Learning Dynamics with Synchronous, Asynchronous and General - PowerPoint PPT Presentation

Formalization Multi-valued Logic ( M VL ) Definition (Atoms) Let V = { v 1 , . . . , v n } be a finite set of n ∈ N variables, and dom : V → N The atoms of M VL are of the form v val where v ∈ V and val ∈ � 0; dom( v ) � . The set of such atoms is denoted by A V dom or simply A . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 10 / 33

Formalization Multi-valued Logic ( M VL ) Definition (Atoms) Let V = { v 1 , . . . , v n } be a finite set of n ∈ N variables, and dom : V → N The atoms of M VL are of the form v val where v ∈ V and val ∈ � 0; dom( v ) � . The set of such atoms is denoted by A V dom or simply A . Definition (Rules) A M VL rule R is defined by: v val 0 ← v val 1 ∧ · · · ∧ v val m = (1) R 0 1 m where ∀ i ∈ � 0; m � , v val i ∈ A are atoms in M VL . i Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 10 / 33

Formalization Multi-valued Logic ( M VL ) Definition (Atoms) Let V = { v 1 , . . . , v n } be a finite set of n ∈ N variables, and dom : V → N The atoms of M VL are of the form v val where v ∈ V and val ∈ � 0; dom( v ) � . The set of such atoms is denoted by A V dom or simply A . Definition (Rules) A M VL rule R is defined by: v val 0 ← v val 1 ∧ · · · ∧ v val m = (1) R 0 1 m where ∀ i ∈ � 0; m � , v val i ∈ A are atoms in M VL . i Left-hand side is called the head of R and is denoted h ( R ) := v val 0 . 0 Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 10 / 33

Formalization Multi-valued Logic ( M VL ) Definition (Atoms) Let V = { v 1 , . . . , v n } be a finite set of n ∈ N variables, and dom : V → N The atoms of M VL are of the form v val where v ∈ V and val ∈ � 0; dom( v ) � . The set of such atoms is denoted by A V dom or simply A . Definition (Rules) A M VL rule R is defined by: v val 0 ← v val 1 ∧ · · · ∧ v val m = (1) R 0 1 m where ∀ i ∈ � 0; m � , v val i ∈ A are atoms in M VL . i var ( h ( R )) := v 0 denotes the variable that occurs in h ( R ). Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 10 / 33

Formalization Multi-valued Logic ( M VL ) Definition (Atoms) Let V = { v 1 , . . . , v n } be a finite set of n ∈ N variables, and dom : V → N The atoms of M VL are of the form v val where v ∈ V and val ∈ � 0; dom( v ) � . The set of such atoms is denoted by A V dom or simply A . Definition (Rules) A M VL rule R is defined by: v val 0 ← v val 1 ∧ · · · ∧ v val m = (1) R 0 1 m where ∀ i ∈ � 0; m � , v val i ∈ A are atoms in M VL . i Right-hand side is called the body of R , written b ( R ) := { v val 1 , . . . , v val m } 1 m Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 10 / 33

Formalization Multi-valued Logic ( M VL ) Definition (Atoms) Let V = { v 1 , . . . , v n } be a finite set of n ∈ N variables, and dom : V → N The atoms of M VL are of the form v val where v ∈ V and val ∈ � 0; dom( v ) � . The set of such atoms is denoted by A V dom or simply A . Definition (Rules) A M VL rule R is defined by: v val 0 ← v val 1 ∧ · · · ∧ v val m = (1) R 0 1 m where ∀ i ∈ � 0; m � , v val i ∈ A are atoms in M VL . i A multi-valued logic program ( M VLP ) is a set of M VL rules. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 10 / 33

Formalization Rules Properties Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 11 / 33

Formalization Rules Properties Definition (Rule Domination) Let R 1 , R 2 be two M VL rules. R 1 dominates R 2 , written R 2 ≤ R 1 if h ( R 1 ) = h ( R 2 ) and b ( R 1 ) ⊆ b ( R 2 ). Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 11 / 33

