An Algebraic Study
Bart Bogaerts, Marc Denecker, Joost Vennekens
- Complete lattice : partially ordered set in which each set S has a greatest lower bound and a least upper bound
- Lattice operator
- Approximation space:
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
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)
- Common concept in mathematics
- Well-understood ...
- ... or not?
- 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)
- Our claim: the induction process is central to our understanding of inductive definitions
- We develop a theory of this induction process
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
The induction process:
Natural inductionsIf 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
If O is a lattice operator, an O-induction is a sequence of lattice elements (xi) such that:
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?
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
- 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?
- Intuition: only derive something it remains derivable.
- Formally, a derivation is safe if: for each O-induction (yj)j≤β in xi:
- 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)
- 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)
- In general, the problem "is safe(O) ≥ x" is co-NP hard and in
(see paper for precise definitions)
- 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
"I (an introspective agent) only know the following:"
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
- 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
Safe Inductions: An Algebraic Study