Generic Trace Theory Ichiro Hasuo, Bart Jacobs and Ana Sokolova SOS - PowerPoint PPT Presentation

Generic Trace Theory Ichiro Hasuo, Bart Jacobs and Ana Sokolova SOS group - Radboud University Nijmegen CMCS’06, Generic traces – p.1/22

Talk about... • systems as coalgebras states • • • • • • CMCS’06, Generic traces – p.2/22

� � � Talk about... • systems as coalgebras states + transitions • � � a � � � � � � � � • � • b c • � ���������� a • • c CMCS’06, Generic traces – p.2/22

Talk about... • systems as coalgebras states + transitions � S, α : S → F S � , for F a functor CMCS’06, Generic traces – p.2/22

Talk about... • systems as coalgebras states + transitions � S, α : S → F S � , for F a functor • semantic relations represent behaviour CMCS’06, Generic traces – p.2/22

Talk about... • systems as coalgebras states + transitions � S, α : S → F S � , for F a functor • semantic relations represent behaviour LT/BT spectrum CMCS’06, Generic traces – p.2/22

Talk about... • systems as coalgebras states + transitions � S, α : S → F S � , for F a functor • semantic relations represent behaviour LT/BT spectrum ... linear-time behaviour via trace semantics CMCS’06, Generic traces – p.2/22

� � � LT/BT spectrum Are these non-deterministic systems equal ? • y • x � a a � � ���� a � � � • • • � c b � ���� � � c b � � • • • • CMCS’06, Generic traces – p.3/22

� � � LT/BT spectrum Are these non-deterministic systems equal ? • y • x � a a � � ���� a � � � • • • � c b � ���� � � c b � � • • • • x and y are: • different wrt. bisimilarity CMCS’06, Generic traces – p.3/22

� � � LT/BT spectrum Are these non-deterministic systems equal ? • y • x � a a � � ���� a � � � • • • � c b � ���� � � c b � � • • • • x and y are: • different wrt. bisimilarity, but • equivalent wrt. trace semantics tr( x ) = tr( y ) = { ab, ac } CMCS’06, Generic traces – p.3/22

Traces - LTS For LTS with explicit termination (NA) trace = the set of all possible linear behaviors CMCS’06, Generic traces – p.4/22

� � � Traces - LTS For LTS with explicit termination (NA) trace = the set of all possible linear behaviors Example: a b �� �� �� �� ���� a � • y • x � tr( x ) = a + · tr( y ) = a + · b ∗ tr( y ) = b ∗ , CMCS’06, Generic traces – p.4/22

Traces - generative For generative probabilistic systems with ex. termination trace = sub-probability distribution over possible linear behaviors CMCS’06, Generic traces – p.5/22

� � � � � � Traces - generative For generative probabilistic systems with ex. termination trace = sub-probability distribution over possible linear behaviors Example: a [ 1 2 ] ���� �� a [ 1 3 ] • y �� �→ 1 • x tr( x ) : � 3 � 1 � � � 3 b [ 1 1 � 3 ] � � 2 � a �→ 1 3 · 1 � � 2 • z � �� �� �� a 2 �→ 1 3 · 1 2 · 1 2 a [1] · · · CMCS’06, Generic traces – p.5/22

Trace of a coalgebra ? CMCS’06, Generic traces – p.6/22

Trace of a coalgebra ? • Power&Turi ’99 - P (1 + Σ × ) • Jacobs ’04 - PF • Hasuo&Jacobs CALCO ’05 - PF , shapely F • Hasuo&Jacobs CALCO Jnr ’05 - DF , shapely F • Generic Trace Theory - T F , order-enriched setting CMCS’06, Generic traces – p.6/22

Trace of a coalgebra ? • Power&Turi ’99 - P (1 + Σ × ) • Jacobs ’04 - PF • Hasuo&Jacobs CALCO ’05 - PF , shapely F • Hasuo&Jacobs CALCO Jnr ’05 - DF , shapely F • Generic Trace Theory - T F , order-enriched setting main idea: coinduction in a Kleisli category CMCS’06, Generic traces – p.6/22

� � � � Coinduction F (beh) F X F Z � � � � � � � ∼ α = X � � � � � � � Z beh system final coalgebra CMCS’06, Generic traces – p.7/22

� � � � Coinduction F (beh) F X F Z � � � � � � � ∼ α = X � � � � � � � Z beh system final coalgebra • finality = ∃ ! (morphism for any F - coalgebra) • beh gives the behavior of the system • this yields final coalgebra semantics CMCS’06, Generic traces – p.7/22

� � � � Coinduction F (beh) F X F Z � � � � � � � ∼ α = X � � � � � � � Z beh system final coalgebra • f.c.s. in Sets = bisimilarity • f.c.s. in a Kleisli category = trace semantics CMCS’06, Generic traces – p.7/22

Types of systems For trace semantics systems are suitably modelled as coalgebras in Sets c → T F X X CMCS’06, Generic traces – p.8/22

Types of systems For trace semantics systems are suitably modelled as coalgebras in Sets c → T F X X monad - branching type CMCS’06, Generic traces – p.8/22

Types of systems For trace semantics systems are suitably modelled as coalgebras in Sets c → T F X X monad - branching type functor - linear i/o type CMCS’06, Generic traces – p.8/22

Types of systems For trace semantics systems are suitably modelled as coalgebras in Sets c → T F X X monad - branching type functor - linear i/o type needed: distributive law FT ⇒ T F CMCS’06, Generic traces – p.8/22

Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) CMCS’06, Generic traces – p.9/22

� � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x � � � ������� � a a � � � � • • � � ������ � � a b � b � � � � � CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � c → PF X X CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � c → PF X X CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � c PF c → PF X → PFPF X X CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � c PF c → PF X → PFPF X X CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � c PF c d.l. X → PF X → PFPF X → PPFF X CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � c PF c d.l. X → PF X → PFPF X → PPFF X CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � c PF c d.l. m.m. X → PF X → PFPF X → PPFF X → PFF X CMCS’06, Generic traces – p.9/22

� � � � � Distributive law is needed since branching is irrelevant: LTS with � - PF = P (1 + Σ × ) • x • x � � � ������� � a a � � � � • • aa ab � � ������ � � a b � b � � � � � � c PF c d.l. m.m. X → PF X → PFPF X → PPFF X → PFF X CMCS’06, Generic traces – p.9/22