Incompleteness Theorems, Large Cardinals, and Automata over Finite Words Equipe de Logique Mathématique Institut de Mathématiques de Jussieu - Paris Rive Gauche CNRS and Université Paris 7 TAMC 2017, Berne
The Axiomatic System ZFC of Set Theory The axioms of ZFC (Zermelo 1908, Fraenkel 1922) express some natural facts that we consider to hold in the universe of sets. These axioms are first-order sentences in the logical language of set theory whose only non logical symbol is the membership binary relation symbol. The Axiom of Extensionality states that two sets x and y are equal iff they have the same elements: The Powerset Axiom states the existence of the set of subsets of a set x.
Models of the Axiomatic System ZFC A model (V, ) of the axiomatic system ZFC is a collection V of objects called sets, equipped with the membership relation, where x y means that the set x is an element of the set y, which satisfies the axioms of ZFC.
The Topological complexity of a 1-counter ω-language depends on the models of ZFC Theorem ( F. 2009 ) There exists a 1-counter Büchi automaton A such that the topological complexity of the ω-language L(A) is not determined by the axiomatic system ZFC. 1 There is a model V 1 of ZFC in which the ω-language L(A) is an analytic but non Borel set. 2 There is a model V 2 of ZFC in which the ω-language L(A) is a G δ -set (i.e. Π 0 2 -set).
The context-free determinacy Theorem (F. 2011) The determinacy of Gale-Stewart games G(L), where L is accepted by a real-time 1-counter Büchi automaton, is equivalent to the effective analytic determinacy, and thus it is not provable in ZFC.
cardinals Two sets A and B have same cardinality is denoted A B. The relation is an equivalence relation. Using the axiom of choice AC, any set A can be well-ordered so there is an ordinal γ such that A γ. In set theory the cardinal of the set A is then formally defined as the smallest such ordinal γ. The infinite cardinals are denoted by ℵ 0, ℵ 1, ℵ 2,..., ℵ α,... The first infinite ordinal is ω and it is the smallest ordinal which is countably infinite so ℵ 0 = ω. The first uncountable ordinal is ω 1, and formally ℵ 1 = ω 1, and so on...
regular cardinals Let α be a limit ordinal, the cofinality of α, denoted cof (α), is the least ordinal β such that there exists a strictly increasing sequence of ordinals (α i ) i<β, of length β, such that i < β α i < α and sup i<β α i = α. This definition is usually extended to 0 and to the successor ordinals: cof (0) = 0 and cof (α + 1) = 1 for every ordinal α. The cofinality of a limit ordinal is always a limit ordinal satisfying: ω cof (α) α. Moreover cof (α) is in fact a cardinal. A cardinal κ is said to be regular iff cof (κ) = κ. Otherwise cof (κ) < κ and the cardinal κ is said to be singular.
inaccessible cardinals A cardinal κ is said to be a (strongly) inaccessible cardinal iff 1 κ > ω, 2 κ is regular, and 3 for all cardinals λ < κ it holds that 2 λ < κ, where 2 λ is the cardinal of P(λ). κ cannot be reached from smaller cardinals by the use of the powerset operation. This is a notion of large cardinal similar to the fact that ω is large with regard to the finite ordinals, i.e. integers.
The cumulative hierarchy The class of sets in a model V of ZF may be stratified in a transfinite hierarchy, called the Cumulative Hierarchy, which is defined by V = α ON V α, where: 1 V 0 = 2 V α = β<α V β, for α a limit ordinal, and 3 V α+1 = P(V α ). If V is a model of ZFC and κ is an inaccessible cardinal in V then V κ is also a model of ZFC.
One cannot prove in ZFC that there exists an inaccessible cardinal If V is a model of ZFC and κ is an inaccessible cardinal in V then V κ is also a model of ZFC. If there exist in V at least n inaccessible cardinals, where n 1 is an integer, and if κ is the n-th inaccessible cardinal, then V κ is also a model of ZFC + There exist exactly n 1 inaccessible cardinals. This implies that one cannot prove in ZFC that there exists an inaccessible cardinal, because if κ is the first inaccessible cardinal in V then V κ is a model of ZFC + There exist no inaccessible cardinals.
