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
⟨L,≤⟩\langle L,\leq\rangle
\bigwedge S
⋀S\bigwedge S
\bigvee S
⋁S\bigvee S
O:L\to L
O:L→LO:L\to L
\langle L^2,\leq, \leq_p \rangle
⟨L2,≤,≤p⟩\langle L^2,\leq, \leq_p \rangle
A:L^2\to L^2
A:L2→L2A: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}
∀x,y:R(x,y)←G(x,y).∀x,y:R(x,y)←G(x,z)∧R(z,y).\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}
∀x,y:R(x,y)←G(x,y).∀x,y:R(x,y)←G(x,z)∧R(z,y).\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)}G=\{(a,b),(b,c),(a,d),(d,e)\}
R=\emptyset
R=∅R=\emptyset
R=\{(a,b)\}
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)\}
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)}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}
x0=⊥xi≤xi+1≤O(xi)∨xi\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}
Even(0).∀x:Even(x+1)←¬Even(x).\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}
O:2N→2N:S↦{0}∪{x+1∣x∉S}\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}
O:2N→2N:S↦{0}∪{x+1∣x∉S}\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
∅→{0}→{0,2}→{0,2,4}→⋯\emptyset \to \{0\} \to \{0,2\}\to \{0,2,4\}\to \cdots
\emptyset \to \{1\} \to \{0,1\}\to \{0,1,3\}\to \cdots
∅→{1}→{0,1}→{0,1,3}→⋯\emptyset \to \{1\} \to \{0,1\}\to \{0,1,3\}\to \cdots
\emptyset \to \mathbb{N}
∅→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)
xi+1≤xi∨O(yβ)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}
O:2N→2N:S↦{0}∪{x+1∣x∉S}\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
∅→{1}→{0,1}→{0,1,3}→⋯\emptyset \to \{1\} \to \{0,1\}\to \{0,1,3\}\to \cdots
\emptyset \to \{0\} \qquad
∅→{0}\emptyset \to \{0\} \qquad
\{1\}\nsubseteq \emptyset \cup O(\{0\}) = \mathbb{N}\setminus \{1\}
{1}⊈∅∪O({0})=N∖{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)
for each induction (yi)i≤β in xi:xi+1≤xi∨O(yβ)\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
Δ2P\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}
q⇔¬Kpr⇔¬Kq\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\}
FΘ:2A→2A:S↦{a∈A∣S 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\}
UΘ:2A→2A:S↦{a∈A∣S 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(UΘ)safe(U_\Theta)
safe(F_\Theta)
safe(FΘ)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
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