diff --git a/Cours/cours3.pdf b/Cours/cours3.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..db06682c24c9b4232d143db698a77e041c70c892
Binary files /dev/null and b/Cours/cours3.pdf differ
diff --git a/Pystan/config/example_interval.py b/Pystan/config/example_interval.py
new file mode 100644
index 0000000000000000000000000000000000000000..8aa35e01334ffe53cbe00789994a19c11f544c59
--- /dev/null
+++ b/Pystan/config/example_interval.py
@@ -0,0 +1,4 @@
+from interval_analysis import analyze
+from example_opsem import factorial
+
+analyze(factorial)
diff --git a/Pystan/config/example_interval_eval_exp.py b/Pystan/config/example_interval_eval_exp.py
new file mode 100644
index 0000000000000000000000000000000000000000..1203cab645be505a71cd14888374824eca277e68
--- /dev/null
+++ b/Pystan/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/Pystan/config/example_interval_reduce.py b/Pystan/config/example_interval_reduce.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d7cdda54e9d02cfd00b41b9a65d67b63f5120b9
--- /dev/null
+++ b/Pystan/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/Pystan/config/example_sign.py b/Pystan/config/example_sign.py
new file mode 100644
index 0000000000000000000000000000000000000000..6aa6de3d4adb559dddcd246d9cdb3159f2f12b67
--- /dev/null
+++ b/Pystan/config/example_sign.py
@@ -0,0 +1,4 @@
+from sign_analysis import analyze
+from example_opsem import factorial
+
+analyze(factorial)
diff --git a/Pystan/config/example_sign_eval_exp.py b/Pystan/config/example_sign_eval_exp.py
new file mode 100644
index 0000000000000000000000000000000000000000..c8bdd995dad92d272d407c6a2c040666e1f2ad29
--- /dev/null
+++ b/Pystan/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/Pystan/config/example_sign_reduce.py b/Pystan/config/example_sign_reduce.py
new file mode 100644
index 0000000000000000000000000000000000000000..17ed63624d1e282d9f535cccecbdf5d04e169e20
--- /dev/null
+++ b/Pystan/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/Pystan/config/example_widen.py b/Pystan/config/example_widen.py
new file mode 100644
index 0000000000000000000000000000000000000000..c9372be7b6162f6f47cd8f7ed8a5c2e240aa187d
--- /dev/null
+++ b/Pystan/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/Pystan/config/interval_analysis.py b/Pystan/config/interval_analysis.py
new file mode 100644
index 0000000000000000000000000000000000000000..9186efbf8ffeedb0fe1e207381313552f1b34163
--- /dev/null
+++ b/Pystan/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/Pystan/config/sign_analysis.py b/Pystan/config/sign_analysis.py
new file mode 100644
index 0000000000000000000000000000000000000000..61bd064cf6d836b3edcbf068f5b739187194d229
--- /dev/null
+++ b/Pystan/config/sign_analysis.py
@@ -0,0 +1,338 @@
+# pyright: strict
+"""
+implements a sign analysis
+"""
+from enum import Enum
+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):
+ raise NotImplementedError
+ case AEVar(var):
+ raise NotImplementedError
+ case AEUop(uop,expr):
+ raise NotImplementedError
+ case AEBop(bop,left_expr,right_expr):
+ raise NotImplementedError
+ case _: pass
+
+def eval_bexp(env: abstract_env, e: BoolExpr) -> bool | BTop | None:
+ """abstract evaluation of a boolean expression"""
+ match e:
+ 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 _: 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:
+ # To be completed (see course's slides)
+ 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/Pystan/tp2.md b/Pystan/tp2.md
new file mode 100644
index 0000000000000000000000000000000000000000..fabc311543f5397040f5eeec390b13df7c11be54
--- /dev/null
+++ b/Pystan/tp2.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` Ã 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)