# Safe Inductions: An Algebraic Study

Bart Bogaerts, Marc Denecker, Joost Vennekens
(KU Leuven)

# Preliminaries

## Preliminaries

• Complete lattice            : partially ordered set in which each set S has a greatest lower bound         and a least upper bound
• Lattice operator
• Approximation space:
• Approximator
\langle L,\leq\rangle
$\langle L,\leq\rangle$
\bigwedge S
$\bigwedge S$
\bigvee S
$\bigvee S$
O:L\to L
$O:L\to L$
\langle L^2,\leq, \leq_p \rangle
$\langle L^2,\leq, \leq_p \rangle$
A:L^2\to L^2
$A:L^2\to L^2$

Approximation Fixpoint Theory defines various fixpoints:

• (partial) grounded fixpoints of O
• the A-well-founded fixpoint of O
• the A-Kripke-Kleene fixpoint of O
• (partial) A-stable fixpoints of O

# Context: ​Inductive Definitions

The Well-Founded Semantics Is the Principle of Inductive Definition, Revisited. M.Denecker and J.Vennekens (KR 2014)

A Logical Study of Some Common Principles of Inductive Definition M. Denecker, B. Bogaerts and J. Vennekens (Under Review)

## Inductive definitions

• Common concept in mathematics
• Well-understood ...
• ... or not?

## Inductive Definitions

• What are the implicit conventions underlying IDs in mathematics?
• Can the (informal) semantics of IDs be formalized

These questions are (partially) answered in the cited papers

We generalize to an algebraic setting (approximation fixpoint theory)

## Inductive definitions

• Our claim: the induction process is central to our understanding of inductive definitions
• We develop a theory of this induction process

## Example

The transitive closure R of a graph G is defined as follows:

• If (x,y) is in G, then (x,y) is in R
• If (x,z) is in G and (z,y) in R, then (x,y) is in R
\begin{array}{l} \forall x, y: R(x,y) \leftarrow G(x,y).\\ \forall x,y: R(x,y) \leftarrow G(x,z)\wedge R(z,y). \end{array}
$\begin{array}{l} \forall x, y: R(x,y) \leftarrow G(x,y).\\ \forall x,y: R(x,y) \leftarrow G(x,z)\wedge R(z,y). \end{array}$

## Example

The induction process:

\begin{array}{l} \forall x, y: R(x,y) \leftarrow G(x,y).\\ \forall x,y: R(x,y) \leftarrow G(x,z)\wedge R(z,y). \end{array}
$\begin{array}{l} \forall x, y: R(x,y) \leftarrow G(x,y).\\ \forall x,y: R(x,y) \leftarrow G(x,z)\wedge R(z,y). \end{array}$
G=\{(a,b),(b,c),(a,d),(d,e)\}
$G=\{(a,b),(b,c),(a,d),(d,e)\}$
R=\emptyset
$R=\emptyset$
R=\{(a,b)\}
$R=\{(a,b)\}$
R=\{(a,b),(b,c),(a,d),(d,e)\}
$R=\{(a,b),(b,c),(a,d),(d,e)\}$
R=\{(a,b),(b,c),(a,d),(d,e), (a,c),(a,e)\}
$R=\{(a,b),(b,c),(a,d),(d,e), (a,c),(a,e)\}$

## Natural inductions

If D is a definition, a natural induction is a sequence (I i) of interpretations such that
• I 0 = ∅
• I i+1=I i∪ A with A a set of atoms such that for each a∈A There is a rule r in D whose body is satisfied in I i with head a

## O-inductions

If O is a lattice operator, an O-induction is a sequence of lattice elements (xi) such that:

\begin{array}{l} x_0=\bot\\ x_i\leq x_{i+1} \leq O(x_i)\lor x_i \end{array}
$\begin{array}{l} x_0=\bot\\ x_i\leq x_{i+1} \leq O(x_i)\lor x_i \end{array}$

## Monotone operators

If O is monotone, all O-inductions converge to lfp(O)

• They uniquely determine a (good) fixpoint of interest
• What if O is non-monotone?

## Example

The even numbers are defined as follows:

• 0 is an even number
• n+1 is an even number if n is not an even number
\begin{array}{l} Even(0). \\ \forall x: Even(x+1) \leftarrow \neg Even(x). \end{array}
$\begin{array}{l} Even(0). \\ \forall x: Even(x+1) \leftarrow \neg Even(x). \end{array}$
\begin{array}{l} O: 2^\mathbb{N} \to 2^\mathbb{N}: S \mapsto\{0\}\cup\{x+1\mid x\notin S\} \end{array}
$\begin{array}{l} O: 2^\mathbb{N} \to 2^\mathbb{N}: S \mapsto\{0\}\cup\{x+1\mid x\notin S\} \end{array}$

