diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000000000000000000000000000000000000..c34d28b93db2c6ac9546c53b0fd5a76c8d526611
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,2 @@
+__pycache__
+.DS_Store
\ No newline at end of file
diff --git a/TP3/config/cfg.py b/TP3/config/cfg.py
new file mode 100644
index 0000000000000000000000000000000000000000..08fa8d04abcddee3e330c6267f0f3a7dc31f2b06
--- /dev/null
+++ b/TP3/config/cfg.py
@@ -0,0 +1,144 @@
+"""Control flow graphs
+
+- Label and its subclasses describe the label on the edges of the CfGs
+- Nodes describes nodes, with an id and its successors
+- Graph describes a graph
+- Cfg describes a cfg, i.e. a graph with initial, final, and error nodes
+- function cfg creates a cfg for the given program
+"""
+from syntax import *
+from dataclasses import dataclass
+
+@dataclass
+class Label:
+ """generic class for edges' labels"""
+ pass
+
+@dataclass
+class LSkip(Label):
+ """label skip (does nothing)"""
+ def __str__(self):
+ return "skip"
+
+@dataclass
+class LSet(Label):
+ """assignment label"""
+ var: str
+ expr: ArithExpr
+ def __str__(self):
+ return f"{self.var} := {self.expr}"
+
+@dataclass
+class LTest(Label):
+ """test label"""
+ cond: BoolExpr
+ def __str__(self):
+ return f"?{self.cond}"
+
+@dataclass
+class LErr(Label):
+ """error label"""
+ err: Expr
+ def __str__(self):
+ return f"Err({self.err})"
+
+class Node:
+ """node of the graph, with its id and its successors"""
+ def __init__(self,nid):
+ self.nid = nid
+ self.succs = []
+ def __str__(self):
+ return f"Node {self.nid}"
+
+class Graph:
+ nb_id: int
+ nodes: list[Node]
+
+ """a graph, i.e. a list of nodes"""
+ def __init__(self):
+ self.nb_nid = 0
+ self.nodes = []
+
+ def new_node(self) -> Node:
+ """creates a new node, with a fresh id and no succs"""
+ node = Node(self.nb_nid)
+ self.nb_nid += 1
+ self.nodes += [node]
+ return node
+
+ def new_edge(self,n1: Node,n2: Node,l: Label) -> None:
+ """creates a new edge.
+ If the nodes are not already in the graph, they are added.
+ In that case, no check is made that a node with the same id
+ is already present or not.
+ """
+ if n1 not in self.nodes:
+ self.nodes += [n1]
+ if n2 not in self.nodes:
+ self.nodes += [n2]
+ n1.succs += [(n2,l)]
+
+def cfg_aux(g:Graph,e:Node,i:Node,instr:Instr) -> Node:
+ """auxiliary function to build a CfG
+ takes as argument a graph, the error node, the node starting
+ the current instruction, and the instruction itself, returns
+ the final node of current instruction
+ """
+ match instr:
+ case ISkip():
+ f = g.new_node()
+ g.new_edge(i,f,LSkip())
+ return f
+ case ISet(var,expr):
+ f = g.new_node()
+ g.new_edge(i,f,LSet(var,expr))
+ g.new_edge(i,e,LErr(expr))
+ return f
+ case ISeq(first,second):
+ m = cfg_aux(g,e,i,first)
+ return cfg_aux(g,e,m,second)
+ case ICond(cond,true_branch,false_branch):
+ t = g.new_node()
+ f = g.new_node()
+ g.new_edge(i,t,LTest(cond))
+ g.new_edge(i,f,LTest(BENeg(cond)))
+ g.new_edge(i,e,LErr(cond))
+ tf = cfg_aux(g,e,t,true_branch)
+ ff = cfg_aux(g,e,f,false_branch)
+ f = g.new_node()
+ g.new_edge(tf,f,LSkip())
+ g.new_edge(ff,f,LSkip())
+ return f
+ case ILoop(cond,body):
+ ib = g.new_node()
+ f = g.new_node()
+ g.new_edge(i,ib,LTest(cond))
+ g.new_edge(i,f,LTest(BENeg(cond)))
+ g.new_edge(i,e,LErr(cond))
+ fb = cfg_aux(g,e,ib,body)
+ g.new_edge(fb,i,LSkip())
+ return f
+ case _: assert False
+
+class Cfg:
+ """describes a CfG"""
+ g: Graph
+ err_node: Node
+ init_node: Node
+ final_node: Node
+ def __init__(self,instr):
+ self.g = Graph()
+ self.err_node = self.g.new_node()
+ self.init_node = self.g.new_node()
+ self.final_node = cfg_aux(self.g,self.err_node,self.init_node,instr)
+
+ def __str__(self):
+ res = ""
+ for node in self.g.nodes:
+ res+=f"{node}:\n"
+ for (dest,lab) in node.succs:
+ res+=f" {lab} --> {dest}\n"
+ res+=f"init node is {self.init_node}\n"
+ res+=f"final node is {self.final_node}\n"
+ res+=f"err node is {self.err_node}\n"
+ return res
diff --git a/TP3/config/constant_propagation.py b/TP3/config/constant_propagation.py
new file mode 100644
index 0000000000000000000000000000000000000000..a530c34071106acc53ced1ee96915ab2cb9deb93
--- /dev/null
+++ b/TP3/config/constant_propagation.py
@@ -0,0 +1,107 @@
+"""
+implements a constant propagation analysis
+"""
+
+from dataclasses import dataclass
+from cfg import Label, LSkip, LSet, LTest, LErr
+from iteration import Transfer
+from syntax import *
+
+@dataclass
+class abstract_value[T]:
+ """None indicates that we don't know the precise value"""
+ value: T | None
+ def __str__(self):
+ return f"{self.value}"
+
+type abstract_env = dict[str, abstract_value[int]]
+"""mapping from variables to abstract values.
+
+As with concrete environment, a variable not in the dictionary will
+lead to an error if we try to obtain its value
+"""
+type state = abstract_env | None
+"""abstract state is either an abstract env or bottom
+"""
+
+def eval_aexp(env: abstract_env, e: ArithExpr) -> abstract_value[int] | None:
+ """evaluate an arithmetic expression in an abstract environment
+ returns None in case of error
+ """
+ match e:
+ case AECst(value): return abstract_value(value)
+ case AEVar(var):
+ if var in env: return env[var]
+ else: return None
+ case AEUop(uop,expr):
+ res = eval_aexp(env,expr)
+ if res is None: return None
+ if res.value is None: return abstract_value(None)
+ if uop == Uop.OPP: return abstract_value(-res.value)
+ return None
+ case AEBop(bop,left_expr,right_expr):
+ v1 = eval_aexp(env,left_expr)
+ v2 = eval_aexp(env,right_expr)
+ if v1 is None or v2 is None: return None
+ if v1.value is None or v2.value is None:
+ return abstract_value(None)
+ match bop:
+ case Bop.ADD: return abstract_value(v1.value+v2.value)
+ case Bop.MUL: return abstract_value(v1.value*v2.value)
+ case Bop.DIV:
+ if v2.value == 0: return None
+ return abstract_value(v1.value//v2.value)
+ case _: assert False
+
+def eval_bexp(env: abstract_env, e: BoolExpr) -> abstract_value[bool] | None:
+ """abstract evaluation of a boolean expression"""
+ match e:
+ case BEPlain(aexpr):
+ res = eval_aexp(env, aexpr)
+ if res is None: return None
+ if res.value is None: return abstract_value(None)
+ return abstract_value(res.value!=0)
+ case BEEq(left_expr,right_expr):
+ v1 = eval_aexp(env, left_expr)
+ v2 = eval_aexp(env, right_expr)
+ if v1 is None or v2 is None: return None
+ if v1.value is None or v2.value is None: return abstract_value(None)
+ return abstract_value(v1.value == v2.value)
+ case BELeq(left_expr,right_expr):
+ v1 = eval_aexp(env, left_expr)
+ v2 = eval_aexp(env, right_expr)
+ if v1 is None or v2 is None: return None
+ if v1.value is None or v2.value is None: return abstract_value(None)
+ return abstract_value(v1.value <= v2.value)
+ case BENeg(expr):
+ v = eval_bexp(env,expr)
+ if v is None: return None
+ if v.value is None: return v
+ return abstract_value(not v.value)
+ case _: assert False
+
+class Constant_propagation(Transfer[state]):
+ variables: frozenset[str]
+ def __init__(self,instr):
+ self.variables = variables_of_instr(instr)
+ def bottom(self) -> state:
+ ...
+ def init_state(self) -> state:
+ ...
+ def join(self,s1:state,s2:state) -> state:
+ ...
+
+ def included(self,s1: state,s2: state) -> bool:
+ ...
+
+ def tr_skip(self,s: state) -> state:
+ ...
+
+ def tr_set(self,s: state,v: str,e: ArithExpr) -> state:
+ ...
+
+ def tr_test(self,s: state,c: BoolExpr) -> state:
+ ...
