workin progress The M in CoPaR From Partition Refjnement to Minimization Hans-Peter Deifel Oberseminar 09.06.2020 1 / 26
The M in CoPaR From Partition Refjnement to Minimization Hans-Peter Deifel Oberseminar 09.06.2020 1 / 26 workin progress
CoPaR Coalgebraic Partition Refjner 2 / 26
Partition Refjnement 3 / 26
Partition Refjnement 3 / 26
Partition Refjnement 3 / 26
Partition Refjnement 3 / 26
Partition Refjnement 3 / 26
Partition Refjnement Difgerent partition refjnement algorithms: Pajge-Tarjan Hopcroft Markov-Chain lumping Color refjnement 4 / 26 ⇒ Used in countless minimization tools
Partition Refjnement Difgerent partition refjnement algorithms: Pajge-Tarjan Hopcroft Markov-Chain lumping Color refjnement 4 / 26 ⇒ Used in countless minimization tools
CoPaR Coalgebraic Partition Refjner 5 / 26
CoM Coalgebraic Minimizer 5 / 26
Coalgebras Labelled Chains Markov Automata Deterministic Systems Transition Set Set Type Functor type functor successor map states 6 / 26 c X FX
Coalgebras Labelled Chains Markov Automata Deterministic Systems Transition type functor successor map states 6 / 26 c X FX Type Functor F : Set → Set 2 × X A FX = P ( A × X ) R ( X ) . . .
Behavioural Equivalence homomorphism Bisimilarity Weighted Equivalence Language Bisimilarity Identifjed by Coalgebra Homomorphism 7 / 26 type functor successor map states c X FX h Fh Y FY d 2 × X A FX = P ( A × X ) R ( X ) . . .
Coalgebraic Partition Refjnement Generic Algorithm Refjnement Interface Partition 8 / 26 F -Coalgebra Functor F for F
Functor Encoding Encoding map Bags 9 / 26 Encoding for Functor F Abstract Type A of labels ♭ X : FX → B ( A × X ) Example: P ( − ) A = 1 ♭ X ( { x 1 , x 2 , x 3 } ) = { ( ∗ , x 1 ) , ( ∗ , x 2 ) , ( ∗ , x 3 ) }
Functor Encoding Encoding map Bags 9 / 26 Encoding for Functor F Abstract Type A of labels ♭ X : FX → B ( A × X ) Example: R ( − ) A = R ♭ X ( f ) = { ( r, x ) | x ∈ X, f ( x ) = r � = 0 }
Functor Encoding Bags Not Natural 9 / 26 Encoding map Encoding for Functor F Abstract Type A of labels ♭ X : FX → B ( A × X ) Let f ∈ R 2 , f (0) = 0 . 5 , f (1) = 0 . 5 ♭ 2 f { (0 . 5 , 0) , (0 . 5 , 1) } R (!) B ( id , !) ♭ 1 / ( ∗ �→ 1 . 0) { (0 . 5 , 1) , (0 . 5 , 1) }
Functor Encoding Encoding map Bags Refjnement Interface 9 / 26 Encoding for Functor F Abstract Type A of labels ♭ X : FX → B ( A × X ) Type W (abstract, could be ints, reals, trees, …) init : F 1 × B A → W update : B A × W → W × F 3 × W
Syntax Input DX q1: {q2: 0.5, q3: 0.5} q2: {q1: 0.5, q3: 0.5} q3: {q3: 1} 10 / 26
Syntax Input DX q1: {q2: 0.5, q3: 0.5} q2: {q1: 0.5, q3: 0.5} q3: {q3: 1} Output Block 0: q1, q2, q3 10 / 26
Syntax Input DX q1: {q2: 0.5, q3: 0.5} q2: {q1: 0.5, q3: 0.5} q3: {q3: 1} Cooler Output DX q1: {q1: 1} 10 / 26
Minimization Two tasks: Computing the (encoding of) the quotient coalgebra Removing unreachable states 11 / 26
Minimization Two tasks: Computing the (encoding of) the quotient coalgebra Removing unreachable states 11 / 26
Quotient Encoding: Goal quotient encoding coalgebra output coalgebra input map 12 / 26 ♭ X c X FX B ( A × X ) q Fq ? Y FY B ( A × Y ) d ♭ Y
Solution: Minimization Interface merge merge and 13 / 26 merge : B ( A ) → B ( A ) B ( A ) fil S · ♭ {} FX B ( A ) Fχ S 1 fil { 1 } · ♭ F 2 B ( A ) {} B ( A ) fil S ( t )( a ) = � x ∈ S t ( a, x )
Minimization Interface: Examples Powerset Monoid-Valued otherwise Polynomial 14 / 26 merge ( ℓ )( ∗ ) = min (1 , ℓ ( ∗ )) � { Σ ℓ } Σ ℓ � = 0 merge ( ℓ ) = {} merge = id
Victory? group monoid-valued ungroup 15 / 26 curry ·B ( swap ) B ( A × q ) ♭ B ( A ) ( Y ) FX B ( A × X ) B ( A × Y ) merge ( Y ) Fq B ( A ) ( Y ) FY B ( A × Y ) ♭ for B -monoid
16 / 26 Encoding Assumption Nope! We additionally need the following assumption on ♭ : ♭ X FX B ( A × X ) fil { x } Fχ { x } B ( A ) fil { 1 } ♭ 2 F 2 B ( A × 2)
Encoding Assumption: Counterexample An encoding for Powerset But: Ruled out by assumption 17 / 26 A = 1 + 1 ♭ 2 ( { x 1 , . . . , x n } ) = ( { ( inj 1 ∗ , x 1 ) . . . , ( inj 1 ∗ , x n ) } ) ♭ X � =2 ( { x 1 , . . . , x n } ) = ( { ( inj 2 ∗ , x 1 ) . . . , ( inj 2 ∗ , x n ) } ) ⇒ Lawful merge possible!
