01 ∧ card(x0) ≈ l10 + l11 ∧ sum(x0) ≈ s00 + s01 ∧ sum(x0) ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

• ∃ l0, l1, ∃ s0, s1,

card(x0) ≈ l0 ∧ card(x0) ≈ l1 ∧ sum(x0) ≈ s0 ∧ sum(x0) ≈ s1 ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Delete Quantifier)

∃ l0, l1, ∃ s0, s1, card(x0) ≈ l0 ∧ card(x0) ≈ l1 ∧ sum(x0) ≈ s0 ∧ sum(x0) ≈ s1 ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Delete Quantifier)

∃ l0, l1, ∃ s0, s1, card(x0) ≈ l0 ∧ card(x0) ≈ l1 ∧ sum(x0) ≈ s0 ∧ sum(x0) ≈ s1 ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

• card(x0) ≈ l0 ∧ card(x0) ≈ l1 ∧ sum(x0) ≈ s0 ∧ sum(x0) ≈ s1

∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Delete Definitions)

card(x0) ≈ l0 ∧ card(x0) ≈ l1 ∧ sum(x0) ≈ s0 ∧ sum(x0) ≈ s1 ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Delete Definitions)

card(x0) ≈ l0 ∧ card(x0) ≈ l1 ∧ sum(x0) ≈ s0 ∧ sum(x0) ≈ s1 ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

• ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11

l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Add Axioms)

∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Add Axioms)

∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

• ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11

l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l00 ≈ 0 → s00 ≈ 0) ∧ . . . ∧ (l11 ≈ 0 → s11 ≈ 0) ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Add Axioms)

∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l00 ≈ 0 → s00 ≈ 0) ∧ . . . ∧ (l11 ≈ 0 → s11 ≈ 0) ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

Remove Quantifier (Add Axioms)

∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11 l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l00 ≈ 0 → s00 ≈ 0) ∧ . . . ∧ (l11 ≈ 0 → s11 ≈ 0) ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

• ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11

l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l00 ≈ 0 → s00 ≈ 0) ∧ . . . ∧ (l11 ≈ 0 → s11 ≈ 0) ∧ (l01 ≈ 1 ∧ l00 ≈ 1) → (s01 ≈ s00) ∧ . . . ∧ (l11 ≈ 1 ∧ l10 ≈ 1) → (s11 ≈ s10) ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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Example

In Summary

∃ x1 (x0 ⊆ x1 → sum(x0) ≈ sum(x0 ∩ x1))

• ∃ l00, l01, l10, l11 ∃ s00, s01, s10, s11

l0 ≈ l00 + l01 ∧ l1 ≈ l10 + l11 ∧ s0 ≈ s00 + s01 ∧ s1 ≈ s10 + s11 ∧ (l00 ≈ 0 → s00 ≈ 0) ∧ . . . ∧ (l11 ≈ 0 → s11 ≈ 0) ∧ (l01 ≈ 1 ∧ l00 ≈ 1) → (s01 ≈ s00) ∧ . . . ∧ (l11 ≈ 1 ∧ l10 ≈ 1) → (s11 ≈ s10) ∧ (l10 ≈ 0 → s10 + s11 ≈ s11)

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(Counter-)Example

Transformation

(card(x) ≤ 5 ∧ sum(x) ≤ 5)

• (l0 ≈ 0 → s0 ≈ 0) ∧ (l1 ≈ 0 → s1 ≈ 0) ∧

(l0 ≈ 1 ∧ l1 ≈ 1) → (s0 ≈ s1) ∧ (l1 ≤ 5 ∧ s1 ≤ 5)

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(Counter-)Example

Transformation

(card(x) ≤ 5 ∧ sum(x) ≤ 5)

• (l0 ≈ 0 → s0 ≈ 0) ∧ (l1 ≈ 0 → s1 ≈ 0) ∧

(l0 ≈ 1 ∧ l1 ≈ 1) → (s0 ≈ s1) ∧ (l1 ≤ 5 ∧ s1 ≤ 5) Elements of sets are natural numbers (F = N) l0 → 0 s0 → 0 l1 → 5 s1 → 5

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(Counter-)Example

Transformation

(card(x) ≤ 5 ∧ sum(x) ≤ 5)

• (l0 ≈ 0 → s0 ≈ 0) ∧ (l1 ≈ 0 → s1 ≈ 0) ∧

(l0 ≈ 1 ∧ l1 ≈ 1) → (s0 ≈ s1) ∧ (l1 ≤ 5 ∧ s1 ≤ 5) Elements of sets are natural numbers (F = N) l0 → 0 s0 → 0 l1 → 5 s1 → 5 Cannot be extended!