+
+ def tr_err(self,s: state,e: Expr) -> state:
+ ...
diff --git a/TP3/config/example_constant_propagation.py b/TP3/config/example_constant_propagation.py
new file mode 100644
index 0000000000000000000000000000000000000000..0bf11f1a7a796124d13f8a3ea0e579835985a6d2
--- /dev/null
+++ b/TP3/config/example_constant_propagation.py
@@ -0,0 +1,29 @@
+from syntax import *
+from cfg import Cfg
+from iteration import fixpoint_iteration
+from constant_propagation import Constant_propagation
+
+x = AEVar("x")
+y = AEVar("y")
+c = AEVar("c")
+one = AECst(1)
+zero = AECst(0)
+
+propagate_compute=ISeq(ISet("x",one),ISet("x", AEBop(Bop.ADD,x,one)))
+same_branch_if=ICond(BELeq(c,zero), ISet("y", one), ISet("y",one))
+equal_branch=ICond(BEEq(c,one),ISkip(),ISet("c",one))
+
+prog = ISeq(ISeq(propagate_compute,same_branch_if),equal_branch)
+
+cfg = Cfg(prog)
+
+cst_propagation = Constant_propagation(prog)
+
+res = fixpoint_iteration(cst_propagation,cfg)
+
+for (node, state) in res.items():
+ print(f"{node}:")
+ if state is None: print(" None")
+ else:
+ for (var, value) in state.items():
+ print(f" {var} -> {value}")
diff --git a/TP3/config/example_interval.py b/TP3/config/example_interval.py
new file mode 100644
index 0000000000000000000000000000000000000000..8aa35e01334ffe53cbe00789994a19c11f544c59
--- /dev/null
+++ b/TP3/config/example_interval.py
@@ -0,0 +1,4 @@
+from interval_analysis import analyze
+from example_opsem import factorial
+
+analyze(factorial)
diff --git a/TP3/config/example_interval_eval_exp.py b/TP3/config/example_interval_eval_exp.py
new file mode 100644
index 0000000000000000000000000000000000000000..1203cab645be505a71cd14888374824eca277e68
--- /dev/null
+++ b/TP3/config/example_interval_eval_exp.py
@@ -0,0 +1,32 @@
+from syntax import *
+from interval_analysis import eval_aexp, eval_bexp, abstract_env, Interval
+
+x: ArithExpr = AEVar("x")
+y: ArithExpr = AEVar("y")
+
+# x - y * (x / y)
+test_aexp: ArithExpr = AEBop(Bop.ADD,x, AEUop(Uop.OPP,AEBop(Bop.MUL,y,AEBop(Bop.DIV,x,y))))
+
+# ! (x - y <= y * (x - x/y))
+test_bexp: BoolExpr = BENeg(BELeq(AEBop(Bop.ADD,x,AEUop(Uop.OPP,y)),test_aexp))
+
+def test(env: abstract_env) -> None:
+ res_a = eval_aexp(env,test_aexp)
+ res_b = eval_bexp(env,test_bexp)
+ print(f'In environment { {var: str(i) for (var,i) in env.items()} }')
+ print(f"{test_aexp} -> {res_a}")
+ print(f"{test_bexp} -> {res_b}")
+
+test({ 'x': Interval(20,20), 'y': Interval(2,2) })
+
+test({ 'x': Interval(10,20), 'y': Interval(1,2) })
+
+test({ 'x': Interval(-30,0), 'y': Interval(-10,0) })
+
+test({ 'x': Interval(-20,20), 'y': Interval(0,0) })
+
+test({ 'x': Interval(0,20), 'y': Interval(2,5) })
+
+test({ 'x': Interval(-10,20), 'y': Interval(-5,-2) })
+
+test({})
diff --git a/TP3/config/example_interval_reduce.py b/TP3/config/example_interval_reduce.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d7cdda54e9d02cfd00b41b9a65d67b63f5120b9
--- /dev/null
+++ b/TP3/config/example_interval_reduce.py
@@ -0,0 +1,39 @@
+from syntax import *
+from interval_analysis import state, abstract_env, Interval, reduce_state
+
+x: ArithExpr = AEVar("x")
+y: ArithExpr = AEVar("y")
+
+test_eq: BoolExpr = BEEq(x,y)
+
+test_neq: BoolExpr = BENeg(test_eq)
+
+test_leq: BoolExpr = BELeq(x,y)
+
+test_gt: BoolExpr = BENeg(test_leq)
+
+def env_str(env: abstract_env) -> str:
+ return '{ ' + ', '.join([ f'{var}: {i}' for (var,i) in env.items()]) + ' }'
+
+def state_str(env: state) -> str:
+ if env is None: return 'None'
+ return env_str(env)
+
+def test(env: abstract_env) -> None:
+ res_eq = reduce_state(env,test_eq)
+ res_neq = reduce_state(env,test_neq)
+ res_leq = reduce_state(env,test_leq)
+ res_gt = reduce_state(env,test_gt)
+ print(f'In environment { env_str(env) }')
+ print(f"{test_eq} -> {state_str(res_eq)}")
+ print(f"{test_neq} -> {state_str(res_neq)}")
+ print(f"{test_leq} -> {state_str(res_leq)}")
+ print(f"{test_gt} -> {state_str(res_gt)}")
+
+test({ 'x': Interval(0,5), 'y': Interval(-5,0) })
+
+test({ 'x': Interval(-5,5), 'y': Interval(3,7) })
+
+test({ 'x': Interval(0,5), 'y': Interval(-10,-5) })
+
+test({ 'x': Interval(0,0), 'y': Interval(0,4) })
diff --git a/TP3/config/example_iteration.py b/TP3/config/example_iteration.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a9a71bb66f797cb0a6ac81b285ed1c2a8da2f75
--- /dev/null
+++ b/TP3/config/example_iteration.py
@@ -0,0 +1,52 @@
+from syntax import *
+from cfg import Cfg
+from iteration import *
+from opsem import eval_bexp, eval_aexp
+
+one = AECst(1)
+zero = AECst(0)
+x = AEVar("x")
+dead_if=ICond(BEPlain(one),ISet("x",one),ISet("y",zero))
+infinite_loop=ILoop(BEPlain(one),ISet("x",AEBop(Bop.ADD,x,one)))
+prog_with_dead_code=ISeq(dead_if,infinite_loop)
+
+class NoInfo(Transfer[frozenset[bool]]):
+ """Almost trivial static analysis
+
+ We only have two elements in our lattice, the empty set and the singleton {True}.
+ The latter is propagated from initial state to any potentially reachable state.
+ In the end, the only nodes still at bottom are the one which are guarded by
+ always-false tests.
+ """
+ type S = frozenset[bool]
+ def bottom(self) -> S: return frozenset()
+ def init_state(self) -> S: return frozenset([True])
+ def join(self, s1:S, s2:S) -> S: return s1 | s2
+ def included(self, s1:S, s2:S) -> bool: return s1.issubset(s2)
+ def tr_skip(self,s:S): return s
+ def tr_set(self, s:S, v:str, e: ArithExpr) -> S: return s
+ def tr_test(self,s:S, c:BoolExpr) -> S:
+ if eval_bexp({},c) is False:
+ return frozenset()
+ else:
+ return s
+ def tr_err(self, s:S, e:Expr) -> S:
+ if isinstance(e,ArithExpr):
+ if eval_aexp({},e) is not None:
+ return frozenset()
+ else:
+ return s
+ else:
+ if isinstance(e,BoolExpr):
+ if eval_bexp({},e) is not None:
+ return frozenset()
+ else:
+ return s
+ else: assert False
+
+cfg = Cfg(prog_with_dead_code)
+print(f"Cfg is {cfg}")
+res = fixpoint_iteration(NoInfo(),cfg)
+unreachable = [node for node, state in res.items() if state == frozenset()]
+for node in unreachable:
+ print(f"{node} is unreachable")
diff --git a/TP3/config/example_opsem.py b/TP3/config/example_opsem.py
new file mode 100644
index 0000000000000000000000000000000000000000..bcc5dca4e8bbb042e3de024382c4cab6ac304b65
--- /dev/null
+++ b/TP3/config/example_opsem.py
@@ -0,0 +1,21 @@
+from syntax import *