Consistency of a theory A (first-order) theory T in the language of set theory is a set of (first-order) sentences, called the axioms of the theory. If T is a theory and ϕ is a sentence then T ϕ iff there is a formal proof of ϕ from T. ( There is a finite sequence of sentences ϕ j, 1 j n, such that ϕ 1 ϕ 2... ϕ n, where ϕ n is the sentence ϕ and for each j [1, n], either ϕ j is in T or ϕ j is a logical axiom or ϕ j follows from ϕ 1, ϕ 2,... ϕ j 1 by usual rules of inference which can be defined purely syntactically. ) The theory T is consistent iff for no (first-order) sentence ϕ does T ϕ and T ϕ. If T is inconsistent, then for every sentence ϕ T ϕ.
Gödel s Second Incompleteness Theorem One can code in a recursive manner the sentences in the language of set theory by finite sequences over the alphabet {0, 1}, by using a classical Gödel numbering of the sentences. The theory T is said to be recursive iff the set of codes of axioms in T is a recursive set of words over {0, 1}. one can also code formal proofs from axioms of a recursive theory T and then Cons(T) is an arithmetical statement. The theory ZFC is recursive. Theorem (Gödel 1931) Let T be a consistent recursive extension of ZF or T = PA. Then T Cons(T ).
First Lemma Lemma Let T be a recursive theory (in the language of set theory or T = PA). Then there exists a Turing machine M T, starting on an empty tape, such that M T halts iff T is inconsistent. The machine M T works as a program which enumerates all the formal proofs from T and enters in an accepting state and then halts iff it finds a proof of the sentence x(x x). If the theory T is consistent the machine will never halt. But if the theory is inconsistent then at some point of the computation the machine finds a proof of x(x x) and halts.
Post Correspondence Problem The Post Correspondence Problem PCP is one of the famous undecidable problems in Theoretical Computer Science and in Formal Language Theory. It follows from the undecidability of the Post Correspondence Problem that many problems about context-free languages are also undecidable. the universality problem, the inclusion and the equivalence problems
Post Correspondence Problem Theorem (Post) Let Γ be an alphabet having at least two elements. Then it is undecidable to determine, for arbitrary n-tuples (x 1, x 2,..., x n ) and (y 1, y 2,..., y n ) of non-empty words in Γ, whether there exists a non-empty sequence of indices i 1, i 2,..., i k such that x i1 x i2 x ik = y i1 y i2 y ik.
Post Correspondence Problem and Set Theory Theorem Let T be a recursive theory (in the language of set theory or T = PA). Then there exist two n-tuples X T = (x 1, x 2,..., x n ) and Y T = (y 1, y 2,..., y n ) of finite words over a finite alphabet Σ, such that there exists a non-empty sequence of indices i 1, i 2,..., i k such that x i1 x i2 x ik = y i1 y i2 y ik iff T is inconsistent.
Post Correspondence Problem and Set Theory T n : ZFC + There exist (at least) n inaccessible cardinals Corollary For every integer n 0, there exist p 1 and two p-tuples X T,n = (x 1,n, x 2,n,..., x p,n ) and Y T,n = (y 1,n, y 2,n,..., y p,n ) of finite words over Σ = {a, b}, such that: P n : there exist no non-empty sequence of indices i 1, i 2,..., i k such that: x i1,nx i2,n x ik,n = y i1,ny i2,n y ik,n iff T n is consistent. In particular, if ZFC + There exist (at least) n inaccessible cardinals is consistent, then P n is provable from ZFC + There exist (at least) n + 1 inaccessible cardinals but not from ZFC + There exist (at least) n inaccessible cardinals.
Sketch of the proof P n is equivalent to Cons(T n ) One can prove from ZFC + There exist (at least) n + 1 inaccessible cardinals that if κ is the n + 1-th inaccessible cardinal, then the set V κ of the cumulative hierarchy is a model of ZFC + There exist n inaccessible cardinals. This implies that T n+1 Cons(T n ) (by Completeness Theorem) Thus T n+1 implies the property P n On the other hand if T n is consistent, then the property P n is not provable from T n. Indeed T n is then a consistent recursive extension of ZFC and thus by Gödel s Second Incompleteness Theorem we know that T n Cons(T n ).
