1. Graph-Based Operational Semantics
Kaminski, Seidl et al. Muscholl
Estonian Summer School on Computer and Systems Science Lecture 2 Arend Rensink, University of Twente

2. Graph-Based Operational Semantics
Seen yesterday
Graphs and morphisms
Graphs: Tuples of nodes, labels, edges
Morphisms: node and edge mappings
State invariants as graph properties
Graph transformation rules
Single-graph and multi-graph representation
Rule applications as pushouts
Negative application conditions
GROOVE tool
State space generation
Graphs as states
Graph productions as transitions
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

3. Graph-Based Language Semantics
Program (textual)
Abstract Syntax Graph
compilation
Program Graph
flow graph construction
Execution Graphs
simulation
Syntax level
Semantic level
Execution Graphs
Execution Graphs
Here, we directly start with flow graphs
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

4. Graph-Based Operational Semantics
Flow graph simulation
Flow type graph
Solution for inheritance?
Multiple “node labels”
Any Procedure-node also labelled Element
Element
true
Procedure
Stop
flow
Predicate
Start
false
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

5. Build instance graphs using grammar
Graph rules for visual “flow language”
Compare: textual (context-free) rules
Elem ::= Seq | Cond | Stop
Seq ::= Proc SEMI Elem
Proc ::= ... | Start
Cond ::= IF Pred THEN Elem ELSE Elem
create Pred
create Stop
merge: creates cycles
create Proc
temporary edge
start graph
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

6. Simulation using a graph grammar
Execution type graph
pc initially points to Start
pc-edge follows flow, true and false
Thread pc
Element
true
Procedure
Stop
flow
Predicate
Start
false
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

7. Graph-Based Operational Semantics
Simulation rules
Initial rule
Create Thread and pc if it does not exist
Flow rules
One per program construct
regular expression
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

8. Extending the language
Boring: simulation copies flow graph!
Exercise: add Fork-statement
to type graph
to construction grammar
to simulation grammar
Effect: Fork creates a new Thread
with pc-edge pointing to well-defined Start
threads are unnamed
no Join possible
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

9. Graph-Based Operational Semantics
Modular extension
Flow graph:
Construction rule:
Element
true
Procedure
Stop
flow
Predicate
Start
false
run
Fork
Simulation rule:
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

10. Textual operational semantics
BNF grammar
Stat ::= Prim | Skip | Seq | Block | If | While
Seq ::= Stat Stat
Block ::= LBRACE Stat RBRACE
If ::= IF Cond Stat ELSE Stat
While ::= WHILE Cond Stat
Derivation rules per statement type
Premisses and conclusion are predicates
Transitions Stat  Stat’
Termination Stat
Premisse1 Premisse2 …
Conclusion
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

11. Rules for the textual language (1)
Basic
Sequential
Block
Prim ! Skip
Skip 
Stat1 ! Stat1’
Stat1 Stat2 ! Stat1’ Stat2
Stat1 Stat2 ! Stat2’
Stat1 Stat2 ! Stat2’
Stat1 Stat2 
Stat1 Stat2 
Stat ! Stat’
{ Stat } ! Stat’
Stat 
{ Stat } 
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

12. Rules for the textual language (2)If
While
nondeterminism: condition not regarded
if Cond Stat1 else Stat2 ! Stat1
if Cond Stat1 else Stat2 ! Stat2
while Cond Stat ! Stat while Cond Stat
while Cond Stat ! Skip
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

13. Example textual semantics
Program:
while (c1) { if (c2) p1 else if (c3) { p2; p3 } else ;
} p4
Semantics w ;w i1 i2 w i1 i2 p1 p4 p2 p3
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

14. Extending the language
How about Fork?
Stat ::= … | Fork
Fork ::= FORK Stat
Extend the semantic predicates
Transitions S1 | S2 | …  S1’ | S2’ | … for vectors of Stats
Less modular than for graph-based rules!
fork Stat ! Skip | Stat
S1 ! S1’ | S1’’
S1 S2 ! S1’ S2 | S1’’
S1 ! S1’
… | S1 | … ! … | S1’ | …
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

15. Example textual semantics
Program:
p1;
fork { if (c1) p2 }
while (c2) p3
Semantics p1 w i w i p2 p3 ;w
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

