A Tableau System for Right Propositional Neighborhood Logic over - PowerPoint PPT Presentation

A Tableau System for Right Propositional Neighborhood Logic over Finite Linear Orders: an Implementation TABLEAUX-2013 Davide Bresolin, Dario Della Monica, Angelo Montanari and Guido Sciavicco University of Verona - Italy, Reykjavik University - Iceland University of Udine - Italy, University of Murcia - Spain

Presentation This is an implementation and experimental work, no new theoretical results here.

Presentation This is an implementation and experimental work, no new theoretical results here. Nevertheless, some experimental work, such as this one, are not free of unexpected problems and difficulties.

Presentation This is an implementation and experimental work, no new theoretical results here. Nevertheless, some experimental work, such as this one, are not free of unexpected problems and difficulties. Moreover, this implementation, although particularly simple, is the only one of its kind.

Time and logics In AI usually time is formalized with languages (logics) that are: point-based, or interval-based.

Time and logics In AI usually time is formalized with languages (logics) that are: point-based, or interval-based. In point-based temporal logics, formulas are interpreted directly over points. In interval-based ones, they are interpreted over intervals (our case). We are interested only in qualitative relationships between intervals, and truth of a formula over an interval does not come from nor influences the truth of the same formula over sub-intervals.

Time and logics In AI usually time is formalized with languages (logics) that are: point-based, or interval-based. In point-based temporal logics, formulas are interpreted directly over points. In interval-based ones, they are interpreted over intervals (our case). We are interested only in qualitative relationships between intervals, and truth of a formula over an interval does not come from nor influences the truth of the same formula over sub-intervals. It is well-known that interval-based logics are much more difficult to deal with.

What is an interval? Definition Given a linear order D = � D , < � : an interval in D is a pair [ d 0 , d 1 ] such that d 0 < d 1 ; I ( D ) is the set of all (strict) intervals on D ; � D , I ( D ) � is an interval structure.

What is an interval? Definition Given a linear order D = � D , < � : an interval in D is a pair [ d 0 , d 1 ] such that d 0 < d 1 ; I ( D ) is the set of all (strict) intervals on D ; � D , I ( D ) � is an interval structure. We consider intervals as pairs of time points. A point d ∈ D belongs to [ d 0 , d 1 ] if d 0 ≤ d ≤ d 1 . Sometimes, the non-strict semantics, where intervals may have coincident endpoints, is considered; nowadays, the common choice is to exclude this possibility, treating points as a different sort, and giving rise to more complex point-interval logics.

Allen’s binary relations There are 13 different binary relations between intervals:

Allen’s binary relations There are 13 different binary relations between intervals: later

Allen’s binary relations There are 13 different binary relations between intervals: later after/meets

Allen’s binary relations There are 13 different binary relations between intervals: later after/meets overlaps

Allen’s binary relations There are 13 different binary relations between intervals: later after/meets overlaps ends/finishes

Allen’s binary relations There are 13 different binary relations between intervals: later after/meets overlaps ends/finishes during

Allen’s binary relations There are 13 different binary relations between intervals: later after/meets overlaps ends/finishes during begins/starts

Allen’s binary relations There are 13 different binary relations between intervals: later � L � after/meets � A � overlaps � O � ends/finishes � E � during � D � begins/starts � B �

Allen’s binary relations There are 13 different binary relations between intervals: later � L � � L � � A � � A � after/meets overlaps � O � � O � ends/finishes � E � � E � during � D � � D � begins/starts � B � � B � together with their inverses.

Allen’s binary relations There are 13 different binary relations between intervals: later � L � � L � � A � � A � after/meets overlaps � O � � O � ends/finishes � E � � E � during � D � � D � begins/starts � B � � B � together with their inverses. Between points we have only three binary ordering relations!

Halpern-Shoham’s modal logic of interval relations Every interval relation gives rise to a modal operator over interval structures.

Halpern-Shoham’s modal logic of interval relations Every interval relation gives rise to a modal operator over interval structures. Thus, a multimodal logic arises: Syntax of Halpern-Shoham’s logic, hereafter called HS : φ ::= p | ¬ φ | φ ∧ ψ | � B � φ | � E � φ | � B � φ | � E � φ | � A � φ | � A � φ.

Halpern-Shoham’s modal logic of interval relations Every interval relation gives rise to a modal operator over interval structures. Thus, a multimodal logic arises: Syntax of Halpern-Shoham’s logic, hereafter called HS : φ ::= p | ¬ φ | φ ∧ ψ | � B � φ | � E � φ | � B � φ | � E � φ | � A � φ | � A � φ. Interval model: M = � I ( D ) , V � , where V : AP �→ 2 I ( D ) .

