Automated Reasoning: 10th International Joint Conference, IJCAR 2020, Paris, France, July 1–4, 2020, Proceedings, Part II

This paper discusses the design of a hierarchy of structures which combine linear algebra with concepts related to limits, like topology and norms, in dependent type theory. This hierarchy is the backbone of a new library of formalized classical analysis, for the Coq proof assistant. It extends the Mathematical Components library, geared towards algebra, with topics in analysis. Issues of a more general nature related to the inheritance of poorer structures from richer ones arise due to this combination. We present and discuss a solution, coined forgetful inheritance, based on packed classes and unification hints.


Tableau Algorithm
I takes as input an ALC concept C in NNF I decides the satisfiability of C with an empty TBox This algorithm is used: I in practice: is implemented by state-of-the-art DL reasoners I for other problems reducible to satisfiability I as well with a non-empty TBox, modulo some adaptations (see later).

Tableau Algorithm: Principle
To determine whether an NNF ALC concept C is satisfiable, we try to construct a model of C : I if we succeed, we have shown that C is satisfiable; Tableau Algorithm: High Level View I initialize a set S of ABoxes, containing a single ABox {C (a 0 )} I at each stage, apply a rule to some A 2 S (See rules on the next slide) I a rule application replaces A by one or two ABoxes I stop applying rules when either: 1. every A 2 S contains a clash, that is an assertion ?(a i ) or two atoms C (a i ) and ¬C (a i ) 2. some A 2 S is clash-free and complete, meaning that no rule can be applied to A Tableau Algorithm, Expansion Rules u-rule: if (C 1 u C 2 )(a) 2 A and {C 1 (a), C 2 (a)} 6 ✓ A replace A with A [ {C 1 (a), C 2 (a)}.

Automated Reasoning
Tableau Algorithm, an Example

Automated Reasoning
Tableau Algorithm, an Example

Automated Reasoning
Tableau Algorithm, an Example

Properties of the Tableau Algorithm
Let us call our tableau algorithm CSat (for concept satisfiability).
To show that CSat is a decision procedure, we must show: I Termination: CSat always terminates I Soundness: if Csat outputs "yes" on input C 0 , the concept C 0 is satisfiable I Completeness: if C 0 is satisfiable, then CSat outputs "yes" on input C 0

Subconcepts (Example)
The subconcepts of For instance, the role depth of is 2.

Graphical Representation of an ABox
We can represent the ABox {R(a, b), S(b, c), S(a, d)} by the following graph: Note that the above representation is tree shaped (and we will say that the corresponding ABox is tree-shaped).

Termination of CSat
Suppose we run CSat starting from S = {{C (a 0 )}}. Let us make the following observations for every ABox A generated by CSat: 2. the set of role assertions in A forms a tree 3. if D(b) 2 A and the unique path from a 0 to b has length k, then depth(D)  depth(C ) k.
I can you give an example where depth(D) < depth(C ) k?
4. for every individual b in A, there are at most |C | individuals c such that R(b, c) 2 A for some R (at most one per existential concept)

Recall of the Expansion Rules
Automated Reasoning

Termination of CSat
We have that: 1. there is a bound on the size of generated ABoxes

CSat only adds assertions to the ABoxes
Any generated ABox will be either complete or contain a clash. Hence CSat terminates.

Soundness of CSat
Let us assume that CSat returns "yes" on input C . Then S must contain a complete and clash-free ABox A.
We define an interpretation I as follows: To show the claim, we prove by induction on the size of concepts that: Automated Reasoning

Soundness of CSat
Automated Reasoning

Completeness of CSat
Suppose that C is satisfiable. This implies that the ABox {C (a)} is satisfiable. Note that the tableau rules are satisfiability-preserving: I if an ABox A is satisfiable and A 0 is the result of applying a rule to A, then A 0 is also satisfiable I if an ABox A is satisfiable and A 1 and A 2 are obtained when applying a rule to A, then either A 1 or A 2 is satisfiable.
We conclude by noticing that an ABox containing a clash is not satisfiable.
Consider the following concept: CSat runs in exponential time and space on this concept. Why? Let's have a look at what happens for small n: I n = 0: 9R.B u 9R.¬B

Complexity of CSat
Consider the following concept: CSat runs in exponential time and space on this concept. Why? Let's have a look at what happens for small n: I n = 0: 9R.B u 9R.¬B

Complexity of CSat
Consider the following concept: CSat runs in exponential time and space on this concept. Why? Let's have a look at what happens for small n: I n = 0: 9R.B u 9R.¬B

Complexity of CSat
Consider the following concept: CSat runs in exponential time and space on this concept. Why? Let's have a look at what happens for small n: I n = 0: 9R.B u 9R.¬B

EL Normal Form
An EL TBox T is in normal form if all its concept inclusions are of one of the following shape: where A, A 1 , A 2 and B are atomic concept names, and R is an atomic role name.

Normalization
The following rules allow to normalize an EL TBox: where C , D, E denote arbitrary EL concepts,Ĉ andD denote EL concepts that are not atomic concepts or >, B an atomic concept, and A a fresh atomic concept.

Automated Reasoning
Normalization -Example Normalize the following concept inclusion: 9R.C u D v 9S.9R.C