Mathematical Tools MPRI 2{6: Abstract Interpretation, application to verification and static analysis Antoine Min´e CNRS, Ecole´ normale sup´erieure course 1, 2012{2013 course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 1 / 44 Order theory course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 2 / 44 Partial orders Partial orders course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 3 / 44 Partial orders Partial orders Given a set X , a relation v2 X × X is a partial order if it is: 1 reflexive: 8x 2 X ; x v x 2 antisymmetric: 8x; y 2 X ; x v y ^ y v x =) x = y 3 transitive: 8x; y; z 2 X ; x v y ^ y v z =) x v z. (X ; v) is a poset (partially ordered set). If we drop antisymmetry, we have a preorder instead. course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 4 / 44 Partial orders Examples of posets (Z; ≤) is a poset (in fact, completely ordered) (P(X ); ⊆) is a poset (not completely ordered) (S; =) is a poset for any S course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 5 / 44 Partial orders Examples of posets (cont.) Given by a Hasse diagram, e.g.: g e f g v g c d f v f ; g e v e; g b d v d; f ; g c v c; e; f ; g b v b; c; d; e; f ; g a a v a; b; c; d; e; f ; g course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 6 / 44 Partial orders Examples of posets (cont.) Infinite Hasse diagram for (N [ f 1 g; ≤): oo 3 2 1 v 1 1 ··· 1 v 1; 2;:::; 1 0 v 0; 1; 2;:::; 1 0 course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 7 / 44 Partial orders Informal uses of posets Posets are a very useful notion to discuss about: logic: ordered by implication =) approximations: v is an information order program verification: program semantics v specification course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 8 / 44 Partial orders (Least) Upper bounds c is an upper bound of a and b if: a v c and b v c c is a least upper bound(lub or join) of a and b if c is an upper bound of a and b for every upper bound d of a and b, c v d The lub is unique and noted a t b. (proof: assume that c and d are both lubs of a and b; by definition of lubs, c v d and d v c; by antisymmetry of v, c = d) Generalized to upper bounds of arbitrary (even infinite) sets t Y ; Y ⊆ X (well-defined, as t is commutative and associative). Similarly, we define greatest lower bounds(glb, meet) a u b, u Y . (a u b v a; b and 8c; c v a; b =) c v a u b) Note: not all posets have lubs, glbs; e.g., (f a; b g; =). course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 9 / 44 Partial orders (Least) Upper bounds upper bounds of a and b a t b upper bound of b a b course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 10 / 44 Partial orders Complete partial order (CPO) C ⊆ X is a chain in (X ; v) if it is totally ordered (8x; y 2 C; x v y _ y v x). A poset (X ; v) is a complete partial order (CPO) if every chain C (including ;) has a least upper bound t C. A CPO has a least element t ;, denoted ?. Examples: (N; ≤) is not complete, but( N [ f 1 g; ≤) is complete. (f x 2 Q j 0 ≤ x ≤ 1 g; ≤) is not complete, but (f x 2 R j 0 ≤ x ≤ 1 g; ≤) is complete. (P(Y ); ⊆) is complete for any Y . course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 11 / 44 Lattices Lattices course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 12 / 44 Lattices Lattices A lattice( X ; v; t; u) is a poset with 1 a lub a t b for every pair of elements a and b; 2 a glb a u b for every pair of elements a and b. Examples: integer intervals( f [a; b] j a; b 2 Z; a ≤ b g [ f ; g; ⊆; t; \) where [a; b] t [a0; b0] def= [min(a; a0); max(b; b0)]. ∗ divisibility( N ; j; gcd; lcm) def where xjy ()9 k 2 N; kx = y. If we drop one condition, we have a (join or meet) semilattice. See Birkhoff [Birk76]. course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 13 / 44 Lattices Example: the divisibility lattice 36 8 12 18 27 4 6 9 2 3 5 1 course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 14 / 44 Lattices Complete lattices A complete lattice( X ; v; t; u; ?; >) is a poset with 1 a lub t S for every set S ⊆ X 2 a glb u S for every set S ⊆ X 3 a least element ? 