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Borel Determinacy Borel Determinacy Serge Grigorieff LIAFA, CNRS & Universit´eDenis Diderot-Paris 7 Groupe de travail Calculabilit´e,Complexit´eet Hasard April 12-18th 2012 1 / 65 Alternations of quantifier versus Games 2 / 65 Alternations of quantifier F (~z) ≡ 9x0 8x1 9x2 8x3 9x4 P(x0;:::; x4;~z) The human mind seems limited in its ability to understand and vizualize beyond four or five alternations of quantifier. Indeed, it can be argued that the inventions, subtheories and central lemmas of various parts of mathematics are devices for assisting the mind in dealing with one or two additional alternations of quantifier. Hartley Rogers \Theory of recursive functions and effective computability" (1967) (cf. page 322 x14.7) Another (partial) explanation: 0 ! complexity ≥ Σ4(! ) ; Higher set theory!!! 3 / 65 Alternations of quantifier versus Games F (~z) ≡ 9x0 8x1 9x2 8x3 9x4 P(x0;:::; x4;~z) Roland Fra¨ıss´e'sidea (1954) Relate F (~z) to a game move 0 : I plays some x Two players 0 move 1 : II plays some x I 1 move 2 : I plays some x and 2 move 3 : II plays some x II 3 move 4 : I plays some x4 Who wins? I wins iff P(x0;:::; x4;~z) holds F (~z) () player I has a winning strategy 4 / 65 Strategies move 0 : I plays some x0 The xi 's in X move 1 : II plays some x1 move 2 : I plays some x2 I wins move 3 : II plays some x3 iff move 4 : I plays some x4 P(x0;:::; x4;~z) 2 Strategy for I = σI : fnilg [ X [ X −! X 2 Strategy for II = σI : X [ X −! X 8 x = σ (nil) < 0 I I follows strategy σI if x2 = σI(x1) : x4 = σI(x1; x3) x1 = σII(x0) II follows strategy σII if x3 = σII(x0; x1) Winning strategy: ALWAYS wins 5 / 65 Alternations of quantifier and games F (~z) ≡9 x0 8x1 9x2 8x3 9x4 P(x0;:::; x4;~z) ≡ player I has a winning strategy for the game where I wins if (x0;:::; x4) 2 P :F (~z) ≡8 x0 9x1 8x2 9x3 8x4 :P(x0;:::; x4;~z) ≡ player II has a winning strategy for the game where I wins if (x0;:::; x4) 2 P Law of Excluded Middle: either F (~z) or :F (~z) Hence either I has a winning strategy or II has a winning strategy 6 / 65 Infinitely many alternations of quantifier 9x0 8x1 9x2 8x3 ::: P((xi )i2N;~z) Moschovakis' game quantifier αG P(α;~z) 8x0 9x1 8x2 9x3 ::: :P((xi )i2N;~z) What does this mean? Infinite game Two players Iand II Rule move 2i : I plays x2i I wins iff move 2i + 1 : II plays x2i+1 (xi )i2N 2 A where A = f(xi )i2N j P((xi )i2N;~z)g 7 / 65 Two players I and II Rule move 2i : I plays x2i I wins iff move 2i + 1 : II plays x2i+1 P((xi )i2N;~z) <! Strategy for I = σI : X −! X <! Strategy for II = σII :(X n fnilg) −! X I follows σI if 8i 2 N x2i = σI((x2j+1)j<i ) II follows σII if 8i 2 N x2i+1 = σII((x2j )j≤i ) Winning strategy: ALWAYS wins 9x0 8x1 9x2 8x3 ::: P((xi )i2N;~z) ≡ I has a winning strategy 8x0 9x1 8x2 9x3 ::: :P((xi )i2N;~z) ≡ II has a winning strategy Need X well-ordered or Axiom of dependent choices 8 / 65 Excluded middle and determinacy 9x0 8x1 9x2 8x3 ::: P((xi )i2N;~z) ≡ I has a winning strategy 8x0 9x1 8x2 9x3 ::: :P((xi )i2N;~z) ≡ II has a winning strategy Fact. 8 : (9x 8x 9x 8x ::: P((x ) ;~z)) < 0 1 2 3 i i2N is equivalent to : 8x0 9x1 8x2 9x3 ::: :P((xi )i2N;~z) if and only the game is determined (one of the players has a winning strategy) 9 / 65 Which sets are determined? 10 / 65 Countable sets are determined Infinite game G(A) Two players I and II Rule move 2i : I plays x2i I wins iff move 2i + 1 : II plays x2i+1 (xi )i2N 2 A Fact. If A ⊂ X ! is countable then II has a winning strategy in G(A) Proof. Diagonal argument. If A = ffi j i 2 Ng, 2i + 1 player II plays x2i+1 6= fi (2i + 1) Are all sets determined? NO (requires the axiom of choice) 11 / 65 Borel subsets of X ! Discrete topology on X Product topology on X ! metrics d(f ; g) = 2− maxfnj8i<n f (i)=g(i)g Basis of clopen sets: the uX ! for u 2 X <! Care. if X uncountable, open sets may be unions of uncountably many clopen sets But metrizability implies closed set are G in X <! :::δ (G = intersection of countably many open sets) :::δ This allows for the usual definition of Borel sets 0 ! Σ::1(X ) = open sets 0 ! S 0 ! Σ::α(X ) = countable unions of sets in β<α Π::β(X ) 0 ! 