Post Correspondence Problem and PA Moreover, since PA is consistent, we also get the following result. Corollary There exist two p-tuples X = (x 1, x 2,..., x p ) and Y = (y 1, y 2,..., y p ) of finite words over Σ = {a, b}, such that: (1) there exist no non-empty sequence of indices i 1, i 2,..., i k such that: x i1 x i2 x ik = y i1 y i2 y ik (2) The property (1) is not provable from PA.
Context-free grammars and Set Theory Theorem Let T be a recursive theory in the language of set theory or T = PA. Then there exists a context-free grammar G T which is unambiguous iff T is consistent. Corollary For every integer n 0, there exists a context-free grammar G n such that G n is unambiguous iff T n is consistent. In particular, if ZFC + There exist (at least) n inaccessible cardinals is consistent, then G n is unambiguous is provable from ZFC + There exist (at least) n + 1 inaccessible cardinals but not from ZFC + There exist (at least) n inaccessible cardinals.
Context-free grammars and Set Theory Corollary For every integer n 0, there exist context-free grammars G 1,n G 2,n, G 3,n, and G 4,n, such that Cons(T n ) is equivalent to each of the following items: (1) L(G 1,n ) L(G 2,n ) = ; (2) L(G 3,n ) = L(G 4,n ); (3) L(G 3,n ) = Γ, for some alphabet Γ. In particular, if ZFC + There exist (at least) n inaccessible cardinals is consistent, then each of the properties of these context-free languages given by Items (1)-(3) is provable from ZFC + There exist (at least) n + 1 inaccessible cardinals but not from ZFC + There exist (at least) n inaccessible cardinals.
Rational relations and Set Theory Theorem Let T be a recursive theory (in the language of set theory or T = PA). Then there exist 2-tape automata A, B, C, and D, accepting finitary rational relations X, Y, Z, L A B, for two alphabets A and B having at least two letters, and such that: Cons(T ) is equivalent to each of the following items: (1) X Y = ; (2) Z = A B ; (3) A B Z ; (4) L is accepted by a deterministic 2-tape automaton; (5) L is accepted by a synchronous 2-tape automaton.
Rational relations and Set Theory Corollary For every integer n 0, there exist 2-tape automata A n, B n, C n, and D n, accepting subsets of A B, for two alphabets A and B having at least two letters, such that Cons(T n ) is equivalent to each of the following items: (1) L(A n ) L(B n ) = ; (2) L(C n ) = A B ; (3) L(D n ) is accepted by a deterministic 2-tape automaton; (4) L(D n ) is accepted by a synchronous 2-tape automaton. In particular, if ZFC + There exist (at least) n inaccessible cardinals is consistent, then each of the properties of these 2-tape automata given by Items (1)-(4) is provable from ZFC + There exist (at least) n + 1 inaccessible cardinals but not from ZFC + There exist (at least) n inaccessible cardinals.
Weighted automata A non-deterministic Z-automaton is a 5-tuple A = (Σ, Q, δ, J, F), where: Σ = {a 1, a 2,..., a k } is a finite input alphabet and the letter a i is associated to a matrice M i Z n n ; Q = {1, 2,..., n} is the state set (and i corresponds to the ith row and column of the matrices); J is the set of initial states and F Q is the set of final states; δ is the set of transitions that provides the rules r (a i m) s, where a i Σ, and m = (M i ) rs is the multiplicity of the rule. A path ( b1 m π = s 1 ) ( b2 m 1 s 2 ) ( bt m 2 s 3 s t ) t s t+1 is a computation of the automaton A reading a word w = b 1 b 2... b t Σ and the muliplicity of this path is equal to π = m 1 m 2... m t Z.
Weighted automata For a word w Σ we denote by Π rs the set of the paths of A reading the word w which go from state r to state s. Then the multiplicity of the word w = a i1 a i2... a it Σ from r to s is the sum A rs (w) = π = (M i1 M i2... M it ) rs π Π rs and we get the multiplicity of w in A from the accepting paths: A(w) = A rs (w) = (M i1 M i2... M it ) rs. r J,s F r J,s F
Theorem Let T be a recursive theory in the language of set theory or T = PA. Then there exists a finite set of matrices M 1, M 2,..., M n Z 3 3, for some integer n 1, such that the subsemigroup of Z 3 3 generated by these matrices does not contain any matrix M with M 13 = 0 if and only if T is consistent.