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Sum Constructive Structures

Definition (sum constructive)

A structure is called sum constructive if and only if its element support F has the following property: For all c ∈ F there exist infinitely many a, b ∈ F, such that a = b and a + b = c.

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Sum Constructive Structures

Definition (sum constructive)

A structure is called sum constructive if and only if its element support F has the following property: For all c ∈ F there exist infinitely many a, b ∈ F, such that a = b and a + b = c. Sum constructive : Z, Q, R, . . . Not sum constructive : N, R+, any finite set, . . .

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(Counter-)Example

Formula

(l0 ≈ 0 → s0 ≈ 0) ∧ (l1 ≈ 0 → s1 ≈ 0) ∧ (l0 ≈ 1 ∧ l1 ≈ 1) → (s0 ≈ s1) ∧ (l1 ≤ 5 ∧ s1 ≤ 5) Elements of sets are integers (F = Z) l0 → 0 s0 → 0 l1 → 5 s1 → 5

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(Counter-)Example

Formula

(l0 ≈ 0 → s0 ≈ 0) ∧ (l1 ≈ 0 → s1 ≈ 0) ∧ (l0 ≈ 1 ∧ l1 ≈ 1) → (s0 ≈ s1) ∧ (l1 ≤ 5 ∧ s1 ≤ 5) Elements of sets are integers (F = Z) l0 → 0 s0 → 0 l1 → 5 s1 → 5 Can be extended!

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(Counter-)Example

Formula

(l0 ≈ 0 → s0 ≈ 0) ∧ (l1 ≈ 0 → s1 ≈ 0) ∧ (l0 ≈ 1 ∧ l1 ≈ 1) → (s0 ≈ s1) ∧ (l1 ≤ 5 ∧ s1 ≤ 5) Elements of sets are integers (F = Z) l0 → 0 s0 → 0 l1 → 5 s1 → 5 Can be extended! x → ∅ x → {5, 1, −1, 2, −2}

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Properties

Theorem (Decision Procedure)

If we are considering sum constructive structures and we have a decision procedure for the resulting arithmetical fragment, we have a decision procedure for BAPAS.

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Properties

Theorem (Decision Procedure)

If we are considering sum constructive structures and we have a decision procedure for the resulting arithmetical fragment, we have a decision procedure for BAPAS.

Complexity

Let F be the input formula.

◮ Size of result is bounded by O(22size(F)) ◮ Time for transformation bounded by O(22size(F)) ◮ Number of quantifier alternations does not change

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Conclusion

◮ Introduced BAPAS as extension of BAPA ◮ Presented a transformation from BAPAS to pure arithmetic ◮ Decision procedure for BAPAS3 ◮ Properties are proven ◮ Implementation is ongoing

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Outlook

◮ min and max as additional bridging function ◮ Developement of a general approach for new bridging functions ◮ Consider intervals instead of sets and try to extend the developed

methods

◮ Consider if the developed methods can be extend for reasoning with

the duration calculus

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The constant U

◮ UA ∈ Pf (F), ◮ MAXC := card(U) ◮ MAXS := sum(U) ◮ MAXCA = cardA(UA) := |UA| ◮ MAXSA = sumA(UA) := e∈UA

e

◮ ∀ x p(x) true iff pA(o) true for all o ∈ P(UA) ◮ ∃ x p(x) true iff pA(o) true for at least one o ∈ P(UA) ◮ o ∈ Pf (F), then ∁Ao := {e | e ∈ UA and e ∈ o}.

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Duration Calculus

◮ Temporal logic ◮ Zhou Chaochen, C. A. R. Hoare, and Anders P. Ravn (1991) ◮ Talks about intervals ◮ Example: (l ≥ 60) → (

• Leak ≤ 0.05 ∗ l)

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