Formalization Rules Properties Definition (Rule Domination) Let R 1 , R 2 be two M VL rules. R 1 dominates R 2 , written R 2 ≤ R 1 if h ( R 1 ) = h ( R 2 ) and b ( R 1 ) ⊆ b ( R 2 ). Proposition (Double domination is equality) If R 1 ≤ R 2 and R 2 ≤ R 1 then R 1 = R 2 . Rules with the most general bodies dominate the other rules. These are the rules we want since they cover the most general cases. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 11 / 33

Formalization Dynamical System Modeling Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 12 / 33

Formalization Dynamical System Modeling Definition (Discrete State) A discrete state s is a function from V to N , i.e., it associates an integer value to each variable in V . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 12 / 33

Formalization Dynamical System Modeling Definition (Discrete State) A discrete state s is a function from V to N , i.e., it associates an integer value to each variable in V . It can be equivalently represented by the set of atoms { v s ( v ) | v ∈ V} and thus we can use classical set operations on it. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 12 / 33

Formalization Dynamical System Modeling Definition (Discrete State) A discrete state s is a function from V to N , i.e., it associates an integer value to each variable in V . It can be equivalently represented by the set of atoms { v s ( v ) | v ∈ V} and thus we can use classical set operations on it. Definition (Transitions) We write S to denote the set of all discrete states, and a couple of states ( s , s ′ ) ∈ S 2 is called a transition. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 12 / 33

Formalization Dynamical System Modeling Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 13 / 33

Formalization Dynamical System Modeling Definition (Rule-state matching) Let s ∈ S . The M VL rule R matches s , written R ⊓ s , if b ( R ) ⊆ s . When matching a state, a rule can be used to realize a transition. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 13 / 33

Formalization Dynamical System Modeling Definition (Rule-state matching) Let s ∈ S . The M VL rule R matches s , written R ⊓ s , if b ( R ) ⊆ s . When matching a state, a rule can be used to realize a transition. Definition (Rule realization) R A rule R realizes the transition ( s , s ′ ), written s → s ′ , if R ⊓ s , h ( R ) ∈ s ′ . − Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 13 / 33

Formalization Dynamical System Modeling Definition (Rule-state matching) Let s ∈ S . The M VL rule R matches s , written R ⊓ s , if b ( R ) ⊆ s . When matching a state, a rule can be used to realize a transition. Definition (Rule realization) R A rule R realizes the transition ( s , s ′ ), written s → s ′ , if R ⊓ s , h ( R ) ∈ s ′ . − Definition (Program realization) P A M VLP P realizes ( s , s ′ ), written s → s ′ , if − R → s ′ . ∀ v ∈ V , ∃ R ∈ P , var ( h ( R )) = v ∧ s − P P It realizes T ⊆ S 2 , written → T , if ∀ ( s , s ′ ) ∈ T , s → s ′ . ֒ − − Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 13 / 33

Formalization Desired Properties In the following, for all sets of transitions T ⊆ S 2 , we denote: fst ( T ) := { s ∈ S | ∃ ( s 1 , s 2 ) ∈ T , s 1 = s } . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 14 / 33

Formalization Desired Properties In the following, for all sets of transitions T ⊆ S 2 , we denote: fst ( T ) := { s ∈ S | ∃ ( s 1 , s 2 ) ∈ T , s 1 = s } . Definition (Conflicts) A M VL rule R conflicts with a set of transitions T ⊆ S 2 when � R ⊓ s ∧ ∀ ( s , s ′ ) ∈ T , h ( R ) / ∈ s ′ � ∃ s ∈ fst ( T ) , . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 14 / 33