Corollary There exists a finite set of matrices M 1, M 2,..., M n Z 3 3, for some integer n 1, such that: (1) the subsemigroup of Z 3 3 generated by these matrices does not contain any matrix M with M 13 = 0, and (2) The property (1) is not provable from PA. Corollary Let T be a recursive theory in the language of set theory or T = PA. Then there exists a 3-state Z-automaton A such that A accepts a word with multiplicity zero iff T is inconsistent.
Weighted automata ans Set Theory Corollary Let T be a recursive theory (in the language of set theory or T = PA). Then there exists two 2-state N-automata A and B such that A and B accept a word w with the same multiplicity iff T is inconsistent.
Weighted automata ans Set Theory Theorem Let T be a recursive theory in the language of set theory or T = PA. Then there exists a finite set of matrices M 1, M 2,..., M n Z 3 3, for some integer n 1, such that the subsemigroup of Z 3 3 generated by these matrices does not contain the zero matrix if and only if T is consistent.
Corollary For every integer p 0, there exists a finite set of matrices M 1, M 2,..., M np Z 3 3, for some integer n p 1, such that the subsemigroup of Z 3 3 generated by these matrices does not contain the zero matrix if and only if T p is consistent. In particular, if ZFC + There exist (at least) p inaccessible cardinals is consistent, then the property The subsemigroup of Z 3 3 generated by the matrices M 1, M 2,..., M np, does not contain the zero matrix is provable from ZFC + There exist (at least) p + 1 inaccessible cardinals but not from ZFC + There exist (at least) p inaccessible cardinals.
This could be compared to the fact that if PA (respectively, ZFC) is consistent then there is a polynomial P(x 1,..., x n ) which has no integer roots, but for which this cannot be proved from PA (respectively, ZFC); this result can be inferred from Matiyasevich s Theorem. See H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical logic. Undergraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994.
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J. Hartmanis. Independence results about context-free languages and lower bounds. Information Processing Letters, 20(5):241 248, 1985. D. Joseph and P. Young. Independence results in computer science? Journal of Computer and System Sciences, 23(2):205 222, 1981. H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical logic. Undergraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1994. A. Horská. Where is the Gödel-point hiding: Gentzen s consistency proof of 1936 and his representation of constructive ordinals. Springer Briefs in Philosophy. Springer, Cham, 2014.
T. Harju. Decision questions on integer matrices. In Proceedings of the International Conference Developments in language theory (Vienna, 2001), volume 2295 of Lecture Notes in Computer Science, pages 57 68. Springer, Berlin, 2002. V. Halava and T. Harju. Mortality in matrix semigroups. American Mathematical Monthly, 108(7):649 653, 2001.
Conclusion 1 A great number of elementary properties of automata over finite words are actually independent from strong set theories. 2 One can effectively construct some automata, like 1-counter or 2-tape automata, for which many elementary properties reflect the scale of a hierarchy of large cardinals axioms like There exist (at least) n inaccessible cardinals for integers n 1. 3 We show how we can use Post Correspondence Problem to get simple combinatorial statements about finite words which are independent from strong set theories. 4 The results of this paper are true for other large cardinals than inaccessible ones. For instance we can replace inaccessible cardinals by hyperinaccessible, hypermahlo, measurable,... and still other ones and obtain similar results.
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Consistency of theories and 1-counter automata Theorem (F. 2014) and there exists a real-time 1-counter Büchi automaton A T, reading words over a finite alphabet Γ, such that Cons(T ) is equivalent to each of the following items: (1 ) L(A T ) Γω ; (2 ) L(A T ) is not ω-regular; (3 ) L(A T ) is not deterministic; (4 ) L(A T ) is Σ1 1 -complete; (5 ) L(A T ) is not Borel; (6 ) L(A T ) is not in the Borel class Σ0 α (for a non-null countable ordinal α); (7 ) L(A T ) is not in the Borel class Π 0 α (for a non-null countable ordinal α); (8 ) L(A T ) is inherently ambiguous; (9 ) L(A T ) has the maximum degree of ambiguity (for acceptance by 1-counter automata or by Turing machines); (10 ) L(A T ) is not an arithmetical set; (11 ) L(A T ) is not an hyperarithmetical set; (12 ) L(A T ) is not in the arithmetical class Σ n (for n 1); (13 ) L(A T ) is not in the arithmetical class Π n (for n 1);
Consistency of theories and 1-counter automata T n : ZFC + There exist (at least) n inaccessible cardinals. We can apply the preceding theorem to the recursive theories T n, and get the real-time 1-counter Büchi automata A Tn and A T n, denoted A n and A n. Theorem (F. 2014) For every integer n 0, there exist two real-time 1-counter Büchi automata A n and A n, reading words over a finite alphabet Γ, such that Cons(T n ) is equivalent to each of the items (1)-(11) and (1 )-(13 ) of the preceding theorem where A T and A T are replaced by A n and A n. In particular, if ZFC + There exist (at least) n inaccessible cardinals is consistent, then each of the properties of A n and A n given by these items (1)-(11) and (1 )-(13 ) is provable from ZFC + There exist (at least) n + 1 inaccessible cardinals but not from ZFC + There exist (at least) n inaccessible cardinals.