4 a greatest element > Notes: 1 implies 2 as u X = t f y j 8x 2 X ; y v x g (and 2 implies 1 as well), 1 and 2 imply 3 and 4: ? = t ; = u X , > = u ; = t X , a complete lattice is also a CPO. course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 15 / 44 Lattices Complete lattice examples real segment [0; 1]:(f x 2 R j 0 ≤ x ≤ 1 g; ≤; max; min; 0; 1) powersets( P(S); ⊆; [; \; ;; S) any finite lattice (t Y and u Y for finite Y ⊆ X are always defined). integer intervals with finite and infinite bounds: (f [a; b] j a 2 Z [ f −∞ g; b 2 Z [ f +1 g; a ≤ b g [ f ; g; ⊆; t; \; ;; [−∞; +1]) def with ti2I [ai ; bi ] = [mini2I ai ; maxi2I bi ]. course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 16 / 44 Lattices Example: the powerset complete lattice Example: (P(f 0; 1; 2 g); ⊆; [; \; ;; f 0; 1; 2 g) f0; 1; 2g f0; 2g f0; 1g f1; 2g f1g 0g f2gf ; course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 17 / 44 Lattices Derivation Given (complete) posets or lattices (X ; vX ;:::), (Y ; vY ;:::) we can derive new ones by: duality( X ; wX ;:::) 0 0 0 8x; x ; x wX x () x vX x adding a least element ? (lifting) (X [ f ? g; v;:::) 0 0 def 0 8x; x ; x v x () x = ?_ x vX x product (X × Y ; v;:::) 0 0 0 0 def 0 0 8x; x ; y; y ; (x; y) v (x ; y ) () x vX x ^ y vY y point-wise lifting by some set S (S ! X ; v;:::) 0 0 def 0 8x; x ; x v x ()8 s 2 S; x(s) vX x (s) sublattice 0 0 (X ; vX ; tX ; uX ) where X ⊆ X is closed by tX and uX course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 18 / 44 Fixpoints Fixpoints course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 19 / 44 Fixpoints Functions A function f :(X ; vX ;:::) ! (Y ; vY ;:::) is monotonic if 0 0 0 8x; x ; x vX x =) f (x) vY f (x ) (aka: increasing, isotone, order-preserving, morphism) strict if f (?X ) = ?Y continuous between CPOs if 8C chain ⊆ X ; f f (c) j c 2 C g is a chain in Y and f (tX C) = tY f f (c) j c 2 C g a (complete) t−morphism between (complete) lattices if 8S ⊆ X ; f (tX S) = tY f f (s) j s 2 S g extensive if X = Y and 8x; x vX f (x) course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 20 / 44 Fixpoints Fixpoints Given f :(X ; v) ! (X ; v) x is a fixpoint of f if f (x) = x x is a prefixpoint of f if x v f (x) x is a postfixpoint of f if f (x) v x We may have several (or none) fixpoints fp(f ) def= f x 2 X j f (x) = x g def lfpx f = minv f y 2 fp(f ) j x v y g if it exists (least fixpoints) def lfp f = lfp? f dually, gfpx f , gfp f (greatest fixpoints) course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 21 / 44 Fixpoints Tarski's fixpoint theorem Tarksi's theorem If f : X ! X is monotonic in a complete lattice X then fp(f ) is a complete lattice. Proved by Knaster and Tarski [Tars55]. course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 22 / 44 Fixpoints Tarski's fixpoint theorem Tarksi's theorem If f : X ! X is monotonic in a complete lattice X then fp(f ) is a complete lattice. Proof: We prove lfp f = u f x j f (x) v x g (meet of postfixpoints). Let f ∗ = f x j f (x) v x g and a = u f ∗. 8x 2 f ∗; a v x (by definition of u) so f (a) v f (x) (as f is monotonic) so f (a) v x (as x is a postfixpoint). We deduce that f (a) v u f ∗, i.e. f (a) v a. course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 22 / 44 Fixpoints Tarski's fixpoint theorem Tarksi's theorem If f : X ! X is monotonic in a complete lattice X then fp(f ) is a complete lattice.