Encoding Assumption: Counterexample An encoding for Powerset But: Ruled out by assumption 17 / 26 A = 1 + 1 ♭ 2 ( { x 1 , . . . , x n } ) = ( { ( inj 1 ∗ , x 1 ) . . . , ( inj 1 ∗ , x n ) } ) ♭ X � =2 ( { x 1 , . . . , x n } ) = ( { ( inj 2 ∗ , x 1 ) . . . , ( inj 2 ∗ , x n ) } ) ⇒ Lawful merge possible!
Victory! group ungroup 18 / 26 B ( A × q ) ♭ B ( A ) ( Y ) FX B ( A × X ) B ( A × Y ) Fq merge ( Y ) B ( A ) ( Y ) FY B ( A × Y ) ♭
Victory! group ungroup 18 / 26 B ( A × q ) ♭ B ( A ) ( Y ) FX B ( A × X ) B ( A × Y ) Fq merge ( Y ) B ( A ) ( Y ) FY B ( A × Y ) ♭
Minimization Two tasks: Computing the (encoding of) the quotient coalgebra Removing unreachable states 19 / 26
Minimization Two tasks: Computing the (encoding of) the quotient coalgebra Removing unreachable states 19 / 26
Reachability: Problems “Two tasks” all the way down: Pointed Coalgebras Computing the Reachable Subcoalgebra 20 / 26
Reachability: Problems “Two tasks” all the way down: Pointed Coalgebras Computing the Reachable Subcoalgebra 20 / 26
Reachability: Problems “Two tasks” all the way down: Pointed Coalgebras Computing the Reachable Subcoalgebra 20 / 26 ⇒ Simple matter of programming™
Reachability: Problems “Two tasks” all the way down: Pointed Coalgebras Computing the Reachable Subcoalgebra 20 / 26 ⇒ Simple matter of programming™
Computing the Reachable Subcoalgebra In Set: Standard graph search on canonical graph Canonical Graph does not factorize through For coalgebra: 21 / 26
Computing the Reachable Subcoalgebra In Set: Standard graph search on canonical graph 21 / 26 Canonical Graph τ X : FX → PX τ X ( t ) = { x ∈ X | 1 t − → FX does not factorize through F ( X \ { x } ) Fi − → FX } c τ C For coalgebra: C − → FC − → P C
Goal We want: Canonical graph of original coalgebra and canonical graph of encoded coalgebra to be the same! 22 / 26 c τ C FC P C τ ♭ B ( A × C )
Should be true™ Questment: Encoding contains canonical graph? 23 / 26 Direction: τ ( t ) ⊆ τ ( ♭ ( t ))
Questment: Encoding contains canonical graph? 23 / 26 Direction: τ ( t ) ⊆ τ ( ♭ ( t )) ⇒ Should be true™
Encoded Coalgebra a b c d 0.5 0.5 0 Canonical Graph a b d 24 / 26 Counterexample for τ ( t ) ⊇ τ ( ♭ ( t )) Functor: R ( − )
Requirement: Sub-Naturality Encoding must be sub-natural: now works! 25 / 26 ♭ X X FX B ( A × X ) f Ff B ( A × f ) ♭ Y Y FY B ( A × Y )
Requirement: Sub-Naturality Encoding must be sub-natural: 25 / 26 ♭ X X FX B ( A × X ) f Ff B ( A × f ) ♭ Y Y FY B ( A × Y ) ⇒ now works!
Question: Connection? Assumption 1 Assumption 2 26 / 26 ♭ X FX B ( A × X ) fil { x } Fχ { x } B ( A ) fil { 1 } ♭ 2 F 2 B ( A × 2) ♭ X X FX B ( A × X ) f Ff B ( A × f ) ♭ Y Y FY B ( A × Y )
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