Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Computing power of Turing machines based on quantum logic Yun Shang AMSS, Chinese Academy of Sciences UH-CAS Workshop on Mathematical Logic 2018-11-02 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Contents 1 Classical theory of computation 2 Theory of quantum automata 1.Quantum automata based on quantum mechanics 2.Quantum automata based on quantum logic 3 Theory of Turing machines based on quantum logic 1.Basic definitions 2.Languages of quantum Turing machines 3.Computing power of Turing machines based on quantum logic Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Classical theory of computation 1 Computability 2 Computational complexity 3 Algorithmic theory 4 Models of computation Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Probabilities representation of quantum mechanics 1 Quantum Turing machine: Feymann (1982), Deutsch(1985); 2 The universal quantum Turing machine: Bernstein and Vazirani (1997) 3 Quantum circuit families: Deutsch (1989), Yao(1993); 4 Quantum randomacess machine; Knill(1996)(classically controlled machine enriched with quantum device 5 Classical controlled quantum Turing machine: Perdrix and Jorrand (2007); Observable quantum Turing machine: Perdrix (2011); 6 Quantum finite state automata: Moore and Crutchfield (2000)(measure once); Kondacs and Watrous (1997)(measure many); Ambainis and R.Freivalds (1998) (one way) ; Yamasaki, Kobayashi, Imai (2001)(two way) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Logical foundation of quantum mechanics{Quantum logic 1 Classical mechanics: Classical logic 2 Closed quantum system: Von Neumann's quantum logic (sharp quantum logic) ! PV measurement 3 Open quantum system: unsharp quantum logic ! POV measurement Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Algebraic model for quantum logic 1 sharp quantum logic ! orthomodular lattice 2 unsharp quantum logic ! effect algebra( Foulis,1994); Quantum MV algebra (Giuntini,1996) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Unsharp quantum logic model Boolean difference poset - difference poset 6 (by F. K^opka) 6 (by F. K^opka) add distributivity ? ? ? MV algebra - effect algebra QMV algebra MV algebra Pi (by C. C. Chang) PP (by C. Giuntini) minus PP PP distributivity ?(by G. Georgescu) ?P P ? ? - pseudo MV algebra - pseudo effect algebra weak QMV algebra ? 6 6(by A. Dvureˇcenskij) 6 pseudo coupled semiring ? 6- ? ? weak pseudo boolean coupled semiring - pseudo difference poset difference poset Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Unsharp quantum logic models Y. Shang, Y. M. Li,Int J Theort. Phys.2004,43(2) Y. Shang, Y. M. Li,and M.Y. Chen,Int J Theort. Phys.2004,43(5) Y. Shang, Y. M. Li,and M.Y. Chen,Int J Theort. Phys. 2003,42(12) Y. Shang, Li Y.,and M.Y. Chen,Int J Theort. Phys.2004,43(12) Y. Shang, and R. Q. Lu. Semirings and pseudo MV algebras. Soft Computing - A Fusion of Foundations, Methodologies and Applications (2007),11. X. Lu, Y. Shang, R. Q. Lu., J. Zhang, and F. F. Ma, Weak QMV algebras and some ring-like structures Soft Computing - A Fusion of Foundations, Methodologies and Applications(2017),21. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Quantum automata based on quantum logic 1 Logical proposition ) element of orthomodular lattice 2 Mingsheng Ying and Daowen Qiu (finite state automata and pushdown automata based on sharp quantum logic) 3 Logical proposition ) element of lattice ordered QMV algebra 4 Yun Shang, Xian Lu and Ruqian Lu (finite state automata and pushdown automata based on unsharp quantum logic) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic M. S. Ying, A theory of computation based on quantum logic(I). 75 pages, 2005, Theoretical computer science. D. W. Qiu, Automata theory based on quantum logic: some characterizations, Information and Computation, 109:2(2004) 179{195. Yun S., Xian L., Ruqian L.,Automata theory based on unsharp quantum logic, Mathematical structures in computer science, 19(2009),737-756. Yun S, Xian L, Ruqian L, A theory of computation based on unsharp quantum logic:finite state automata and pushdown automata,Theoretical computer science 434(2012),53-86. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Difference between classical computation and quantum computation We find the essential difference between automata theory based on quantum logic and classical automata theory. That is the universal validity of many fundamental properties of automata depend heavily not only on the distributive law but also on the non-contradiction law. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Classical Sharp Unsharp Automata Quantum Automata Quantum Automata 6 6 6 Boolean Orthomodular a a 6= -a Lattice-ordered Algebra Lattice QMV Algebra 6 6 (a b) ^ (a c) 6= a (b ^ c) (a _ b) ^ (a _ c) 6= a _ (b ^ c) Boolean a a 6= -a Algebra MV Algebra Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Physical implication For sharp quantum automata, in order to preserve the classical properties of automata, the underlying logic should degenerate to be a boolean algebra. It requires that the simple observables corresponding to projection operators are mutually commutative. For unsharp quantum automata, in order to preserve the classical properties of automata, the simple observables corresponding to effects need and only need to be coexistent. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic What about Turing machines based on quantum logic? Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Orthomodular lattices and boolean algebras 1 An orthcomplemented lattice is an orthomodular lattice if a ≤ c ) c = a _ (a ^ c0): (Orthomodular law) 2 An orthcomplemented lattice is a boolean algebra if a ^ (b _ c) = (a ^ b) _ (a ^ c): (Distributive law) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic S-algebras A supplement algebra (S-algebra for short) is an algebraic structure 0 E = (E; ; ; 0; 1) consisting of set M with two constant elements 0 0; 1, a unary operation and a binary operation on M satisfying the following axioms: (S1) a b = b a. (S2) a (b c) = (a b) c. 0 (S3) a a = 1. (S4) a 0 = a. (S5) a00 = a. (S6) a 1 = 1. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic MV algebras and QMV algebras An multiple-valued (MV) algebra (Chang, 1957) is an S-algebra that satisfies: 0 0 0 0 (MV) (a b) b = (a b ) a A quantum MV (QMV) algebra (Giuntini,1996) is an S-algebra that satisfies: 0 0 (QMV) a [(a e b) e (c e a )] = (a b) e (a c) where 0 0 0 a b = (a b ) 0 a e b = (a ⊕ b ) b Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Classical Turing machines Definition A non-deterministic Turing machine (NTM) is a 7-tuple: M = (Q; Σ; Γ; δ; B; q0;F ), where 1. Q is a finite nonempty state set. 2. Σ is the finite set of input symbols. 3. Γ is the complete set of tape symbols; Σ ⊆ Γ=B. 4. F ⊆ Q is the set of final states. 5. δ ⊆ Q × Γ × Q × Γ × fR; Lg. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Classical Turing machines Classical Turing machines ··· B a b c d B ··· q0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Classical Turing machines ··· B a0 b c d B ··· q1 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Classical Turing machines ··· B a0 b0 c d B ··· q2 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic Classical Turing machines ··· B a0 b0 c0 d B ··· q3 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic L-valued Turing machines (LNTMs) 1 I(q0) = x x ··· B a b c d B ··· z y q0 0 Ref. Y.M. Li, P. Li, Turing machine based on quantum logic. Chinese Journal of Computers (35)2012 1407-1420. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic L-valued Turing machines (LNTMs) 0 1 δ(q0; a; q1; a ;R) = y x ··· B a0 b c d B ··· z y q1 0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic L-valued Turing machines (LNTMs) 0 1 δ(q1; b; q2; b ;R) = z x ··· B a0 b0 c d B ··· z y q2 0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic L-valued Turing machines (LNTMs) 0 1 δ(q2; c; q3; c ;R) = 1 x ··· B a0 b0 c0 d B ··· z y q3 0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic L-valued Turing machines (LNTMs) 1 T (q3) = z x ··· B a0 b0 c0 d B ··· z y q3 0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic L-valued non-deterministic Turing machines (LNTMs) Definition An L-valued non-deterministic Turing machines (LNTM) is a 7-tuple: M = (Q; Σ; Γ; δ; B; I; T ), where 1 Q is a finite nonempty set of state.