+from opsem import eval_instr
+
+x = AEVar("x")
+y = AEVar("y")
+zero = AECst(0)
+one = AECst(1)
+xneg = BELeq(x, zero)
+bound_check = ICond(xneg,ISet("x",one),ISkip())
+init_y = ISet("y",one)
+body = ISeq(ISet("y",AEBop(Bop.MUL,y,x)),ISet("x",AEBop(Bop.ADD,x,AEUop(Uop.OPP,one))))
+loop = ILoop(BENeg(BELeq(x,zero)),body)
+factorial = ISeq(ISeq(bound_check,init_y),loop)
+
+print(f"prog is\n{factorial}")
+
+env = { "x" : 4 }
+envf = eval_instr(env,factorial)
+
+print(f"init is \n{env}")
+print(f"result is\n{envf}")
diff --git a/TP3/config/example_sign.py b/TP3/config/example_sign.py
new file mode 100644
index 0000000000000000000000000000000000000000..6aa6de3d4adb559dddcd246d9cdb3159f2f12b67
--- /dev/null
+++ b/TP3/config/example_sign.py
@@ -0,0 +1,4 @@
+from sign_analysis import analyze
+from example_opsem import factorial
+
+analyze(factorial)
diff --git a/TP3/config/example_sign_eval_exp.py b/TP3/config/example_sign_eval_exp.py
new file mode 100644
index 0000000000000000000000000000000000000000..c8bdd995dad92d272d407c6a2c040666e1f2ad29
--- /dev/null
+++ b/TP3/config/example_sign_eval_exp.py
@@ -0,0 +1,28 @@
+from syntax import *
+from sign_analysis import eval_aexp, eval_bexp, abstract_env, sign
+
+x: ArithExpr = AEVar("x")
+y: ArithExpr = AEVar("y")
+
+# y * (x - x / y)
+test_aexp: ArithExpr = AEBop(Bop.MUL,y,AEBop(Bop.ADD,x, AEUop(Uop.OPP,AEBop(Bop.DIV,x,y))))
+
+# ! (x - y <= y * (x - x/y))
+test_bexp: BoolExpr = BENeg(BELeq(AEBop(Bop.ADD,x,AEUop(Uop.OPP,y)),test_aexp))
+
+def test(env: abstract_env) -> None:
+ res_a = eval_aexp(env,test_aexp)
+ res_b = eval_bexp(env,test_bexp)
+ print(f'In environment { {var: str(sign) for (var,sign) in env.items()} }')
+ print(f"{test_aexp} -> {res_a}")
+ print(f"{test_bexp} -> {res_b}")
+
+test({ 'x': sign.SPOS, 'y': sign.SPOS })
+
+test({ 'x': sign.NEG, 'y': sign.NEG })
+
+test({ 'x': sign.INT, 'y': sign.Z })
+
+test({ 'x': sign.POS, 'y': sign.NZ })
+
+test({})
diff --git a/TP3/config/example_sign_reduce.py b/TP3/config/example_sign_reduce.py
new file mode 100644
index 0000000000000000000000000000000000000000..17ed63624d1e282d9f535cccecbdf5d04e169e20
--- /dev/null
+++ b/TP3/config/example_sign_reduce.py
@@ -0,0 +1,33 @@
+from syntax import *
+from sign_analysis import state, abstract_env, sign, reduce_state
+
+x: ArithExpr = AEVar("x")
+y: ArithExpr = AEVar("y")
+
+test_eq: BoolExpr = BEEq(x,y)
+
+test_neq: BoolExpr = BENeg(test_eq)
+
+test_leq: BoolExpr = BELeq(x,y)
+
+test_gt: BoolExpr = BENeg(test_leq)
+
+def test(env: abstract_env) -> None:
+ res_eq = reduce_state(env,test_eq)
+ res_neq = reduce_state(env,test_neq)
+ res_leq = reduce_state(env,test_leq)
+ res_gt = reduce_state(env,test_gt)
+ print(f'In environment { {var: str(sign) for (var,sign) in env.items()} }')
+ print(f"{test_eq} -> {res_eq}")
+ print(f"{test_neq} -> {res_neq}")
+ print(f"{test_leq} -> {res_leq}")
+ print(f"{test_gt} -> {res_gt}")
+
+test({ 'x': sign.POS, 'y': sign.NEG })
+
+test({ 'x': sign.NZ, 'y': sign.NEG })
+
+test({ 'x': sign.POS, 'y': sign.SNEG })
+
+test({ 'x': sign.Z, 'y': sign.POS })
+
diff --git a/TP3/config/example_widen.py b/TP3/config/example_widen.py
new file mode 100644
index 0000000000000000000000000000000000000000..c9372be7b6162f6f47cd8f7ed8a5c2e240aa187d
--- /dev/null
+++ b/TP3/config/example_widen.py
@@ -0,0 +1,61 @@
+from iteration import Transfer, fixpoint_iteration
+from syntax import *
+from cfg import Cfg
+from example_opsem import factorial
+
+@dataclass
+class Number_step:
+ bound: int | None
+ def __str__(self):
+ if self.bound is None: return "+∞"
+ else: return str(self.bound)
+
+type S = Number_step | None
+
+class Analysis(Transfer[S]):
+ def bottom(self) -> S:
+ return None
+
+ def init_state(self) -> S:
+ return Number_step(0)
+
+ def join(self,s1:S,s2:S) -> S:
+ if s1 is None: return s2
+ if s2 is None: return s1
+ if s1.bound is None: return s1
+ if s2.bound is None: return s2
+ return Number_step(max(s1.bound, s2.bound))
+
+ def widen(self, s1:S, s2:S) -> S:
+ if s1 is None: return s2
+ if s2 is None: return s1
+ if s1.bound is None: return s1
+ if s2.bound is None: return s2
+ if s1.bound < s2.bound: return Number_step(None)
+ return s1
+
+ def included(self,s1: S,s2: S) -> bool:
+ if s1 is None: return True
+ if s2 is None: return False
+ if s2.bound is None: return True
+ if s1.bound is None: return False
+ return s1.bound <= s2.bound
+
+ def tr_skip(self,s: S) -> S:
+ if s is None: return None
+ if s.bound is None: return s
+ return Number_step(s.bound + 1)
+
+ def tr_set(self,s: S,v: str,e: ArithExpr) -> S:
+ return self.tr_skip(s)
+
+ def tr_test(self,s: S,c: BoolExpr) -> S:
+ return self.tr_skip(s)
+
+ def tr_err(self,s: S,e: Expr) -> S:
+ return self.tr_skip(s)
+
+res = fixpoint_iteration(Analysis(),Cfg(factorial))
+
+for (node, nb) in res.items():
+ print(f'{node} -> {nb}')
diff --git a/TP3/config/interval_analysis.py b/TP3/config/interval_analysis.py
new file mode 100644
index 0000000000000000000000000000000000000000..9186efbf8ffeedb0fe1e207381313552f1b34163
--- /dev/null
+++ b/TP3/config/interval_analysis.py
@@ -0,0 +1,406 @@
+# pyright: strict
+
+"""
+implements a constant propagation analysis
+"""
+
+from dataclasses import dataclass
+from cfg import Cfg
+from iteration import Transfer, fixpoint_iteration
+from syntax import *
+
+@dataclass
+class Interval:
+ """ potentially unbounded interval
+
+ None in either bound means that we don't have info.
+ Hence `Interval(None,None)` denotes all integers
+ """
+ lower_bound: int | None
+ upper_bound: int | None
+ def __str__(self):
+ if self.lower_bound is None:
+ l = "-∞"
+ else:
+ l = str(self.lower_bound)
+ if self.upper_bound is None:
+ u = "+∞"
+ else:
+ u = str(self.upper_bound)
+ return f"[{l}; {u}]"
+
+class Top:
+ def __str__(self): return "TOP"
+
+@dataclass
+class BTop:
+ """class used for the evaluation of boolean expressions
+
+ BTop(False) indicates that we don't know if the result is True or False, but
+ that the evaluation cannot lead to an error
+ BTop(True) indicates that we neither know the result nor whether an error can occur
+ """
+ has_error: bool
+ def __str__(self):
+ if self.has_error: return "TOP (maybe error)"
+ else: return "TOP (no error)"
+
+type abstract_env = dict[str, Interval]
+"""mapping from variables to abstract values.