16. Difference visual/textual semantics
Aspect
Textual
Visual
Derivation method
proof tree (derivation)
pushout (one-shot)
Syntax
term: manipulated
graph: fixed
Context in rules?
no (context free, compositional)
yes (context sensitive, whole-world)
Mathematical structure
ad hoc (non-uniform)
graph (uniform)
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

17. Overall framework for GT semantics
Abstract Syntax Graph:
control flow added - variables resolved
Run-Time Graph:
- data values - heap structure - stack structure
fixed
points to
adapted
Operational rules manipulate run-time graph
Track program (control) location
Perform data calculations
Add & delete heap slots
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

18. Graph-Based Operational Semantics
Modelling data
Simple structures
Singly-linked lists
Global variables: x, y etc
Dereferencing: x^, y^ etc
Assignments (= procedure nodes), e.g.:
x^ = y (assigns y to x’s out-reference)
x = y^ (halts if y has no out-reference)
x = new (assigns new node to x)
Comparisons (in predicates), e.g.:
x == y
x == null (true if x points to node for null)
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

19. Example: List reversal
Program
while (v != null) { t = w; w = v; v = v^; w^ = t; }
Global variables
v: contains the initial list
w: should contain the result list, initially null
t: auxiliary variable, initially null
Initial state
Example: v w t
Nil
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

20. Representation choices
Variables become nodes (not edges)
Value represented by outgoing pointer
Advantages:
Uniformity with list nodes
One rule for assignment rather than per variable
Special node for null
Alternative: null value = “no pointer”
Advantages
No case distinctions in rules, e.g. comparison
Missing pointer always an error (except for null)
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

21. Extended syntax type graph
Element
true
flow
false
Proc
Stop
Pred
Compare
condition
Assign
left right
Target
left
right
Expr
Indir
Dir
Create
Deref
Alias
Null Eq
Neq
Ident id id
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

22. Graph-Based Operational Semantics
Example syntax rules
Elem-Pred
Elem-Proc
Proc-Assign
Elem-merge
Compare-Eq
Target-Dir
Expr-Alias
Expr-Create
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

23. List reversal syntax graph
Complete
Flow
Assigns
Assigns with variables
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

24. Graph-Based Operational Semantics
Language choices
Assignment and expression distinct
Alternative: Assign a subcategory of Expr
Conditions and expressions distinct
For simplicity
Occur in distinct contexts (no boolean vars)
Alternative: Condition as boolean expression
No nested expressions
Extension rather straightforward
Identifiers already resolved
One node per variable
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

25. Typing the run-time graphs
Variable for instantiated Idents
Run-time variable distinct from syntax!
Global variables: one instance per Ident
Pointer id from Var to corresponding Ident
Object for structures on the heap
Outgoing pointer val to next Object
Special object Nil (distinct from syntax!)
Var also modelled as a special Object
Auxiliary nodes for expression values
Pointer at from Aux to corresponding Expr
Alternative: introduce a stack
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

26. Run-time type graph: Expressions
abstract
Store
val
0..1
no incoming val
no outgoing val
Var
Nil
Aux
Obj
Expr
(from syntax) at
Ident
(from syntax) id
singleton
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

27. Run-time type graph: Conditions
no confusion due to distinct contexts
Aux
val
Condition
(from syntax) at
Boolean
singletons
False
True
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

28. List reversal initial state
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

29. Generic representation choices
No arrows from syntax to run-time
Use singletons where required
Auxiliary nodes for intermediate values
Extra indirectional layer
High-level graph structure
Program graph: syntax, flow, name resolution
Value graph: heap, global values & variables
Frame graph: stack, local variables, threads
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

30. Operational rules: Initialisation
init True
init False
init Nil
init Var
init Thread
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

31. Operational rules: Expressions
Alias
Deref
Create
Null
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

32. Other operational rules
Assign-Dir
Condition-Eq (failed)
Pred-False
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics

33. Graph-Based Operational Semantics
Seen today
Visual and textual operational semantics
With and without parallelism
Statements and expressions with data
Type graph and instances
Graph structure
Program graph
Frame graph
Value graph
Operational semantics
One or two rules per syntax construct
Completely modular
ESSCASS Lecture 2, 26 August 2007
Graph-Based Operational Semantics