Formal semantics of HS � B � : M , [ d 0 , d 1 ] � � B � φ iff there exists d 2 such that d 0 ≤ d 2 < d 1 and M , [ d 0 , d 2 ] � φ . � B � : M , [ d 0 , d 1 ] � � B � φ iff there exists d 2 such that d 1 < d 2 and M , [ d 0 , d 2 ] � φ . current interval: φ � B � φ : φ � B � φ :

Formal semantics of HS � B � : M , [ d 0 , d 1 ] � � B � φ iff there exists d 2 such that d 0 ≤ d 2 < d 1 and M , [ d 0 , d 2 ] � φ . � B � : M , [ d 0 , d 1 ] � � B � φ iff there exists d 2 such that d 1 < d 2 and M , [ d 0 , d 2 ] � φ . � E � : M , [ d 0 , d 1 ] � � E � φ iff there exists d 2 such that d 0 < d 2 ≤ d 1 and M , [ d 2 , d 1 ] � φ . � E � : M , [ d 0 , d 1 ] � � E � φ iff there exists d 2 such that d 2 < d 0 and M , [ d 2 , d 1 ] � φ . current interval: φ � E � φ : φ � E � φ :

Formal semantics of HS � B � : M , [ d 0 , d 1 ] � � B � φ iff there exists d 2 such that d 0 ≤ d 2 < d 1 and M , [ d 0 , d 2 ] � φ . � B � : M , [ d 0 , d 1 ] � � B � φ iff there exists d 2 such that d 1 < d 2 and M , [ d 0 , d 2 ] � φ . � E � : M , [ d 0 , d 1 ] � � E � φ iff there exists d 2 such that d 0 < d 2 ≤ d 1 and M , [ d 2 , d 1 ] � φ . � E � : M , [ d 0 , d 1 ] � � E � φ iff there exists d 2 such that d 2 < d 0 and M , [ d 2 , d 1 ] � φ . � A � : M , [ d 0 , d 1 ] � � A � φ iff there exists d 2 such that d 1 < d 2 and M , [ d 1 , d 2 ] � φ . � A � : M , [ d 0 , d 1 ] � � A � φ iff there exists d 2 such that d 2 < d 0 and M , [ d 2 , d 0 ] � φ . current interval: φ � A � φ : φ � A � φ :

Formal semantics of HS - contd’ � L � : M , [ d 0 , d 1 ] � � L � φ iff there exists d 2 , d 3 such that d 1 < d 2 < d 3 and M , [ d 2 , d 3 ] � φ . � L � : M , [ d 0 , d 1 ] � � L � φ iff there exists d 2 , d 3 such that d 2 < d 3 < d 0 and M , [ d 2 , d 3 ] � φ . current interval: φ � L � φ : φ � L � φ :

Formal semantics of HS - contd’ � L � : M , [ d 0 , d 1 ] � � L � φ iff there exists d 2 , d 3 such that d 1 < d 2 < d 3 and M , [ d 2 , d 3 ] � φ . � L � : M , [ d 0 , d 1 ] � � L � φ iff there exists d 2 , d 3 such that d 2 < d 3 < d 0 and M , [ d 2 , d 3 ] � φ . � D � : M , [ d 0 , d 1 ] � � D � φ iff there exists d 2 , d 3 such that d 0 < d 2 < d 3 < d 1 and M , [ d 2 , d 3 ] � φ . � D � : M , [ d 0 , d 1 ] � � D � φ iff there exists d 2 , d 3 such that d 2 < d 0 < d 1 < d 3 and M , [ d 2 , d 3 ] � φ . current interval: φ � D � φ : φ � D � φ :

Formal semantics of HS - contd’ � L � : M , [ d 0 , d 1 ] � � L � φ iff there exists d 2 , d 3 such that d 1 < d 2 < d 3 and M , [ d 2 , d 3 ] � φ . � L � : M , [ d 0 , d 1 ] � � L � φ iff there exists d 2 , d 3 such that d 2 < d 3 < d 0 and M , [ d 2 , d 3 ] � φ . � D � : M , [ d 0 , d 1 ] � � D � φ iff there exists d 2 , d 3 such that d 0 < d 2 < d 3 < d 1 and M , [ d 2 , d 3 ] � φ . � D � : M , [ d 0 , d 1 ] � � D � φ iff there exists d 2 , d 3 such that d 2 < d 0 < d 1 < d 3 and M , [ d 2 , d 3 ] � φ . � O � : M , [ d 0 , d 1 ] � � O � φ iff there exists d 2 , d 3 such that d 0 < d 2 < d 1 < d 3 and M , [ d 2 , d 3 ] � φ . � O � : M , [ d 0 , d 1 ] � � O � φ iff there exists d 2 , d 3 such that d 2 < d 0 < d 3 < d 1 and M , [ d 2 , d 3 ] � φ . current interval: φ � O � φ : φ � O � φ :

The zoo of fragments of HS Technically, there are 2 12 = 4096 fragments of HS.