+
+As with concrete environment, a variable not in the dictionary will
+lead to an error if we try to obtain its value
+"""
+
+type state = abstract_env | None
+"""abstract state is either an abstract env or bottom
+"""
+
+def interval_leq(i1: Interval, i2: Interval) -> bool:
+ if i2.lower_bound is None:
+ if i2.upper_bound is None:
+ return True
+ if i1.upper_bound is None:
+ return False
+ return i1.upper_bound <= i2.upper_bound
+ if i2.upper_bound is None:
+ if i1.lower_bound is None:
+ return False
+ return i1.lower_bound >= i2.lower_bound
+ if i1.lower_bound is None or i1.upper_bound is None:
+ return False
+ return i1.lower_bound >= i2.lower_bound and i1.upper_bound <= i2.upper_bound
+
+def interval_join(i1: Interval, i2: Interval) -> Interval:
+ if i1.lower_bound is None or i2.lower_bound is None:
+ l = None
+ else:
+ l = min(i1.lower_bound,i2.lower_bound)
+ if i1.upper_bound is None or i2.upper_bound is None:
+ u = None
+ else:
+ u = max(i1.upper_bound,i2.upper_bound)
+ return Interval(l,u)
+
+def interval_meet(i1: Interval, i2: Interval) -> Interval | None:
+ if i1.lower_bound is None:
+ l = i2.lower_bound
+ elif i2.lower_bound is None:
+ l = i1.lower_bound
+ else:
+ l = max(i1.lower_bound,i2.lower_bound)
+ if i1.upper_bound is None:
+ u = i2.upper_bound
+ elif i2.upper_bound is None:
+ u = i1.upper_bound
+ else:
+ u = min(i1.upper_bound,i2.upper_bound)
+ if l is not None and u is not None and l > u: return None
+ return Interval(l,u)
+
+def interval_widen(i1: Interval, i2: Interval) -> Interval:
+ if i1.lower_bound is None or i2.lower_bound is None:
+ l = None
+ elif i1.lower_bound <= i2.lower_bound:
+ l = i1.lower_bound
+ else:
+ l = None
+ if i1.upper_bound is None or i2.upper_bound is None:
+ u = None
+ elif i1.upper_bound >= i2.upper_bound:
+ u = i1.upper_bound
+ else:
+ u = None
+ return Interval(l,u)
+
+def has_strict_positive_val(i: Interval) -> bool:
+ return i.upper_bound is None or i.upper_bound > 0
+
+def has_strict_negative_val(i: Interval) -> bool:
+ return i.lower_bound is None or i.lower_bound < 0
+
+def contains_zero(i: Interval) -> bool:
+ if i.lower_bound is None:
+ return i.upper_bound is None or i.upper_bound >= 0
+ if i.upper_bound is None:
+ return i.lower_bound <= 0
+ return i.lower_bound <= 0 and i.upper_bound >= 0
+
+def is_zero(i: Interval) -> bool:
+ return i.lower_bound == 0 and i.upper_bound == 0
+
+def interval_opp(i: Interval) -> Interval:
+ if i.lower_bound is None:
+ u = None
+ else:
+ u = -i.lower_bound
+ if i.upper_bound is None:
+ l = None
+ else:
+ l = -i.upper_bound
+ return Interval(l,u)
+
+def interval_add(i1: Interval, i2: Interval) -> Interval:
+ raise NotImplementedError
+
+def interval_mul_pos(i1: Interval, i2: Interval) -> Interval:
+ assert (i1.lower_bound is not None and i1.lower_bound >= 0)
+ assert (i2.lower_bound is not None and i2.lower_bound >= 0)
+ l = i1.lower_bound * i2.lower_bound
+ if i1.upper_bound is None or i2.upper_bound is None:
+ u = None
+ else:
+ u = i1.upper_bound * i2.upper_bound
+ return Interval(l,u)
+
+def interval_opt_join(i1: Interval | None, i2: Interval) -> Interval:
+ if i1 is None: return i2
+ return interval_join(i1,i2)
+
+def interval_mul(i1: Interval, i2: Interval) -> Interval:
+ i1_neg = interval_meet(i1,Interval(None,0))
+ i2_neg = interval_meet(i2,Interval(None,0))
+ i1_pos = interval_meet(i1,Interval(0,None))
+ i2_pos = interval_meet(i2,Interval(0,None))
+ if i1_neg is None or i2_pos is None:
+ res = None
+ else:
+ res = interval_opp(interval_mul_pos(interval_opp(i1_neg),i2_pos))
+ if i1_pos is not None and i2_neg is not None:
+ res1 = interval_opp(interval_mul_pos(i1_pos,interval_opp(i2_neg)))
+ res = interval_opt_join(res, res1)
+ if i1_neg is not None and i2_neg is not None:
+ res1 = interval_mul_pos(interval_opp(i1_neg),interval_opp(i2_neg))
+ res = interval_opt_join(res,res1)
+ if i1_pos is not None and i2_pos is not None:
+ res1 = interval_mul_pos(i1_pos, i2_pos)
+ res = interval_opt_join(res, res1)
+ # at least one of the 4 cases is not empty
+ assert res is not None
+ return res
+
+def interval_div(i1: Interval, i2: Interval) -> Interval | Top | None:
+ raise NotImplementedError
+
+def eval_aexp(env: abstract_env, e: ArithExpr) -> Interval | Top | None:
+ """evaluate an arithmetic expression in an abstract environment
+ returns None in case of error
+ """
+ match e:
+ case AECst(value): return Interval(value,value)
+ case AEVar(var):
+ if var in env: return env[var]
+ else: return None
+ case AEUop(uop,expr):
+ res = eval_aexp(env,expr)
+ if res is None or isinstance(res,Top): return res
+ if uop == Uop.OPP: return interval_opp(res)
+ return None
+ case AEBop(bop,left_expr,right_expr):
+ v1 = eval_aexp(env,left_expr)
+ v2 = eval_aexp(env,right_expr)
+ if v1 is None or v2 is None: return None
+ if isinstance(v1,Top) or isinstance(v2,Top):
+ return Top()
+ match bop:
+ case Bop.ADD: return interval_add(v1,v2)
+ case Bop.MUL: return interval_mul(v1,v2)
+ case Bop.DIV: return interval_div(v1,v2)
+ case _: pass
+
+def eval_bexp(env: abstract_env, e: BoolExpr) -> bool | BTop | None:
+ """abstract evaluation of a boolean expression"""
+ match e:
+ case BEPlain(aexpr):
+ res = eval_aexp(env, aexpr)
+ if res is None: return None
+ if isinstance(res,Top): return BTop(True)
+ if res.lower_bound == 0 and res.upper_bound == 0: return True
+ if not interval_leq(Interval(0,0), res): return False
+ return BTop(False)
+ case BEEq(left_expr,right_expr):
+ v1 = eval_aexp(env, left_expr)
+ v2 = eval_aexp(env, right_expr)
+ if v1 is None or v2 is None: return None
+ if isinstance(v1,Top) or isinstance(v2,Top): return BTop(True)
+ if v1.lower_bound is None or v2.lower_bound is None or v1.upper_bound is None or v2.upper_bound is None:
+ return BTop(False)
+ if v1.lower_bound > v2.upper_bound or v1.upper_bound < v2.lower_bound:
+ return False
+ if v1.lower_bound == v1.upper_bound and v2.lower_bound == v2.upper_bound and v1.lower_bound == v2.lower_bound:
+ return True
+ return BTop(False)
+ case BELeq(left_expr,right_expr):
+ v1 = eval_aexp(env, left_expr)
+ v2 = eval_aexp(env, right_expr)
+ if v1 is None or v2 is None: return None
+ if isinstance(v1,Top) or isinstance(v2,Top): return BTop(True)
+ if v1.upper_bound is None:
+ if v1.lower_bound is None:
+ return BTop(False)
+ if v2.upper_bound is None:
+ return BTop(False)
+ if v2.upper_bound < v1.lower_bound:
+ return False
+ return BTop(False)
+ if v2.lower_bound is None:
+ return BTop(False)
+ if v1.upper_bound <= v2.lower_bound:
+ return True
+ if v1.lower_bound is None:
+ return BTop(False)
+ if v2.upper_bound is None:
+ return BTop(False)
+ if v2.upper_bound < v1.lower_bound:
+ return False
+ return BTop(False)
+ case BENeg(expr):
+ v = eval_bexp(env,expr)
+ if v is None: return None
+ if isinstance(v,BTop): return v
+ return not v
+ case _: pass
+
+def reduce_eq(s:state, x:str, i: Interval) -> state:
+ raise NotImplementedError
+
+def reduce_neq(s:state, x: str, i: Interval) -> state:
+ raise NotImplementedError
+
+def reduce_leq(s:state,x:str,upper_bound: int) -> state:
+ raise NotImplementedError
+
+def reduce_geq(s: state, x: str, lower_bound: int) -> state:
+ raise NotImplementedError
+
+def reduce_state(s: state,c: BoolExpr) -> state:
+ if s is None: return s
+ match c:
+ case BEEq(AEVar(x), AEVar(y)):
+ vx = eval_aexp(s,AEVar(x))
+ vy = eval_aexp(s,AEVar(y))
+ if vx is None or vy is None: return None
+ if not isinstance(vx,Top):
+ s = reduce_eq(s,y,vx)
+ if not isinstance(vy,Top):
+ s = reduce_eq(s,x,vy)
+ return s
+ case BEEq(AEVar(x), right_expr):
+ v = eval_aexp(s,right_expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ return reduce_eq(s,x,v)
+ case BEEq(left_expr, AEVar(y)):
+ v = eval_aexp(s,left_expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ return reduce_eq(s,y,v)
+ case BELeq(AEVar(x), AEVar(y)):
+ vx = eval_aexp(s,AEVar(x))
+ vy = eval_aexp(s,AEVar(y))
+ if vx is None or vy is None: return None
+ if not isinstance(vy,Top) and vy.upper_bound is not None:
+ s = reduce_leq(s,x,vy.upper_bound)
+ if not isinstance(vx,Top) and vx.lower_bound is not None:
+ s = reduce_geq(s,y,vx.lower_bound)
+ return s
+ case BELeq(AEVar(x),right_expr):
+ v = eval_aexp(s,right_expr)
+ if v is None: return None
+ if isinstance(v,Top) or v.upper_bound is None: return s
+ return reduce_leq(s,x,v.upper_bound)
+ case BELeq(left_expr,AEVar(y)):
+ v = eval_aexp(s,left_expr)
+ if v is None: return None
+ if isinstance(v,Top) or v.lower_bound is None: return s
+ return reduce_geq(s,y,v.lower_bound)
+ case BENeg(BEEq(AEVar(x), AEVar(y))):
+ vx = eval_aexp(s,AEVar(x))
+ vy = eval_aexp(s,AEVar(y))
+ if vx is None or vy is None: return None
+ if not isinstance(vx,Top):
+ s = reduce_neq(s,y,vx)
+ if not isinstance(vy,Top):
+ s = reduce_neq(s,x,vy)
+ return s
+ case BENeg(BEEq(AEVar(x), right_expr)):
+ v = eval_aexp(s,right_expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ return reduce_neq(s,x,v)
+ case BENeg(BEEq(left_expr, AEVar(y))):
+ v = eval_aexp(s,left_expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ return reduce_neq(s,y,v)
+ case BENeg(BELeq(AEVar(x), AEVar(y))):
+ vx = eval_aexp(s,AEVar(x))
+ vy = eval_aexp(s,AEVar(y))
+ if vx is None or vy is None: return None
+ if not isinstance(vx,Top) and vx.upper_bound is not None:
+ s = reduce_leq(s,y,vx.upper_bound - 1)
+ if not isinstance(vy,Top) and vy.lower_bound is not None:
+ s = reduce_geq(s,x,vy.lower_bound + 1)
+ return s
+ case BENeg(BELeq(AEVar(x),right_expr)):
+ v = eval_aexp(s,right_expr)
+ if v is None: return None
+ if isinstance(v,Top) or v.lower_bound is None: return s
+ return reduce_geq(s,x,v.lower_bound + 1)
+ case BENeg(BELeq(left_expr,AEVar(y))):
+ v = eval_aexp(s,left_expr)
+ if v is None: return None
+ if isinstance(v,Top) or v.upper_bound is None: return s
+ return reduce_leq(s,y,v.upper_bound - 1)
+ case _: return s
+
+class Interval_analysis(Transfer[state]):
+ variables: frozenset[str]
+ def __init__(self,instr: Instr):
+ self.variables = variables_of_instr(instr)
+ def bottom(self) -> state:
+ return None
+ def init_state(self) -> state:
+ return dict.fromkeys(self.variables, Interval(None,None))
+ def join(self,s1:state,s2:state) -> state:
+ if s1 is None: return s2
+ if s2 is None: return s1
+ res: abstract_env = {}
+ for var in self.variables:
+ v1 = s1[var]
+ v2 = s2[var]
+ v = interval_join(v1,v2)
+ res[var] = v
+ return res
+
+ def widen(self, s1:state, s2:state) -> state:
+ raise NotImplementedError
+
+ def included(self,s1: state,s2: state) -> bool:
+ if s1 is None: return True
+ if s2 is None: return False
+ for var in self.variables:
+ if not interval_leq(s1[var],s2[var]): return False
+ return True
+
+ def tr_skip(self,s: state) -> state:
+ return s
+
+ def tr_set(self,s: state,v: str,e: ArithExpr) -> state:
+ raise NotImplementedError
+
+ def tr_test(self,s: state,c: BoolExpr) -> state:
+ raise NotImplementedError
+
+ def tr_err(self,s: state,e: Expr) -> state:
+ raise NotImplementedError
+
+def analyze(i: Instr) -> None:
+ cfg = Cfg(i)
+ res = fixpoint_iteration(Interval_analysis(i), cfg)
+ for node in cfg.g.nodes:
+ print(f"State at {node}:")
+ s = res[node]
+ if s is not None:
+ for (v, s) in s.items():
+ print(f" {v}: {s}")
diff --git a/TP3/config/iteration.py b/TP3/config/iteration.py
new file mode 100644
index 0000000000000000000000000000000000000000..f91169661c566371dc324842b44f562fe95b4d76
--- /dev/null
+++ b/TP3/config/iteration.py
@@ -0,0 +1,46 @@
+"""Fixpoint iterator over a CfG
+
+- Transfer is a base class that must be instantiated for
+each static analysis.