Formalization Desired Properties In the following, for all sets of transitions T ⊆ S 2 , we denote: fst ( T ) := { s ∈ S | ∃ ( s 1 , s 2 ) ∈ T , s 1 = s } . Definition (Conflicts) A M VL rule R conflicts with a set of transitions T ⊆ S 2 when � R ⊓ s ∧ ∀ ( s , s ′ ) ∈ T , h ( R ) / ∈ s ′ � ∃ s ∈ fst ( T ) , . Definition (Concurrent rules) Two M VL rules R and R ′ are concurrent when R ⊓ R ′ ∧ var ( h ( R ))= var ( h ( R ′ )) ∧ h ( R ) � = h ( R ′ ). Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 14 / 33

Formalization Definition (Complete program) A M VLP P is complete if ∀ s ∈ S , ∀ v ∈ V , ∃ R ∈ P , R ⊓ s ∧ var ( h ( R )) = v . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 15 / 33

Formalization Definition (Complete program) A M VLP P is complete if ∀ s ∈ S , ∀ v ∈ V , ∃ R ∈ P , R ⊓ s ∧ var ( h ( R )) = v . A complete program realize atleast one transition for each state s ∈ S . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 15 / 33

Formalization Definition (Complete program) A M VLP P is complete if ∀ s ∈ S , ∀ v ∈ V , ∃ R ∈ P , R ⊓ s ∧ var ( h ( R )) = v . A complete program realize atleast one transition for each state s ∈ S . Definition (Consistent program) A M VLP P is consistent with a set of transitions T if P does not contains any rule R conflicting with T . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 15 / 33

Formalization Definition (Complete program) A M VLP P is complete if ∀ s ∈ S , ∀ v ∈ V , ∃ R ∈ P , R ⊓ s ∧ var ( h ( R )) = v . A complete program realize atleast one transition for each state s ∈ S . Definition (Consistent program) A M VLP P is consistent with a set of transitions T if P does not contains any rule R conflicting with T . Let s ∈ fst ( T ), a program consistent with T will only realize the transitions ( s , s ′ ) ∈ T . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 15 / 33

Formalization Optimal M VLP Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 16 / 33

Formalization Optimal M VLP Definition (Suitable program) Let T ⊆ S 2 . A M VLP P is suitable for T when: P is consistent with T , Cover no negative example P realizes T , Cover all positive example P is complete Cover all state space ∀ R not conflicting with T , ∃ R ′ ∈ P s.t. R ≤ R ′ . Cover all hypotheses Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 16 / 33

Formalization Optimal M VLP Definition (Suitable program) Let T ⊆ S 2 . A M VLP P is suitable for T when: P is consistent with T , Cover no negative example P realizes T , Cover all positive example P is complete Cover all state space ∀ R not conflicting with T , ∃ R ′ ∈ P s.t. R ≤ R ′ . Cover all hypotheses Definition (Optimal program) If in addition, ∀ R ∈ P , all the rules R ′ belonging to a M VLP suitable for T are such that R ≤ R ′ implies R ′ ≤ R then P is called optimal. An optimal program is the set of all rules that are not dominated by any consistent rules. Contains all minimal hypotheses Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 16 / 33

Formalization Optimal M VLP Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 17 / 33

Formalization Optimal M VLP Proposition Let T ⊆ S 2 . The M VLP optimal for T is unique and denoted P O ( T ) . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 17 / 33

Formalization Optimal M VLP Proposition Let T ⊆ S 2 . The M VLP optimal for T is unique and denoted P O ( T ) . Troll mode on: does it works if it is the empty set? :p Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 17 / 33

Formalization Optimal M VLP Proposition Let T ⊆ S 2 . The M VLP optimal for T is unique and denoted P O ( T ) . Troll mode on: does it works if it is the empty set? :p Proposition P O ( ∅ ) = { v val ← ∅ | v val ∈ A} . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 17 / 33

