Infinite and Standard Computation with Unconventional and Quantum Methods Using Automata
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The document discusses Konstantinos Giannakis's dissertation defense on unconventional computing methods using automata. It summarizes the dissertation's structure, including discussions on standard computation, infinite computation using automata, membrane computing using novel membrane automata definitions, and quantum computing using periodic quantum automata. It provides examples and definitions related to each of these topics.
![ω-regular path queries (cont.)
∙ Use of ω-regular expressions that correspond to the regular
expressions used in regular path queries.
∙ These expressions can be translated into equivalent Büchi
automata.
∙ E.g. the query asking for the mitochondria that performs
infinitely often the fusion process could have the following form:
SELECT *
WHERE
{
< mitochondrion > [(performs)|(isNeighbour∗
/performs)]ω
<
mitochondrion >.
}
21](https://image.slidesharecdn.com/phddefense-160720150852/85/Infinite-and-Standard-Computation-with-Unconventional-and-Quantum-Methods-Using-Automata-22-320.jpg)
![the corresponding automaton
s0 s1
a
b
a
b
Figure: A Büchi automaton A describing the ω-regular expression
underneath the simplest example of mitochondrial fusion ([(a)|(b∗
a)]ω
).
Symbol Explanation
a performs
b isNeighbour
22](https://image.slidesharecdn.com/phddefense-160720150852/85/Infinite-and-Standard-Computation-with-Unconventional-and-Quantum-Methods-Using-Automata-23-320.jpg)
![another example query
∙ An example query of protein folding/misfolding which
represents the amyloid aggregation could have the following
form:
SELECT *
WHERE
{
< protein >
(folding|misfolding)∗
|[translocates|mitochondrion]ω
}
24](https://image.slidesharecdn.com/phddefense-160720150852/85/Infinite-and-Standard-Computation-with-Unconventional-and-Quantum-Methods-Using-Automata-25-320.jpg)
![our approach
∙ Every ambient ≡ membrane subsystem.
∙ Hierarchical structure of ambients ≡ membrane-like segments.
∙ Biomolecular rules from Bioambient calculus to P automata
rewriting rules.
∙ Actions altering ambients’ state (entry, exit, or merge).
Initial configuration:
[[[[[[]AO1[]K]PM1M2]RM1M2]GM1M2]OMOM1M2 [[[[[]BO1]PO1]RO1]GO1]IMOM1M2]skin/cell
Final configuration:
[[]PM1M2[]RM1M2[]GM1M2[]OMOM1M2[]PO1[]RO1[]GO1[]IMOM1M2[]K[]AO1[]BO1]skin/cell
33](https://image.slidesharecdn.com/phddefense-160720150852/85/Infinite-and-Standard-Computation-with-Unconventional-and-Quantum-Methods-Using-Automata-34-320.jpg)
![the production of the the protein mfn1-mfn2
Initial config.
consecutive use of appropriate rule
−−−−−−−−−−−−−−−−−−→ final config. and halt.
∙ Initial configuration: [[[[]PM1M2]RM1M2]GM1M2]OMOM1M2
∙ Final configuration: []PM1M2[]RM1M2[]GM1M2[]OMOM1M2
∙ Halting configuration through consecutive exo operations.
[[[[]PM1M2]RM1M2]GM1M2]OMOM1M2
exo
−−→ [[[]PM1M2]RM1M2]GM1M2[]OMOM1M2
exo
−−→
[[]PM1M2]RM1M2[]GM1M2[]OMOM1M2
exo
−−→ []PM1M2[]RM1M2[]GM1M2[]OMOM1M2
∙ Similarly for the rest of the model.
34](https://image.slidesharecdn.com/phddefense-160720150852/85/Infinite-and-Standard-Computation-with-Unconventional-and-Quantum-Methods-Using-Automata-35-320.jpg)
![dirac symbolism bra-ket notation
∙ Introduced by Paul Dirac.
∙ State 0 is represented as ket |0⟩ and state 1 as ket |1⟩.