+- fixpoint_iteration computes a fixpoint over a cfg for the given
+transfer instance
+"""
+from cfg import *
+from typing import Protocol
+
+class Transfer[S](Protocol):
+ """lists the functions that must be implemented for a static analysis over Lattice S"""
+ def bottom(self) -> S:
+ """the empty state, associated to unvisited nodes"""
+ ...
+ def init_state(self) -> S:
+ """the state associated to the initial node"""
+ ...
+ def join(self,s1:S,s2:S) -> S:
+ """join of two states when merging branches"""
+ ...
+ def included(self,s1: S,s2: S) -> bool:
+ """return True when the first state is included in the second"""
+ ...
+ def tr_skip(self,s: S) -> S:
+ """transfer state across a Skip transition
+
+ Almost always transfer the state unchanged
+ """
+ return s
+ def tr_set(self,s: S,v: str,e: ArithExpr) -> S:
+ """transfer state across an assignment"""
+ ...
+ def tr_test(self,s: S,c: BoolExpr) -> S:
+ """transfer state across a test transition"""
+ ...
+ def tr_err(self,s: S,e: Expr) -> S:
+ """transfer state across a transition to error state"""
+ ...
+
+def fixpoint_iteration[T](transfer: Transfer[T], cfg: Cfg) -> dict[Node,T]:
+ """computes the fixpoint for the given transfer functions over the given CfG
+
+ output maps each node to the computed state
+ """
+ raise NotImplementedError
diff --git a/TP3/config/opsem.py b/TP3/config/opsem.py
new file mode 100644
index 0000000000000000000000000000000000000000..1313e985e2c11ca43cc8fcaa926eec02eba135f1
--- /dev/null
+++ b/TP3/config/opsem.py
@@ -0,0 +1,50 @@
+"""
+Operational semantics for the Why language
+
+Provides evaluation functions for the three syntactic classes,
+based on a environment mapping variables to integer values.
+These functions return None when an error occur somewhere (Big step semantics)
+"""
+
+from syntax import *
+
+def eval_aexp(env: dict[str,int], exp: ArithExpr) -> int | None:
+ """evaluates an arithmetic expression"""
+ match exp:
+ case AECst(value):
+ raise NotImplementedError
+ case AEVar(var):
+ raise NotImplementedError
+ case AEUop(uop,expr):
+ raise NotImplementedError
+ case AEBop(bop,left_expr,right_expr):
+ raise NotImplementedError
+ case _: assert False
+
+def eval_bexp(env: dict[str,int], exp:BoolExpr) -> bool | None:
+ """evaluates a boolean expression"""
+ match exp:
+ case BEPlain(aexpr):
+ raise NotImplementedError
+ case BEEq(left_expr,right_expr):
+ raise NotImplementedError
+ case BELeq(left_expr,right_expr):
+ raise NotImplementedError
+ case BENeg(expr):
+ raise NotImplementedError
+ case _: assert False
+
+def eval_instr(env: dict[str,int], instr: Instr) -> dict[str,int] | None:
+ """evaluates an instruction"""
+ match instr:
+ case ISkip():
+ raise NotImplementedError
+ case ISet(var,expr):
+ raise NotImplementedError
+ case ISeq(first,second):
+ raise NotImplementedError
+ case ICond(cond,true_branch,false_branch):
+ raise NotImplementedError
+ case ILoop(cond,body):
+ raise NotImplementedError
+ case _: assert False
diff --git a/TP3/config/sign_analysis.py b/TP3/config/sign_analysis.py
new file mode 100644
index 0000000000000000000000000000000000000000..19439ad4196f7b5c80560aed07cf07b701aec4cd
--- /dev/null
+++ b/TP3/config/sign_analysis.py
@@ -0,0 +1,448 @@
+# pyright: strict
+"""
+implements a sign analysis
+"""
+from enum import Enum, auto
+from dataclasses import dataclass
+from cfg import Cfg
+from iteration import Transfer, fixpoint_iteration
+from syntax import *
+
+class sign(Enum):
+ SNEG = auto()
+ SPOS = auto()
+ Z = auto()
+ NEG = auto()
+ POS = auto()
+ NZ = auto()
+ INT = auto()
+
+class Top:
+ def __str__(self): return "TOP"
+
+@dataclass
+class BTop:
+ """class used for the evaluation of boolean expressions
+
+ BTop(False) indicates that we don't know if the result is True or False, but
+ that the evaluation cannot lead to an error
+ BTop(True) indicates that we neither know the result nor whether an error can occur
+ """
+ has_error: bool
+ def __str__(self):
+ if self.has_error: return "TOP (maybe error)"
+ else: return "TOP (no error)"
+
+type abstract_env = dict[str, sign]
+"""mapping from variables to abstract values.