## Example

\begin{array}{l} O: 2^\mathbb{N} \to 2^\mathbb{N}: S \mapsto\{0\}\cup\{x+1\mid x\notin S\} \end{array}
$\begin{array}{l} O: 2^\mathbb{N} \to 2^\mathbb{N}: S \mapsto\{0\}\cup\{x+1\mid x\notin S\} \end{array}$
\emptyset \to \{0\} \to \{0,2\}\to \{0,2,4\}\to \cdots
$\emptyset \to \{0\} \to \{0,2\}\to \{0,2,4\}\to \cdots$
\emptyset \to \{1\} \to \{0,1\}\to \{0,1,3\}\to \cdots
$\emptyset \to \{1\} \to \{0,1\}\to \{0,1,3\}\to \cdots$
\emptyset \to \mathbb{N}
$\emptyset \to \mathbb{N}$

## Non-monotone operator

• No more convergence guaranteed
• Problem? Some derivations happen too soon.
• Before it is safe to derive them.
• E.g., in the previous, Even(1) is derived based on the absence of Even(0) before the latter is "fixed"
• I.e., Even(1) does not remain derivable
• Solution? Only derive facts when it is safe to do so. How to formalize this?

## Safety

• Intuition: only derive something it remains derivable.
• Formally, a derivation is safe if: for each O-induction (yj)j≤β in xi:
x_{i+1}\leq x_i \lor O(y_\beta)
$x_{i+1}\leq x_i \lor O(y_\beta)$

## Example

\begin{array}{l} O: 2^\mathbb{N} \to 2^\mathbb{N}: S \mapsto\{0\}\cup\{x+1\mid x\notin S\} \end{array}
$\begin{array}{l} O: 2^\mathbb{N} \to 2^\mathbb{N}: S \mapsto\{0\}\cup\{x+1\mid x\notin S\} \end{array}$
\emptyset \to \{1\} \to \{0,1\}\to \{0,1,3\}\to \cdots
$\emptyset \to \{1\} \to \{0,1\}\to \{0,1,3\}\to \cdots$
$\emptyset \to \{0\} \qquad$
\{1\}\nsubseteq \emptyset \cup O(\{0\}) = \mathbb{N}\setminus \{1\}
$\{1\}\nsubseteq \emptyset \cup O(\{0\}) = \mathbb{N}\setminus \{1\}$
\text{for each induction }(y_i)_{i\leq\beta}\text{ in }x_i: x_{i+1}\leq x_i \lor O(y_\beta)
$\text{for each induction }(y_i)_{i\leq\beta}\text{ in }x_i: x_{i+1}\leq x_i \lor O(y_\beta)$

# Properties

## Properties

• All safe O-inductions converge to a single lattice point.
• We denote it safe(O)
• If O is monotone, safe(O)=lfp(O)
• If O is anti-monotone, safe(O)=lfp(O2)

## Properties

• The Kripke-Kleene fixpoint approximtes safe(O)
• The well-founded fixpoint approximtes safe(O)
• If (xi,yi) is a well-founded induction, then (xi) is a safe O-induction.
• Well-founded inductions provide a cheap (polynomial) approximation of safe(O)

## Complexity

• In general, the problem "is safe(O) ≥ x" is co-NP hard and in
(see paper for precise definitions)
\Delta^P_2
$\Delta^P_2$

## Applications

• Logic programming (this paper)
• Abstract argumentation (extended version)
• Autoepistemic logic (this paper)
• Here, safe inductions solve a known problem with stratification
• Default logic
• Active integrity constraints (thanks to previous paper)

# Application: Autoepistemic Logic

## Autoepistemic Logic

"I (an introspective agent) only know the following:"

\begin{array}{ll} q \Leftrightarrow \neg Kp\\ r \Leftrightarrow \neg Kq \end{array}
$\begin{array}{ll} q \Leftrightarrow \neg Kp\\ r \Leftrightarrow \neg Kq \end{array}$

The safely defined semantics respects stratification

# Application: Abstract Argumentation

## Dung's argumentation frameworks

• Argumentation framework Θ=⟨A,R⟩
• A is a set of arguments
• R is an attack relation
• Two common operators:
•
•
• The following coincide:
• The grounded extension of Θ
•
•
F_\Theta: 2^A\to 2^A: S \mapsto \{a \in A \mid S \text{ defends } a\}
$F_\Theta: 2^A\to 2^A: S \mapsto \{a \in A \mid S \text{ defends } a\}$
U_\Theta: 2^A\to 2^A: S \mapsto \{a \in A \mid S \text{ does not attack } a\}
$U_\Theta: 2^A\to 2^A: S \mapsto \{a \in A \mid S \text{ does not attack } a\}$
safe(U_\Theta)
$safe(U_\Theta)$
safe(F_\Theta)
$safe(F_\Theta)$

## Conclusion

• Lift safe inductions to algebraic setting
• Study relationship with existing types of fixpoints
• Use them to solve a problem in autoepistemic logic
• Pave the way to porting them to other fields too

By krr

IJCAI'17

• 750