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Bounded Arithmetic Descriptive Complexity (Proof Systems) (Finite Model Theory) the Unrestricted Years… “Is It Even Possible ?”

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Bounded Arithmetic Descriptive Complexity (Proof Systems) (Finite Model Theory) the Unrestricted Years… “Is It Even Possible ?” Expressing vs. Proving Relating forms of complexity in logic Antonina Kolokolova, Memorial U. of Newfoundland Sendai, July 22nd, 2011 St. John’s, Newfoundland Click to edit Master text styles Second level Third level Fourth level Fifth level How hard is it ? To find ? What is the path from s to t in G? To prove ? To describe ? T ` “Reachability is transitive” φ (G) = “Graph G is connected” How hard is it ? To find ? Algorithms and complexity To prove ? To describe ? Bounded arithmetic Descriptive complexity (proof systems) (finite model theory) The unrestricted years… “ Is it even possible ?” Gödel, Turing Computability Arithmetic Kleene,Tarski T r a k Finite model theory h t e n b r o t The restricted times… “Everything should be efficient !” Algorithms/complexity Bounded arithmetic Descriptive complexity Complexity theory Computation with limited resources. Complexity theory Bound time by a polynomial in input length: decidable becomes polynomial-time decidable. Bound the size of a certificate by a polynomial: r.e. (semi-decidable) becomes NP (non-deterministic polynomial-time computable). Bound other kinds of resources (e.g., space): log-space (L), constant-depth circuits (AC0) Complexity classes co-r.e. r.e. n AC0: constant time PH parallel computation (addition). co-NP n NL: non-deterministic NP log-space (reachability). P n P: polynomial-time computable (most NL algorithms). L n PH: polynomial-time AC0 hierarchy (generalizes NP). Descriptive complexity Expressing properties. Descriptive complexity Express a property of a class of structures as a formula φ of some restricted type. Example: 3-colourability. Graph is a structure with one binary predicate E. φ = 9 R 9 G 9 B 8 u,v (R(u)Ç G(u) Ç B(u)) Æ (E(u,v) → “u and v have different colour”) Descriptive complexity Application: database theory Formula becomes query, structure becomes database. Given a query Q, is there a tuple in the database S on which Q holds? Q=“Return sets of 4 vertices that are all connected to each other.” Descriptive complexity co-r.e. r.e. n AC0 : first-order logic. n NL : second-order 2CNF. PH SO n P : second-order Horn. SO9 SO8 n NP : second-order 9 logic; co-NP NP all of second-order logic SO-H P orn gives PH. S NL O-2 CNF L AC0 FO Descriptive complexity co-r.e. r.e. n The vocabulary has to include arithmetic (<,+,*) PH SO n Proofs of equivalence are SO9 SO8 similar to Trakhtenbrot’s co-NP NP theorem. SO-H n P orn Two logics are equivalent S iff the corresponding NL O-2 CNF complexity classes are. L AC0 FO Trakhtenbrot’s theorem “First-order logic is undecidable over finite n Take a first-order formula with free variables Statei(t,x) andstructures” Symbolj(t,x). n Encode the statement “Statei and Symbolj correspond to correct computation ending in accept” n Now, finding a model where this formula holds is finding an accepting computation. Bounded arithmetic Reasoning about computation. Bounded arithmetic Fragments of arithmetic where all quantifiers are bounded. Power of a theory of arithmetic ´ how complex are functions it proves total. Two theories are the same (=conservative) if they prove each other’s theorems. Two different theories can talk about the same class of functions. Bounded reverse mathematics (Cook, Nguyen) What is the computational complexity of concepts needed to prove a given mathematical statement? -- Here the systems are much weaker than in Reverse Mathematics! Power of reasoning Systems of arithmetic are related to propositional proof There are direct translations ofsystems. the form “a theory proves soundness of a proof system, and each of the proofs in the theory can be done in the proof system”. AC0 theory corresponds to Bounded Depth Frege proof system; P-theory to Extended Frege. Bounded arithmetic First approach: base theories on bounded arithmetic hierarchy formulas. Does it produce the polynomial-time hierarchy? Buss’ thesis. Result: NP-theory captures functions of P. If it proves that a function is in NP Å co-NP, the function is in P. Fine-tuning to small feasible complexity classes? First vs. Second-order First-order: Buss’s basic theories Si2, Ti2. Have x#y = 2|x|*|y| in the language. Do not capture AC0. Second-order: First, Buss’s theories for PSPACE and beyond (with x#y). By Razborov-Takeuti’s RSUV isomorphism, removing x#y and adding second sort (strings) get two-sorted theory Vi1 for the same class. Sorts are strings and numbers indexing string positions. No operations on strings other than length and index. Complexity classes co-r.e. r.e. n AC0: constant time PH parallel computation. n NL: non-deterministic co-NP log-space. NP n P: polynomial-time P