+
+As with concrete environment, a variable not in the dictionary will
+lead to an error if we try to obtain its value
+"""
+type state = abstract_env | None
+"""abstract state is either an abstract env or bottom
+"""
+
+def sign_leq(v1: sign, v2:sign) -> bool:
+ """order relation on the sign lattice"""
+ if v1 == v2: return True
+ match v1,v2:
+ case _, sign.INT: return True
+ case sign.INT, _: return False
+ case sign.SNEG,(sign.NEG | sign.NZ): return True
+ case sign.SPOS,(sign.POS | sign.NZ): return True
+ case sign.Z,(sign.POS | sign.NEG): return True
+ case sign.NEG,sign.NZ: return True
+ case sign.POS,sign.NZ: return True
+ case _: return False
+
+def sign_join(v1: sign, v2:sign) -> sign:
+ """computes least upper bound"""
+ match v1,v2:
+ case sign.INT, _: return sign.INT
+ case _, sign.INT: return sign.INT
+ case sign.NZ, (sign.NZ | sign.SPOS | sign.SNEG): return sign.NZ
+ case sign.NZ,_: return sign.INT
+ case sign.NEG, (sign.NEG | sign.SNEG | sign.Z): return sign.NEG
+ case sign.NEG, _: return sign.INT
+ case sign.POS, (sign.POS | sign.SPOS | sign.Z): return sign.POS
+ case sign.POS,_: return sign.INT
+ case sign.SNEG, sign.SNEG: return sign.SNEG
+ case sign.SNEG, (sign.Z | sign.NEG): return sign.NEG
+ case sign.SNEG, (sign.SPOS | sign.NZ): return sign.NZ
+ case sign.SNEG, _: return sign.INT
+ case sign.SPOS, sign.SPOS: return sign.SPOS
+ case sign.SPOS, (sign.POS | sign.Z): return sign.POS
+ case sign.SPOS, (sign.SNEG | sign.NZ): return sign.NZ
+ case sign.SPOS, _: return sign.INT
+ case sign.Z, sign.SNEG: return sign.NEG
+ case sign.Z, sign.SPOS: return sign.POS
+ case sign.Z, sign.NZ: return sign.INT
+ case sign.Z, _: return v2
+
+def sign_meet(s1: sign, s2:sign) -> sign | None:
+ """computes greatest lower bound. Can be None (i.e. bottom)"""
+ match s1,s2:
+ case sign.INT, _: return s2
+ case _, sign.INT: return s1
+ case sign.SNEG, (sign.NEG | sign.NZ | sign.SNEG): return s1
+ case (sign.NEG | sign.NZ | sign.SNEG), sign.SNEG: return s2
+ case sign.SNEG, _: return None
+ case _, sign.SNEG: return None
+ case sign.SPOS, (sign.POS | sign.NZ | sign.SPOS): return s1
+ case (sign.POS | sign.NZ | sign.SPOS), sign.SPOS: return s2
+ case sign.SPOS, _: return None
+ case _, sign.SPOS: return None
+ case sign.Z, (sign.NEG | sign.POS): return s1
+ case (sign.NEG | sign.POS), sign.Z: return s2
+ case sign.Z, sign.Z: return s1
+ case sign.Z, _: return None
+ case _, sign.Z: return None
+ case sign.NEG, sign.NEG: return s1
+ case sign.POS, sign.POS: return s1
+ case sign.NEG, sign.POS: return sign.Z
+ case sign.POS, sign.NEG: return sign.Z
+ case sign.NZ, sign.NEG: return sign.SNEG
+ case sign.NEG, sign.NZ: return sign.SNEG
+ case sign.NZ, sign.POS: return sign.SPOS
+ case sign.POS, sign.NZ: return sign.SPOS
+ case sign.NZ, sign.NZ: return sign.NZ
+
+def sign_opp(s:sign) -> sign:
+ """unary minus"""
+ match s:
+ case sign.SNEG: return sign.SPOS
+ case sign.SPOS: return sign.SNEG
+ case sign.POS: return sign.NEG
+ case sign.NEG: return sign.POS
+ case _: return s
+
+def sign_add(s1:sign,s2:sign) -> sign:
+ """addition"""
+ match (s1,s2):
+ case sign.INT,_: return sign.INT
+ case _,sign.INT: return sign.INT
+ case sign.Z,_: return s2
+ case _,sign.Z: return s1
+ case sign.SNEG, (sign.SNEG | sign.NEG): return sign.SNEG
+ case (sign.SNEG | sign.NEG), sign.SNEG: return sign.SNEG
+ case sign.NEG, sign.NEG: return sign.NEG
+ case sign.SPOS, (sign.SPOS | sign.POS): return sign.SPOS
+ case (sign.SPOS | sign.POS), sign.SPOS: return sign.SPOS
+ case sign.POS, sign.POS: return sign.POS
+ case _: return sign.INT
+
+def sign_mul(s1: sign, s2: sign) -> sign:
+ """multiplication"""
+ match s1,s2:
+ case sign.INT,_: return sign.INT
+ case _,sign.INT: return sign.INT
+ case sign.Z,_: return sign.Z
+ case _,sign.Z: return sign.Z
+ case sign.SPOS, _: return s2
+ case _, sign.SPOS: return s1
+ case sign.SNEG, _: return sign_opp(s2)
+ case _,sign.SNEG: return sign_opp(s1)
+ case sign.NZ, _: return sign.INT
+ case _, sign.NZ: return sign.INT
+ case sign.NEG, sign.NEG: return sign.POS
+ case sign.NEG, sign.POS: return sign.NEG
+ case sign.POS, sign.NEG: return sign.NEG
+ case sign.POS, sign.POS: return sign.POS
+
+def sign_div(s1: sign, s2: sign) -> sign | Top | None:
+ """division: can lead to errors even if operands are plain sign"""
+ match s1, s2:
+ case _,sign.Z: return None
+ case _,sign.INT: return Top()
+ case _,sign.POS: return Top()
+ case _,sign.NEG: return Top()
+ case sign.Z,_: return sign.Z
+ case sign.INT,_: return sign.INT
+ case sign.NZ,_: return sign.INT
+ case _,sign.NZ: return sign.INT
+ case (sign.POS | sign.SPOS), sign.SPOS: return sign.POS
+ case (sign.NEG | sign.SNEG), sign.SPOS: return sign.NEG
+ case (sign.POS | sign.SPOS), sign.SNEG: return sign.NEG
+ case (sign.NEG | sign.SNEG), sign.SNEG: return sign.POS
+
+def eval_aexp(env: abstract_env, e: ArithExpr) -> sign | Top | None:
+ """evaluate an arithmetic expression in an abstract environment
+ returns None in case of error
+ """
+ match e:
+ case AECst(value):
+ if value < 0:
+ return sign.NEG
+ elif value > 0:
+ return sign.POS
+ elif value == 0:
+ return sign.Z
+ else:
+ return None
+ case AEVar(var):
+ if var in env:
+ return env[var]
+ else :
+ return None
+ case AEUop(_,expr):
+ s = eval_aexp(env, expr)
+ if s is None:
+ return None
+ elif isinstance(s, Top):
+ return Top()
+ else:
+ return sign_opp(s)
+ case AEBop(bop,left_expr,right_expr):
+ l = eval_aexp(env, left_expr)
+ r = eval_aexp(env, right_expr)
+ if l is None or r is None:
+ return None
+ if isinstance(l, Top) or isinstance(r, Top):
+ return Top()
+ match bop:
+ case Bop.ADD: return sign_add(l, r)
+ case Bop.MUL: return sign_mul(l, r)
+ case Bop.DIV: return sign_div(l, r)
+ case _: pass
+
+def eval_bexp(env: abstract_env, e: BoolExpr) -> bool | BTop | None:
+ """abstract evaluation of a boolean expression"""
+ match e:
+ case BEPlain(aexpr):
+ res = eval_aexp(env, aexpr)
+ if res is None:
+ return None
+ if isinstance(res, Top) :
+ return BTop(True)
+ if res == sign.Z:
+ return False
+ if res == sign.NZ or res == sign.SNEG or res == sign.SPOS :
+ return True
+ return BTop(False)
+ case BEEq(left_expr,right_expr):
+ l = eval_aexp(env, left_expr)
+ r = eval_aexp(env, right_expr)
+ if l is None or r is None:
+ return None
+ if isinstance(l, Top) or isinstance(r, Top) :
+ return BTop(True)
+ match l,r:
+ case sign.SNEG, (sign.Z | sign.POS | sign.SPOS) : return False
+ case (sign.Z | sign.POS | sign.SPOS), sign.SNEG : return False
+ case sign.NEG, sign.SPOS : return False
+ case sign.SPOS, sign.NEG : return False
+ case sign.Z, sign.Z : return True
+ case sign.Z, (sign.SNEG | sign.SPOS | sign.NZ) : return False
+ case (sign.SNEG | sign.SPOS | sign.NZ), sign.Z : return False
+ case _ : return BTop(False)
+ case BELeq(left_expr,right_expr):
+ l = eval_aexp(env, left_expr)
+ r = eval_aexp(env, right_expr)
+ if l is None or r is None:
+ return None
+ if isinstance(l, Top) or isinstance(r, Top) :
+ return BTop(True)
+ match l,r:
+ case sign.SNEG, (sign.Z | sign.POS | sign.SPOS) : return True
+ case sign.NEG, sign.SPOS : return True
+ case sign.Z, sign.SNEG : return False
+ case sign.Z, sign.SPOS : return True
+ case sign.POS, sign.SNEG : return False
+ case sign.SPOS, (sign.SNEG | sign.NEG | sign.Z) : return False
+ case _ : return BTop(False)
+ case BENeg(expr):
+ b = eval_bexp(env, expr)
+ if b is None:
+ return None
+ if isinstance(b, BTop):
+ return b
+ return not b
+ case _: pass
+
+def reduce_eq_sign(s: state, x: str, expr: ArithExpr) -> state:
+ """reduce the value associated to x in s under the hypothesis that x == expr"""
+ if s is None: return None
+ v = eval_aexp(s,expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ res = sign_meet(s[x],v)
+ if res is None: return None
+ return s | { x: res }
+
+def reduce_neq_sign(s: state, x: str, expr: ArithExpr) -> state:
+ """reduce the value associated to x in s under the hypothesis that x != expr"""
+ if s is None: return None
+ v = eval_aexp(s,expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ match s[x],v:
+ case sign.POS, sign.Z: return s | { x: sign.SPOS }
+ case sign.NEG, sign.Z: return s | { x: sign.SNEG }
+ case sign.INT, sign.Z: return s | { x: sign.NZ }
+ case _: return s
+
+def reduce_upper_bound(s: state, x: str, expr: ArithExpr) -> state:
+ """reduce the value associated to x in s under the hypothesis that x <= expr"""