0 ! Π::α(X ) = complements of sets in Σ::α(X ) 12 / 65 Borel determinacy Theorem. (Donald Martin, 1975) All Borel subsets of X ! are determined (whatever big is X ) Find simple winning strategies in G(A)? 1;S Alas. best (general) complexity is ∆2 if A is Borel with code S 1;S Upper bound proof. The set of ws for I is Π1 : σI is ws ≡ 8g σI ? g 2 A 1;S 1;S and every Π1 family contains some ∆2 set (cf. Rogers x16.7 Coro. XLV(c), p. 430) 13 / 65 Determinacy in classical mathematics • 1953, Gale & Stewart Boolean combinations of open subsets of X ! 0 ! 0 ! • 1955, Philip Wolfe Σ::2(X ) and Π::2(X ) sets 0 ! 0 ! • 1964, Morton Davis Σ::3(X ) and Π::3(X ) sets Results proved in 2d-order arithmetic ≡ mathematics of N and P(N) ≡ classical set theory for mathematicians (with N and P(N) one can encode reals, continuous functions,. ) 14 / 65 Determinacy in higher set theory 1 ! • 1970, Donald Martin Σ: 1(! ) in ZF + large cardinal axiom 1 ! Σ: 1(X ) in ZF + stronger large cardinal axiom 0 ! 0 ! • 1972, Jeff B. Paris Σ: 4(X ) and Π: 4(X ) sets in ZF (set theory with cardinal (2@0 )+ is enough hence 3rd-order arithmetic is enough) • 1975, Donald Martin Borel subsets of X ! in ZF • 1985, Donald Martin Much simpler proof (by far. ) in ZF Higher set theory (in ZF) is required!!! • 1971, Harvey Friedman 0 ! For Σ: 5(! ) and beyond, 2d-order arithmetic NOT ENOUGH 0 ! For Σ: 5+α(! ) need α iterations of set exponentiation • ∼2010, Donald Martin 0 ! For Σ: 4(! ) 2d-order arithmetic NOT ENOUGH 15 / 65 A few simple results about determinacy and strategies 16 / 65 Determinacy and complementation If A ⊆ X ! then the shift of A is XA = f(x; x0; x1; x2;:::) j (x0; x1; x2;:::) 2 A)g Let A ⊆ P(X !) be closed under shift: A 2 A =) xA 2 A 8A 2 A A is determined () 8A 2 A X ! n A is determined I has a ws in G(XA) =) II has a ws in G(X ! n A) II has a ws in G(XA) =) I has a ws in G(X ! n A) 17 / 65 Winning strategies viewed as trees <! Strategy σI for I ≡ tree SσI ⊆ X of all plays when I follows σ I u 2 SσI ^ juj even =) 9!x ux 2 SσI u 2 SσI ^ juj odd =) 8x ux 2 SσI (Thus, I always has exactly one possible move and there is no constraint for II-moves) <! Strategy σII for II ≡ tree SσII ⊆ X of all plays when II follows σ II u 2 SσII ^ juj odd =) 9!x ux 2 SσII u 2 SσII ^ juj even =) 8x ux 2 SσII (Thus, II always has exactly one possible move and there is no constraint for I-moves) σI winning for I () [SσI ] ⊆ A ! σII winning for II () [SσII ] ⊆ X n A ([S] = set of infinite branches of S) 18 / 65 Non deterministic winning strategies <! ND strategy σI for I ≡ tree SσI ⊆ X of all plays when I follows σ I SσI is pruned: 8u 2 SσI 9x ux 2 SσI u 2 SσI ^ juj odd =) 8x ux 2 SσI (Thus, I always has some move and there is no constraint for II-moves) <! ND strategy σII for II ≡ tree SσII ⊆ X of all plays when II follows σ II SσI is pruned: 8u 2 SσII 9x ux 2 SσI u 2 SσII ^ juj even =) 8x ux 2 SσII (Thus, II always has some move and there is no constraint for I-moves) σI winning for I () [SσI ] ⊆ A ! σII winning for II () [SσII ] ⊆ X n A 19 / 65 Winning strategies and positions <! ! ! u 2 X A ⊆ X Au = A \ clopen set uX Fact. If juj is odd (next move for II) then II has no winning strategy in G(Au) iff 8x 2 X II has no winning strategy in G(Aux ) (No \miracle" move x for player II) 20 / 65 Winning and Defensive strategies <! ! ! u 2 X A ⊆ X Au = A \ clopen set uX Fact. Let juj even (next move for player I) II has no winning strategy in G(Au) iff 9x 2 X II has no winning strategy in G(Aux ) (Player I has a move x so that II still has no ws afterwards) Always choosing such an x = Defensive strategy for player I CARE: defensive strategy 6) winning strat. 21 / 65 Gale & Stewart's results about 0 ! Σ::: 1(X ) they contain many core ideas of the theory 22 / 65 0 ! 0 ! Determinacy of Σ:: 1(X ) and Π:: 1(X ) Theorem.
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