∙ Every ket corresponds to a vector in a Hilbert space. For
example:
|0⟩ =
[
1
0
]
, |1⟩ =
[
0
1
]
. (2)
45](https://image.slidesharecdn.com/phddefense-160720150852/85/Infinite-and-Standard-Computation-with-Unconventional-and-Quantum-Methods-Using-Automata-46-320.jpg)
![to quantum ω-automata definition
◦ A simple periodic, one-way quantum ω-automaton is a tuple (Q,
Σ, Uδ, q0, π0, F, P, Acc) where:
1. Q is a finite set of states,
2. Σ is the input alphabet,
3. Ua is the n × n unitary matrix that describes the transitions among
the states for each symbol a ∈ Σ,
4. q0 ∈ Q is the initial (pure) state,
5. π0 is the initial vector,
6. F ∈ Q is the set of final states,
7. P is the set [P0, P1, . . . , Pn] of the projection matrices of states, and
8. Acc is an acceptance condition.
54](https://image.slidesharecdn.com/phddefense-160720150852/85/Infinite-and-Standard-Computation-with-Unconventional-and-Quantum-Methods-Using-Automata-55-320.jpg)
![periodic quantum automaton
∙ A simple m-periodic, 1-way quantum ω-automaton with periodic
measurements is a tuple (Q, Σ, Uδ, q0, m, π0, F, P, Acc) where:
1. Q is a finite set of states,
2. Σ is the input alphabet,
3. Uα : Q × Σ −→ C[0,1] is the n × n unitary matrix that describes the
transitions among the states for each symbol a ∈ Σ,
4. q0 ∈ Q is the (pure) initial state,
5. m ∈ N defines the measurement period,
6. π0 is the vector of the initial pure state q0,
7. F ∈ Q is the set of final states,
8. P is the set [P0, P1, . . . , Pn] of the projection matrices of states, and
9. Acc is the almost-sure periodic quantum acceptance condition.
56](https://image.slidesharecdn.com/phddefense-160720150852/85/Infinite-and-Standard-Computation-with-Unconventional-and-Quantum-Methods-Using-Automata-57-320.jpg)
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Infinite and Standard Computation with Unconventional and Quantum Methods Using Automata
- 1. infinite and standard computation with unconventional and quantum methods using automata Dissertation Defense Konstantinos GIannakis [email protected] July, 14 2016 Department of Informatics, Ionian University Supervisor: Dr. Theodore Andronikos Advisory Committee: Dr. Spyros Sioutas and Dr. Michail Stefanidakis Department of Informatics, Ionian University 0
- 2. the dissertation’s story ∙ At first we studied infinite computation with focus on automata. ∙ We then proceeded to the probabilistic versions of the above models. ∙ Probabilistic computation led us to the search for alternative models. ∙ Unconventional means of computing, mainly quantum automata. 1
- 3. motivation and contribution ∙ Moore’s law. ∙ The lack of quantum variants for specific classes of automata. (i.e. ω-automata) ∙ The power of specific quantum algorithms. ∙ most notably the algorithms of Deutsch, Shor, and Grover. ∙ The traces of infinite behaviour in nature and data volumes. 2
- 4. dissertation’s structure ⊙ Standard (or Classical) and Probabilistic Computing ∙ Introductory points ⊙ Infinite Computation ∙ Associating automata variants to queries. ⊙ Membrane Computing ∙ Novel definitions of membrane automata. ∙ Use of actual models to implement our methodology. ⊙ Quantum Computing ∙ Proposal of the periodic quantum automata. ∙ Association of these automata with strategies over quantum games. 3
- 6. a little history ∙ Initiated before WWII, mainly in the ’30s. ∙ Alan Turing is regarded as the pioneering figure. ∙ Other great scientists of the same period: Kurt Gödel, Alonzo Church, Stephen Kleene and many more. ∙ Automata and their theory were developed at the early ’40s and ’50s. 5
- 7. automata ∙ Branch of the Theory of Computation dealing with specialized, abstract computational models. ∙ Finite-state automata. ∙ Finite memory. ∙ Basic properties: states, transitions, and the input/output. 6
- 8. dfa και nfa ∙ For a given symbol there is only one state that can be visited. ∙ (Q, Σ, δ : Q x Σ → Q, q0, F) q0 1 1 0 0 q1 ∙ For a given symbol there is none, one, or more states that can be visited, even for the same symbol. ∙ (Q, Σ, δ : Q x Σ ∪ {ϵ} → P(Q), q0, F) q0 q1 Άλλες ρυθμίσειςΆλλες ρυθμίσεις 1 1 q2 0,1 0,1 ∙ For every NFA there is an equivalent DFA. ∙ The translation of an NFA to a DFA can cause an exponential growth to the number of states! 7