+ if s is None: return None
+ v = eval_aexp(s,expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ upper = sign.INT
+ match v:
+ case sign.NEG | sign.SNEG:
+ upper = v
+ case sign.Z:
+ upper = sign.NEG
+ case _: pass
+ res = sign_meet(s[x],upper)
+ if res is None: return None
+ return s | { x: res }
+
+def reduce_strict_upper_bound(s: state, x: str, expr: ArithExpr) -> state:
+ """reduce the value associated to x in s under the hypothesis that x < expr"""
+ if s is None: return None
+ v = eval_aexp(s,expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ match v:
+ case sign.NEG | sign.SNEG | sign.Z:
+ upper = sign.SNEG
+ case _:
+ upper = sign.INT
+ res = sign_meet(s[x],upper)
+ if res is None: return None
+ return s | { x: res }
+
+def reduce_lower_bound(s: state, x: str, expr: ArithExpr) -> state:
+ """reduce the value associated to x in s under the hypothesis that x >= expr"""
+ if s is None: return None
+ v = eval_aexp(s,expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ lower = sign.INT
+ match v:
+ case sign.POS | sign.SPOS:
+ lower = v
+ case sign.Z:
+ lower = sign.POS
+ case _: pass
+ res = sign_meet(s[x],lower)
+ if res is None: return None
+ return s | { x: res }
+
+def reduce_strict_lower_bound(s: state, x: str, expr: ArithExpr) -> state:
+ """reduce the value associated to x in s under the hypothesis that x > expr"""
+ if s is None: return s
+ v = eval_aexp(s,expr)
+ if v is None: return None
+ if isinstance(v,Top): return s
+ match v:
+ case sign.POS | sign.SPOS | sign.Z:
+ lower = sign.SPOS
+ case _:
+ lower = sign.INT
+ res = sign_meet(s[x],lower)
+ if res is None: return None
+ return s | { x: res }
+
+def reduce_state(s: state,c: BoolExpr) -> state:
+ if s is None: return s
+ match c:
+ case BEPlain(aexpr):
+ if isinstance(aexpr, AEVar):
+ return reduce_state(s, BENeg(BEEq(aexpr, AECst(0))))
+ else: return s
+ case BEEq(left_expr,right_expr):
+ if isinstance(left_expr, AEVar) and isinstance(right_expr, AEVar):
+ return reduce_eq_sign(reduce_eq_sign(s, left_expr.var, right_expr), right_expr.var, left_expr)
+ if isinstance(left_expr, AEVar):
+ return reduce_eq_sign(s, left_expr.var, right_expr)
+ if isinstance(right_expr, AEVar):
+ return reduce_eq_sign(s, right_expr.var, left_expr)
+ return s
+ case BELeq(left_expr,right_expr):
+ if isinstance(left_expr, AEVar) and isinstance(right_expr, AEVar):
+ return reduce_lower_bound(reduce_upper_bound(s, left_expr.var, right_expr), right_expr.var, left_expr)
+ if isinstance(left_expr, AEVar):
+ return reduce_upper_bound(s, left_expr.var, right_expr)
+ if isinstance(right_expr, AEVar):
+ return reduce_lower_bound(s, right_expr.var, left_expr)
+ return s
+ case BENeg(expr):
+ match expr:
+ case BEPlain(aexpr):
+ if isinstance(aexpr, AEVar):
+ return reduce_eq_sign(s, aexpr.var, AECst(0))
+ else: return s
+ case BEEq(left_expr,right_expr):
+ if isinstance(left_expr, AEVar) and isinstance(right_expr, AEVar):
+ return reduce_neq_sign(reduce_neq_sign(s, left_expr.var, right_expr), right_expr.var, left_expr)
+ if isinstance(left_expr, AEVar):
+ return reduce_neq_sign(s, left_expr.var, right_expr)
+ if isinstance(right_expr, AEVar):
+ return reduce_neq_sign(s, right_expr.var, left_expr)
+ return s
+ case BELeq(left_expr,right_expr):
+ if isinstance(left_expr, AEVar) and isinstance(right_expr, AEVar):
+ return reduce_strict_upper_bound(reduce_strict_lower_bound(s, left_expr.var, right_expr), right_expr.var, left_expr)
+ if isinstance(left_expr, AEVar):
+ return reduce_strict_lower_bound(s, left_expr.var, right_expr)
+ if isinstance(right_expr, AEVar):
+ return reduce_strict_upper_bound(s, right_expr.var, left_expr)
+ return s
+ case BENeg(expr):
+ return reduce_state(s, c)
+ case _: return s
+ case _: return s
+
+class Sign_interp(Transfer[state]):
+ variables: frozenset[str]
+ def __init__(self,instr: Instr):
+ self.variables = variables_of_instr(instr)
+ def bottom(self) -> state:
+ return None
+ def init_state(self) -> state:
+ return dict.fromkeys(self.variables, sign.INT)
+ def join(self,s1:state,s2:state) -> state:
+ if s1 is None: return s2
+ if s2 is None: return s1
+ res: abstract_env = {}
+ for var in self.variables:
+ v1 = s1[var]
+ v2 = s2[var]
+ res[var] = sign_join(v1,v2)
+ return res
+
+ def included(self,s1: state,s2: state) -> bool:
+ if s1 is None: return True
+ if s2 is None: return False
+ for var in self.variables:
+ if not sign_leq(s1[var], s2[var]): return False
+ return True
+
+ def tr_skip(self,s: state) -> state:
+ return s
+
+ def tr_set(self,s: state,v: str,e: ArithExpr) -> state:
+ raise NotImplementedError
+
+ def tr_test(self,s: state,c: BoolExpr) -> state:
+ raise NotImplementedError
+
+ def tr_err(self,s: state,e: Expr) -> state:
+ if s is None: return s
+ if isinstance(e,ArithExpr):
+ raise NotImplementedError
+ if isinstance(e,BoolExpr):
+ raise NotImplementedError
+
+def analyze(i: Instr) -> None:
+ cfg = Cfg(i)
+ res = fixpoint_iteration(Sign_interp(i), cfg)
+ for node in cfg.g.nodes:
+ print(f"State at {node}:")
+ s = res[node]
+ if s is not None:
+ for (v, s) in s.items():
+ print(f" {v}: {s}")
diff --git a/TP3/config/syntax.py b/TP3/config/syntax.py
new file mode 100644
index 0000000000000000000000000000000000000000..c0938bda1f9a7554cc4db3a13763d0d68732c5bc
--- /dev/null
+++ b/TP3/config/syntax.py
@@ -0,0 +1,185 @@
+"""Nodes of the abstract syntax tree of the program:
+
+- Uop/Bop represent unary/binary arithmetic operator
+- ArithExpr and its children describe arithmetic expressions
+- BoolExpr and its children describe boolean expressions
+- Instr and its children describe instruction
+
+"""
+from enum import Enum
+from dataclasses import dataclass
+
+class Uop(Enum):
+ """unary operators. Currently only unary minus"""
+ OPP = 1
+ def __str__(self):
+ if self is Uop.OPP:
+ return "-"
+ return ""
+
+class Bop(Enum):
+ """binary operators: addition, multiplication, and division"""
+ ADD = 1
+ MUL = 2
+ DIV = 3
+ def __str__(self):
+ match self:
+ case Bop.ADD: return "+"
+ case Bop.MUL: return "*"
+ case Bop.DIV: return "/"
+
+class Expr:
+ """generic class for expressions, either arithmetic or boolean"""
+ pass
+
+@dataclass
+class ArithExpr(Expr):
+ """generic class for arithmetic expressions"""
+ pass
+
+@dataclass
+class AECst(ArithExpr):
+ """integer constants"""
+ value: int
+ def __str__(self):
+ return f"{self.value}"
+
+@dataclass
+class AEVar(ArithExpr):
+ """variables"""
+ var: str
+ def __str__(self):
+ return self.var
+
+@dataclass
+class AEUop(ArithExpr):
+ """unary operation"""
+ uop: Uop
+ expr: ArithExpr
+ def __str__(self):
+ return f"{self.uop}{self.expr}"
+
+@dataclass
+class AEBop(ArithExpr):
+ """binary operation"""
+ bop: Bop
+ left_expr: ArithExpr
+ right_expr: ArithExpr
+ def __str__(self):
+ #TODO: only put parentheses if necessary
+ return f"({self.left_expr}) {self.bop} ({self.right_expr})"
+
+@dataclass
+class BoolExpr(Expr):
+ """generic class for boolean expressions"""
+ pass
+
+@dataclass
+class BEPlain(BoolExpr):
+ """arithmetic expression (true if not 0)"""
+ aexpr: ArithExpr
+ def __str__(self):
+ return f"{self.aexpr}"
+
+@dataclass
+class BEEq(BoolExpr):
+ """equality between two arithmetic expression"""
+ left_expr: ArithExpr
+ right_expr:ArithExpr
+ def __str__(self):
+ return f"{self.left_expr} = {self.right_expr}"
+
+@dataclass
+class BELeq(BoolExpr):
+ """less or equal"""
+ left_expr: ArithExpr
+ right_expr: ArithExpr
+ def __str__(self):
+ return f"{self.left_expr} <= {self.right_expr}"
+
+@dataclass
+class BENeg(BoolExpr):
+ """Negation"""
+ expr: BoolExpr
+ def __str__(self):
+ return f"!{self.expr}"
+
+class Instr:
+ """generic class for instruction"""
+ pass
+
+@dataclass
+class ISkip(Instr):
+ """Skip (do nothing)"""
+ def __str__(self):
+ return "skip"
+
+@dataclass
+class ISet(Instr):
+ """assignment to a variable"""
+ var: str
+ expr: ArithExpr
+ def __str__(self):
+ return f"{self.var} := {self.expr}"
+
+@dataclass
+class ISeq(Instr):
+ """sequence of two instructions"""
+ first: Instr
+ second: Instr
+ def __str__(self):
+ return f"{self.first};\n{self.second}"
+
+@dataclass
+class ICond(Instr):
+ """if then else"""
+ cond: BoolExpr
+ true_branch: Instr
+ false_branch: Instr
+ def __str__(self):
+ return f"if {self.cond} then\n{self.true_branch}\nelse\n{self.false_branch}\nfi"
+
+@dataclass
+class ILoop(Instr):
+ """while loop"""
+ cond: BoolExpr
+ body: Instr