Extensions There are similar results for: infinitary rational relations accepted by 2-tape Büchi automata. timed languages accepted by timed automata. Even in the finitary case, some similar results hold! The results of this paper are true for other large cardinals than inaccessible ones. For instance we can replace inaccessible cardinals by hyperinaccessible, Mahlo, hypermahlo, measurable, and still other ones and obtain similar results.
Extensions Same methods yield related results: we can construct, for a given theory T in the language of set theory and a given first-order sentence Φ in the language of set theory, a 1-counter Büchi automaton (or a 2-tape Büchi automaton) A 1 (respectively, A 2, A 3 ) such that L(A 1 ) (respectively, L(A 2 ), L(A 3 )) is Borel (and deterministic, ω-regular, unambiguous,... ) if and only if the sentence Φ is provable from T, (respectively, Φ is provable from T, Φ is independent from T ). Similar results hold for weaker arithmetical theories.
Extensions Example: P= NP can be expressed by a first-order sentence Ψ in the language of set theory. one can construct a 2-tape Büchi automaton A 1 (respectively, A 2, A 3 ) such that L(A 1 ) (respectively, L(A 2 ), L(A 3 )) is Borel if and only if the sentence Ψ is provable from T, (respectively, Ψ is provable from T, Ψ is independent from T ). Since the P= NP? problem is one of the millennium problems for the solution of which one million dollars is offered by the Clay Institute, this is the sum one can win by proving that the infinitary rational relation L(A 1 ) (or L(A 2 ) or L(A 3 )) is Borel!
Büchi acceptance condition An automaton A reading infinite words over the alphabet Σ is equipped with a finite set of states K and a set of final states F K. A run of A reading an infinite word σ Σ ω is said to be accepting iff there is some state q f F appearing infinitely often during the reading of σ. An infinite word σ Σ ω is accepted by A if there is (at least ) one accepting run of A on σ. An ω-language L Σ ω is accepted by A if it is the set of infinite words σ Σ ω accepted by A.
Context free or regular ω-languages ( Cohen and Gold 1977; Linna 1976 ) Let L Σ ω. Then the following propositions are equivalent : L is accepted by a Büchi pushdown automaton. L = 1 i n U i.v ω i, for some context free finitary languages U i and V i. L is a context free ω-language. A similar theorem holds if we: omit the pushdown stack and replace context free by regular, or replace pushdown and context-free by 1-counter.
Topological Complexity of ω-languages Non deterministic Büchi (or Muller) Turing machines accept effective analytic sets (Staiger). The class Effective-Σ 1 1 is the class of projections of arithmetical sets. There are some Σ 1 1-complete sets, hence in particular non-borel sets, in the class Effective-Σ 1 1. Theorem [Ressayre and F. 2003] There are some non-borel context-free (and even 1-counter) ω-languages. [F. 2005] 1-counter ω-languages have the same topological complexity as ω-languages accepted by non deterministic Büchi Turing machines.
Games with non-recursive strategies when they exist Theorem ( F. 2011 ) There exists a 1-counter Büchi automaton A such that: (1) (ZFC+ ω L 1 < ω 1 ): Player 1 has a winning strategy σ in the game G(L(A)). But σ cannot be recursive and not even hyperarithmetical. (2) ( ZFC + ω L 1 = ω 1 ): the game G(L(A)) is not determined. Moreover these are the only two possibilities: there are no models of ZFC in which Player 2 has a winning strategy.