Formalization Optimal M VLP Proposition Let T ⊆ S 2 . The M VLP optimal for T is unique and denoted P O ( T ) . Troll mode on: does it works if it is the empty set? :p Proposition P O ( ∅ ) = { v val ← ∅ | v val ∈ A} . Yeah ! And this property is the starting point of the learning algorithm. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 17 / 33

Formalization Optimal M VLP Proposition Let T ⊆ S 2 . The M VLP optimal for T is unique and denoted P O ( T ) . Troll mode on: does it works if it is the empty set? :p Proposition P O ( ∅ ) = { v val ← ∅ | v val ∈ A} . Yeah ! And this property is the starting point of the learning algorithm. Proposition Let T ⊆ S 2 . If P is a M VLP suitable for T, then P O ( T ) = { R ∈ P | ∀ R ′ ∈ P , R ≤ R ′ = ⇒ R ′ ≤ R } Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 17 / 33

Formalization Optimal M VLP Proposition Let T ⊆ S 2 . The M VLP optimal for T is unique and denoted P O ( T ) . Troll mode on: does it works if it is the empty set? :p Proposition P O ( ∅ ) = { v val ← ∅ | v val ∈ A} . Yeah ! And this property is the starting point of the learning algorithm. Proposition Let T ⊆ S 2 . If P is a M VLP suitable for T, then P O ( T ) = { R ∈ P | ∀ R ′ ∈ P , R ≤ R ′ = ⇒ R ′ ≤ R } We can obtain the optimal program from any suitable program by simply removing the dominated rules. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 17 / 33

Learning Process Outline Motivations 1 Formalization 2 Learning Process 3 Semantics 4 Evaluation 5 Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 18 / 33

Learning Process Learning Process Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 19 / 33

Learning Process Learning Process How to make a minimal modifications of a M VLP in order to be suitable with a new set of transitions? Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 19 / 33

Learning Process Learning Process How to make a minimal modifications of a M VLP in order to be suitable with a new set of transitions? Definition (Rule least specialization) Let R be a M VL rule and s ∈ S such that R ⊓ s . The least specialization of R by s is: L spe ( R , s ) := { h ( R ) ← b ( R ) ∪ { v val } | v val ∈ A ∧ v val �∈ s ∧ ∀ val ′ ∈ N , v val ′ �∈ b ( R ) } . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 19 / 33

Learning Process Learning Process How to make a minimal modifications of a M VLP in order to be suitable with a new set of transitions? Definition (Rule least specialization) Let R be a M VL rule and s ∈ S such that R ⊓ s . The least specialization of R by s is: L spe ( R , s ) := { h ( R ) ← b ( R ) ∪ { v val } | v val ∈ A ∧ v val �∈ s ∧ ∀ val ′ ∈ N , v val ′ �∈ b ( R ) } . Thus, a M VLP can be revised to only realizes given transitions from s . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 19 / 33

Learning Process Learning Process How to make a minimal modifications of a M VLP in order to be suitable with a new set of transitions? Definition (Rule least specialization) Let R be a M VL rule and s ∈ S such that R ⊓ s . The least specialization of R by s is: L spe ( R , s ) := { h ( R ) ← b ( R ) ∪ { v val } | v val ∈ A ∧ v val �∈ s ∧ ∀ val ′ ∈ N , v val ′ �∈ b ( R ) } . Thus, a M VLP can be revised to only realizes given transitions from s . Definition (Program least revision) Let P be a M VLP , s ∈ S and T ⊆ S 2 such that fst ( T ) = { s } . Let R P := { R ∈ P | R conflicts with T } . The least revision of P by T is L rev ( P , T ) := ( P \ R P ) ∪ � L spe ( R , s ). R ∈ R P Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 19 / 33

Learning Process Learning Process Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 20 / 33

Learning Process Learning Process Guess what? Least revision can conserves suitability :) Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 20 / 33