computable. n PH: polynomial-time NL hierarchy. L AC0 Descriptive complexity approach Build theories from logics of known descriptive To create a theory, take basic axioms complexityof arithmetic, and add an axiom stating “all objects definable in logic L exist”. For levels of PH, get the same theories as before. For non-deterministic classes, so far provably get the functions in the deterministic level of PH. Systems of bounded arithmetic co-r.e. r.e. First-order formulas give a theory for AC0. Second-order 2CNF give a theory for NL. Second-order Horn give a (minimal) theory PH SO for P. SO9 SO8 co-NP NP V-Horn SO -Ho P rn V- SO -2 NL CN KromL F AC0 V FO 0 Systems of bounded arithmetic co-r.e. r.e. The correspondences are not automatic: recall that a system based on NP formulas captured functions in P. PH SO Need additional conditions on provability of properties. SO9 SO8 co-NP NP V-Horn SO -Ho P rn V- SO -2 NL CN KromL F AC0 V FO 0 Closure properties We want robust definitions of complexity classes. Closure under first-order operations: AND, OR, NOT (hardest one), bounded quantification, and function composition. NP is not known to be closed under complementation. However, P is robust. Closure properties should be “easy” to prove. Closure properties If proving that a class is closed can be done inside the class, then the resulting system of Holds for AC0arithmetic from the captures definitions. that class. For P, need to formalize algorithms. [Cook, K ‘01,’03] Surprisingly, proof that NL=coNL can be done with NL reasoning. [Cook, K’04] LogCFL done from its circuit (SAC1) definition (Kuroda) Current proofs for SL (undirected graph reachability) cannot be formalized in an SL theory. [K’05]. Work in progress: in which theory can SL=L be formalized? Are there easier (non-algebraic) proofs? Proof idea Translate logics from descriptive complexity setting to the language of arithmetic. Define class of theories based on the logics, and show that basic properties (e.g., induction) hold. Introduce functions into the theory by defining their bit graphs by formulas (not the usual recursion-theoretic definitions). Generalize Buss’ witnessing theorem to apply to this setting (complicated base case). Other approaches Constructing systems by adding to V0 an axiom asserting the existence of a solution to a complete problem (Nguyen/Cook). E.g., based on versions of reachability problems Different minimal theories for P, NL, L, etc. Universally axiomatizable theories Applicable to small circuit classes such as TC0 Bounded reverse math Which principles does a theory prove? Weak theories cannot prove much: AC0-theory does not prove Pigeon Hole Principle, although a TC0 theory does. PH-theory formalizes a lot of mathematics (some algebra, probabilistic reasoning…). Most work done for P-theory and above, or AC0 theory; for small classes the area is wide open, although there has been some recent work (Nguyen, etc). More in the book (and Phuong Nguyen’s thesis) Jordan Curve Theorem (on grid graph) is in V0(2) Finite Szpilrajn theorem in VTC0 PigeonHole Principle in VTC0 and not in V0. Open: V0(m) ` PHP? Open: Cayley-Hamilton Theorem (work by Soltys for some algebra) Breaking cryptography with bounded arithmetic If a P-theory proves Fermat’s little theorem, then RSA can be broken. Breaking RSA Suppose a P-theory proves (1 · a < n) Æ (an-1 ≠ 1 (mod n)) ! 9 d (1 <d<n d | n) All existential statements that are theorems of a P-theory can be witnessed by poly-time functions. Then some f(a,n) computes d. If n is composite, the hypothesis holds for >half of values of a (except Carmichael numbers, which can be factored in polynomial time). Probabilistic polynomial-time algorithm for factoring: choose a at random, compute f(a,n). This breaks RSA. Provability of separations Is P vs. NP provable? Is it independent of arithmetic altogether (like the Continuum Hypothesis) ? Diagonalization does not suffice [BGS]. Any proof that is “natural” does not work [RR] Any “algebrizing” proof does not work [AW] Independence of PA1 implies existence of a fast algorithm for SAT[Ben-David/Halevi]. Provability of separations What do we need to know about P and NP to prove separations? [AIV] define systems with limited knowledge of polynomial-time functions (Cobham’s axioms without minimality). Natural way to formalize oracle results as independence (both diagonalization [AIV] and algebrization [IKK]). Conclusion: need locality of computation. Provability of separations Maybe it is easier to separate theories than classes? Ajtai showed that Parity Principle is not provable in an AC0 theory. The proof uses heavy model-theoretic machinery: forcing, non-standard models of arithmetic. Furst, Saxe, Sipser proved that Parity function is not computable by AC0 circuits. Conclusions There is a natural connection between the realms of descriptive complexity and bounded arithmetic, each of which is closely related to complexity theory.
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