+ def __str__(self):
+ return f"while {self.cond} do\n{self.body}\ndone"
+
+def variables_of_aexpr(e: ArithExpr) -> frozenset[str]:
+ """return the set of variables appearing in the expression"""
+ match e:
+ case AECst(_): return frozenset()
+ case AEVar(var): return frozenset(var)
+ case AEUop(expr=e): return variables_of_aexpr(e)
+ case AEBop(left_expr=le,right_expr=re):
+ return variables_of_aexpr(le) | variables_of_aexpr(re)
+ case _: assert False
+
+def variables_of_bexpr(e: BoolExpr) -> frozenset[str]:
+ """return the set of variable appearing in the expression"""
+ match e:
+ case BEPlain(expr): return variables_of_aexpr(expr)
+ case BEEq(left_expr=le,right_expr=re):
+ return variables_of_aexpr(le) | variables_of_aexpr(re)
+ case BELeq(left_expr=le,right_expr=re):
+ return variables_of_aexpr(le) | variables_of_aexpr(re)
+ case BENeg(expr):
+ return variables_of_bexpr(expr)
+ case _: assert False
+
+def variables_of_instr(instr: Instr) -> frozenset[str]:
+ """return the set of variables appearing in a program"""
+ match instr:
+ case ISkip(): return frozenset()
+ case ISet(var=v,expr=e):
+ return frozenset([v]) | variables_of_aexpr(e)
+ case ISeq(first=i1,second=i2):
+ return variables_of_instr(i1) | variables_of_instr(i2)
+ case ICond(cond=c,true_branch=tb,false_branch=fb):
+ vbranches = variables_of_instr(tb) | variables_of_instr(fb)
+ return variables_of_bexpr(c) | vbranches
+ case ILoop(cond=c,body=b):
+ return variables_of_bexpr(c) | variables_of_instr(b)
+ case _ : assert False
diff --git a/TP3/tp.md b/TP3/tp.md
new file mode 100644
index 0000000000000000000000000000000000000000..fabc311543f5397040f5eeec390b13df7c11be54
--- /dev/null
+++ b/TP3/tp.md
@@ -0,0 +1,110 @@
+# Analyse Statique de Programmes -- TP Analyse Statique
+
+CentraleSupélec
+
+Enseignant: Virgile Prevosto
+
+## Préliminaires
+
+Ce TP est la suite directe du [précédent](tp.md), et vise à implémenter une
+analyse des signes et une analyse des intervalles en plus de la propagation de
+constante vue précédemment. En particulier, il est impératif d'avoir terminé
+l'implémentation du calcul de point fixe (c'est-à-dire avoir complété la
+fonction `fixpoint_iteration` du fichier [`iteration.py`](config/iteration.py))
+pour pouvoir exécuter les exemples proposés dans ce TP.
+
+### Rappel Docker
+L'image est ici: https://github.com/Frederic-Boulanger-UPS/docker-webtop-3asl et peut être utilisée soit localement, soit depuis `MyDocker`. De plus le répertoire `config` est monté automatiquement dans le container docker si vous utilisez les scripts associé ([PowerShell](https://github.com/Frederic-Boulanger-UPS/docker-webtop-3asl/blob/main/start-3asl.ps1) pour Windows ou [sh](https://github.com/Frederic-Boulanger-UPS/docker-webtop-3asl/blob/main/start-3asl.sh) pour Linux/macOS/BSD). Ces scripts devraient automatiquement ouvrir un onglet de votre navigateur web avec une session IceWM. Si ce n'est pas le cas, vous pouvez le faire manuellement: http://localhost:3000
+
+NB: l'interpréteur Python est `python3` et non `python`.
+
+### Typage statique
+Les fichiers proposés ont des annotations de types permettant de vérifier que
+les programmes sont bien typés avant de les exécuter. On pourra utiliser
+l'outil [`pyright`](https://microsoft.github.io/pyright/#/) pour la vérification. Il n'est pas présent sur l'image
+Docker, mais peut s'installer via les commandes suivantes:
+```sh
+sudo apt update
+sudo apt install pipx
+sudo apt install pyright
+```
+
+Il dispose également d'un plugin vscode.
+
+## Treillis des signes
+
+Dans un premier temps, on va s'intéresser à l'analyse des signes telle que vue en cours. Le but de cet exercice est de
+compléter le fichier [`sign_analysis.py`](config/sign_analysis.py). Les différentes valeurs possibles sont définies dans
+l'énumération `sign`, complétée par un élément `Top()` (une erreur a pu avoir lieu, mais n'est pas certaine), et `None` qui
+indique comme précédemment l'absence de résultat, c'est à dire le fait qu'on a certainement une erreur. Une variante de `Top()`,
+`BTop(True|False)` sera utilisée pour l'évaluation abstraite des expressions booléennes.
+
+Comme pour la propagation de constante, les environnements abstraits sont des dictionnaires associant un `sign` à un nom de variable,
+et un état est soit un environnement, soit `None` si on n'est jamais passé par le nœud correspondant.
+
+On se donne également les opérations de treillis sur les `sign` (relation d'ordre, `join` et `meet`, ce dernier pouvant renvoyer
+`None`), ainsi que les opérations arithmétiques abstraites.
+
+### Évaluation des expressions
+
+Compléter les fonctions `eval_aexp` et `eval_bexp`, et interpréter les résultats affichés par
+[`example_sign_eval_exp.py`](config/example_sign_eval_exp.py).
+
+NB: les opérations arithmétiques données travaillent uniquement sur des `sign`. C'est donc à `eval_aexp` de prendre en compte les
+cas `None` et `Top`.
+
+### Réduction d'un état
+
+Compléter la fonction `reduce_state` qui étant donné un état et une expression booléenne essaie de réduire cet état sous l'hypothèse que
+l'expression s'évalue à `True`. On pourra s'aider des fonctions définies au-dessus, qui prennent en argument un état, un nom de
+variable `x`, et une expression arithmétique `expr`, et réduisent si possible la valeur associée à `x` en fonction de la comparaison
+effectuée et de la valeur de `expr`. Le fichier [`example_sign_reduce.py`](config/example_sign_reduce.py) contient un certain nombre
+de tests pour cette fonction.
+
+### Fonctions de transfert
+
+Compléter les fonctions de transfert de la classe `Sign_interp` et tester le résultat avec [`example_sign.py`](config/example_sign.py),
+qui effectue l'analyse des signes pour le calcul de la factorielle
+
+## Treillis des intervalles, début
+
+On s'intéresse maintenant au treillis des intervalles. Le fichier [`interval_analysis.py`](config/interval_analysis.py) suit
+la même structure que pour le treillis des signes, si ce n'est que les valeurs abstraites sont maintenant des objets de la classe
+`Interval`, comportant deux champs, `lower_bound` et `upper_bound`, donc chacun peut être un entier ou `None`, ce qui indique qu'il
+n'y a pas de borne.
+
+On demande cette fois-ci de compléter les fonctions `interval_add` (addition d'intervalles) et `interval_div` (division d'intervalle).
+Pour `interval_div`, on pourra notamment utiliser les fonctions `is_zero` et `contains_zero` qui renvoient `True` si leur argument est
+respectivement le singleton `0` ou un intervalle contenant `0`. On pourra ensuite tester ces fonctions avec
+[`example_interval_eval_exp.py`](config/example_interval_eval_exp.py)
+
+Compléter ensuite les fonctions `reduce_eq`, `reduce_neq`, `reduce_leq` et `reduce_geq`, qui réduisent
+la valeur d'une variable `x` sous l'hypothèse qu'elle est respectivement égale, différente, inférieure ou égale, ou supérieur ou égale à un entier donné. On pourra ensuite tester ces fonctions avec [`example_interval_reduce.py`](config/example_interval_reduce.py)
+
+## Ajout de l'élargissement
+
+Pour l'instant, notre fonction d'itération ne fait pas d'élargissement, donc l'analyse d'intervalle risque de ne pas terminer.
+Modifier dans le fichier [`cfg.py`](config/cfg.py) la classe `LSkip` de manière à ce qu'elle ait un champ indiquant s'il s'agit
+d'un arc de retour vers le nœud initial d'une boucle:
+
+```python
+@dataclass
+class LSkip(Label):
+ """label skip (does nothing)"""
+ is_back_edge: bool
+ def __str__(self):
+ return "skip"
+```
+
+Dans ce même fichier, mettez à jour `cfg_aux` pour créer des labels `LSkip` avec le booléen adéquat
+
+Modifier également le fichier [`iteration.py`](config/iteration.py) de la manière suivante:
+- ajouter une fonction `widen` à la class `Transfer`, qui par défaut fait un simple `self.join`
+- modifier `fixpoint_iteration` selon la version mise à jour de l'algorithme telle que montrée en cours.
+
+On pourra tester l'algorithme avec le fichier [`example_widen.py`](config/example_widen.py), dont l'analyse compte le nombre maximum
+d'arêtes rencontrées depuis le début du programme.
+
+## Treillis des intervalles, version finale
+
+Compléter la fonction `widen` et les fonctions de transfert de la classe `Interval_analysis`, et faire l'analyse d'intervalle du calcul de la factorielle avec le fichier [`example_interval.py`](config/example_interval.py)