Learning Process Learning Process Guess what? Least revision can conserves suitability :) Theorem Let s ∈ S and T , T ′ ⊆ S 2 such that | fst ( T ′ ) | = 1 ∧ fst ( T ) ∩ fst ( T ′ ) = ∅ . L rev ( P O ( T ) , T ′ ) is a M VLP suitable for T ∪ T ′ . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 20 / 33

Learning Process Learning Process Guess what? Least revision can conserves suitability :) Theorem Let s ∈ S and T , T ′ ⊆ S 2 such that | fst ( T ′ ) | = 1 ∧ fst ( T ) ∩ fst ( T ′ ) = ∅ . L rev ( P O ( T ) , T ′ ) is a M VLP suitable for T ∪ T ′ . In association with previous results it gives a method to iteratively compute P O ( T ) for any T ⊆ S 2 , starting from P O ( ∅ ). Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 20 / 33

Learning Process GULA: General Usage LFIT Algorithm Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 21 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 21 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 21 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 21 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } a=0 a=1 Observations Positive Negative Positive Negative example example example example 00 11 10 10 00 10 00 01 00 01 11 00 01 10 00 10 11 01 11 11 01 01 11 10 Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 21 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } a=0 a=1 Observations Positive Negative Positive Negative example example example example 00 11 10 10 00 10 00 01 00 01 11 00 01 10 00 10 11 01 11 11 01 01 11 10 Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 21 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val ◮ Extract each rule R of P v val that matches s : M v val := { R ∈ P | b ( R ) ⊆ s } , P v val := P v val \ M v val . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val ◮ Extract each rule R of P v val that matches s : M v val := { R ∈ P | b ( R ) ⊆ s } , P v val := P v val \ M v val . ◮ For each rule R ∈ M v val Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val ◮ Extract each rule R of P v val that matches s : M v val := { R ∈ P | b ( R ) ⊆ s } , P v val := P v val \ M v val . ◮ For each rule R ∈ M v val ⋆ Compute its least specialization P ′ = L spe ( R , s ). Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val ◮ Extract each rule R of P v val that matches s : M v val := { R ∈ P | b ( R ) ⊆ s } , P v val := P v val \ M v val . ◮ For each rule R ∈ M v val ⋆ Compute its least specialization P ′ = L spe ( R , s ). ⋆ Remove all the rules in P ′ dominated by a rule in P v val . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val ◮ Extract each rule R of P v val that matches s : M v val := { R ∈ P | b ( R ) ⊆ s } , P v val := P v val \ M v val . ◮ For each rule R ∈ M v val ⋆ Compute its least specialization P ′ = L spe ( R , s ). ⋆ Remove all the rules in P ′ dominated by a rule in P v val . ⋆ Remove all the rules in P v val dominated by a rule in P ′ . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val ◮ Extract each rule R of P v val that matches s : M v val := { R ∈ P | b ( R ) ⊆ s } , P v val := P v val \ M v val . ◮ For each rule R ∈ M v val ⋆ Compute its least specialization P ′ = L spe ( R , s ). ⋆ Remove all the rules in P ′ dominated by a rule in P v val . ⋆ Remove all the rules in P v val dominated by a rule in P ′ . ⋆ Add all remaining rules in P ′ to P v val . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val ◮ Extract each rule R of P v val that matches s : M v val := { R ∈ P | b ( R ) ⊆ s } , P v val := P v val \ M v val . ◮ For each rule R ∈ M v val ⋆ Compute its least specialization P ′ = L spe ( R , s ). ⋆ Remove all the rules in P ′ dominated by a rule in P v val . ⋆ Remove all the rules in P v val dominated by a rule in P ′ . ⋆ Add all remaining rules in P ′ to P v val . P := P ∪ P v val Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Learning Process GULA: General Usage LFIT Algorithm GULA: INPUT: a set of atoms A and a set of transitions T ⊆ S 2 . For each atom v val ∈ A Extract all states from which no transition to v val exist: Neg v val := { s | ∄ ( s , s ′ ) ∈ T , v val ∈ s ′ } Initialize P v val := { v val ← ∅} For each state s ∈ Neg v val ◮ Extract each rule R of P v val that matches s : M v val := { R ∈ P | b ( R ) ⊆ s } , P v val := P v val \ M v val . ◮ For each rule R ∈ M v val ⋆ Compute its least specialization P ′ = L spe ( R , s ). ⋆ Remove all the rules in P ′ dominated by a rule in P v val . ⋆ Remove all the rules in P v val dominated by a rule in P ′ . ⋆ Add all remaining rules in P ′ to P v val . P := P ∪ P v val OUTPUT: P O ( T ) := P . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 22 / 33

Semantics Outline Motivations 1 Formalization 2 Learning Process 3 Semantics 4 Evaluation 5 Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 23 / 33

Semantics Where is the semantics gone? Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 24 / 33

Semantics Where is the semantics gone? The formalization of M VLP is independant of the semantics that produced its transitions. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 24 / 33

Semantics Where is the semantics gone? The formalization of M VLP is independant of the semantics that produced its transitions. Definition (Semantics) Let A V dom be a set of atoms and S the corresponding set of states. A semantics (on A V dom ) is a function that associates, to each complete M VLP P , a set of transitions T ⊆ S 2 so that: fst ( T ) = S . Equivalently, � � a semantics can be seen as a function of c - M VLP → ( S → ℘ ( S ) \ ∅ ) where c - M VLP is the set of complete M VLP s and ℘ is the power set operator. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 24 / 33

Semantics Definition (Synchronous semantics) The synchronous semantics T syn is defined by: T syn : P �→ { ( s , s ′ ) ∈ S 2 | s ′ ⊆ { h ( R ) ∈ A | R ∈ P , R ⊓ s }} Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 25 / 33

Semantics Definition (Synchronous semantics) The synchronous semantics T syn is defined by: T syn : P �→ { ( s , s ′ ) ∈ S 2 | s ′ ⊆ { h ( R ) ∈ A | R ∈ P , R ⊓ s }} Definition (Asynchronous semantics) The asynchronous semantics T asyn is defined by: \{ h ( R ) } ) ∈ S 2 | R ∈ P ∧ R ⊓ s ∧ h ( R ) / T asyn : P �→ { ( s , s \ ∈ s } ∪ { ( s , s ) ∈ S 2 | ∀ R ∈ P , R ⊓ s = ⇒ h ( R ) ∈ s } . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 25 / 33

Semantics Definition (Synchronous semantics) The synchronous semantics T syn is defined by: T syn : P �→ { ( s , s ′ ) ∈ S 2 | s ′ ⊆ { h ( R ) ∈ A | R ∈ P , R ⊓ s }} Definition (Asynchronous semantics) The asynchronous semantics T asyn is defined by: \{ h ( R ) } ) ∈ S 2 | R ∈ P ∧ R ⊓ s ∧ h ( R ) / T asyn : P �→ { ( s , s \ ∈ s } ∪ { ( s , s ) ∈ S 2 | ∀ R ∈ P , R ⊓ s = ⇒ h ( R ) ∈ s } . Definition (General semantics) The general semantics T gen is defined by: \ r ) ∈ S 2 | r ⊆ { h ( R ) ∈ A | R ∈ P ∧ R ⊓ s } ∧ T gen : P �→ { ( s , s \ ∀ v val 1 , v val 2 ∈ r , v 1 = v 2 = ⇒ val 1 = val 2 } . 1 2 Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 25 / 33

Semantics Definition (Synchronous semantics) The synchronous semantics T syn is defined by: T syn : P �→ { ( s , s ′ ) ∈ S 2 | s ′ ⊆ { h ( R ) ∈ A | R ∈ P , R ⊓ s }} Definition (Asynchronous semantics) The asynchronous semantics T asyn is defined by: \{ h ( R ) } ) ∈ S 2 | R ∈ P ∧ R ⊓ s ∧ h ( R ) / T asyn : P �→ { ( s , s \ ∈ s } ∪ { ( s , s ) ∈ S 2 | ∀ R ∈ P , R ⊓ s = ⇒ h ( R ) ∈ s } . Definition (General semantics) The general semantics T gen is defined by: \ r ) ∈ S 2 | r ⊆ { h ( R ) ∈ A | R ∈ P ∧ R ⊓ s } ∧ T gen : P �→ { ( s , s \ ∀ v val 1 , v val 2 ∈ r , v 1 = v 2 = ⇒ val 1 = val 2 } . 1 2 Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 25 / 33

Semantics Semantic free modeling Finally, we can state that the definitions and method developed in the previous section are independent of those three semantics. Theorem (Semantics-free correctness) Let P be a M VLP such that P is complete. T syn ( P ) = T syn ( P O ( T syn ( P ))) , T asyn ( P ) = T asyn ( P O ( T asyn ( P ))) , T gen ( P ) = T gen ( P O ( T gen ( P ))) . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 26 / 33

Semantics Semantic free modeling Finally, we can state that the definitions and method developed in the previous section are independent of those three semantics. Theorem (Semantics-free correctness) Let P be a M VLP such that P is complete. T syn ( P ) = T syn ( P O ( T syn ( P ))) , T asyn ( P ) = T asyn ( P O ( T asyn ( P ))) , T gen ( P ) = T gen ( P O ( T gen ( P ))) . Whatever the semantic which produced T , given the optimal M VLP of T we can reproduce exactly T with the same semantic. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 26 / 33

Semantics GULA: General Usage LFIT Algorithm And GULA can learn such an optimal M VLP from T . Theorem ( GULA Termination, soundness, completeness, optimality) Let T ⊆ S 2 . The call GULA ( A , T ) terminates and GULA ( A , T ) = P O ( T ) . Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 27 / 33

Semantics GULA: General Usage LFIT Algorithm And GULA can learn such an optimal M VLP from T . Theorem ( GULA Termination, soundness, completeness, optimality) Let T ⊆ S 2 . The call GULA ( A , T ) terminates and GULA ( A , T ) = P O ( T ) . Making the algorithm semantic-free atleast for those three semantics. Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 27 / 33

Semantics GULA: General Usage LFIT Algorithm And GULA can learn such an optimal M VLP from T . Theorem ( GULA Termination, soundness, completeness, optimality) Let T ⊆ S 2 . The call GULA ( A , T ) terminates and GULA ( A , T ) = P O ( T ) . Making the algorithm semantic-free atleast for those three semantics. Theorem (Semantic-freeness) Let P be a M VLP such that P is complete. GULA ( A , T syn ( P )) = P O ( T syn ( P )) GULA ( A , T asyn ( P )) = P O ( T asyn ( P )) GULA ( A , T gen ( P )) = P O ( T gen ( P )) Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 27 / 33

Semantics GULA: General Usage LFIT Algorithm And GULA can learn such an optimal M VLP from T . Theorem ( GULA Termination, soundness, completeness, optimality) Let T ⊆ S 2 . The call GULA ( A , T ) terminates and GULA ( A , T ) = P O ( T ) . Making the algorithm semantic-free atleast for those three semantics. Theorem (Semantic-freeness) Let P be a M VLP such that P is complete. GULA ( A , T syn ( P )) = P O ( T syn ( P )) GULA ( A , T asyn ( P )) = P O ( T asyn ( P )) GULA ( A , T gen ( P )) = P O ( T gen ( P )) Victory! In theory, but how does it scale in practice? Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 27 / 33

Evaluation Outline Motivations 1 Formalization 2 Learning Process 3 Semantics 4 Evaluation 5 Ribeiro et al (LS2N, IRISA, NII) GULA: semantic free dynamics learning 4th September 2018, ILP 28 / 33