- 10. logic and probabilities Γένεση 18:32 Καί είπεν ο Αβραάμ, Ας μή παροξυνθή ο Κύριός μου, εάν λαλήσω έτι άπαξ; εάν ευρεθώσιν εκεί δέκα; καί είπε, Δέν θέλω απολέσει αυτήν χάριν τών δέκα. Genesis 18:32 Then he said, “Oh may the Lord not be angry, and I shall speak only this once; suppose ten are found there?” And He said, ”I will not destroy it on account of the ten.” Subjective vs Objective aspect of probability 9
- 11. probabilistic computation * Introducing probabilities attached to the system’s input. * Result depends not only on the input, but also on coin flips. P(w) := ∑ p∈Acc(w) n∏ i=1 (pi, wi, pi+1) ∑ p∈Run(w) n∏ i=1 (pi, wi, pi+1) . (1) 10
- 12. pfa (probabilistic finite automata) ∙ Introduced by Rabin (1963) ∙ Similar to simple Markov chains 11
- 14. ω-computability receiving infinite inputs ∙ ω-automata. ∙ Infinite input ∙ Acceptance conditions ∙ E.g. Büchi automata. ∙ Büchi acceptance condition. ∙ They accept the runs ρ for which In(ρ) ∩ F ̸= ∅ (F ⊆ Q). ∙ Extension of the simple automata with infinite inputs 13
- 15. ω-computability ∙ An ω-automaton is a tuple (Q, Σ, δ, q0, Acc) where Q is a finite set of states, Σ is the alphabet, δ : Q × Σ −→ P(Q) is the transition function (P(Q) is the powerset of Q), q0 is the initial state, and Acc determines the acceptance condition. ∙ The acceptance condition Acc declares how the infinite runs are accepted by the automaton. ∙ The class of languages recognized from (almost) all the above machines are the ω-regular languages. 14
- 16. automata and queries ∙ There are queries that cannot be expressed neither by standard SPARQL nor its expansion with regular path queries. ∙ These queries assume that the Web of Linked Data can be infinite and therefore classic queries expressed in terms of SPARQL need to be revisited. ∙ We propose a novel method associating ω-automata, to each query on this infinite structure. ∙ We call these queries ω-regular path queries. ∙ They are of the form a|(ab)ω . 15
- 17. infinite horizon ∙ Two SPARQL query semantics ∙ Full-web semantics. ∙ the scope of each query is the full set of LD on the Web. ∙ Reachability-based semantics. ∙ restricted scope of SPARQL queries to data reachable through specific links, using a given initial set of URIs. 16
- 18. queries on infinite graphs of graphs ∙ We assume infinite web of LD. ∙ Reachability is guaranteed (remember, we refer to Linked Data). ∙ Acceptance is defined by a set of final states. ∙ Infinite data/queries -> infinite computation 17
- 19. the process for the association s1 s0 s2 s3 s4 o w o u o u o Figure: It accepts the ω-regular language described by the ω-regular expression u(wou|(ou)?)∗ (ou)ω . Symbol Explanation u Encoding of URI o Symbol #, separator of URIs w Symbol read when the query returns a result 18
- 20. ω-regular path queries Definition ω-regular path queries are particular path queries where the regular expression is replaced by an ω-regular expression. These queries concern cases that include infiniteness in the expected results. Mitochondrion2 Mitochondrion1 Mitochondrion3 Fusion Fission isNeighbour performs performs performs performs performs performs isNeighbour produces produces 19
- 21. ω-regular path queries (cont.) ∙ We assume that the life-cycle of a mitochondrial population undergoes infinitely many fusion and fission processes. ∙ It fits to the notion of eventual computability. ∙ The information from the previous Figure can be represented in the well-known triple form of (subject, predicate, object) that follows: (Mitochondrion1, performs, Fusion) (Mitochondrion1, performs, Fission) (Mitochondrion2, performs, Fusion) (Mitochondrion2, performs, Fission) (Mitochondrion3, performs, Fusion) (Mitochondrion3, performs, Fission) (Mitochondrion1, isNeighbour, Mitochondrion2) (Mitochondrion2, isNeighbour, Mitochondrion3) … … … ∙ A query could be asking for the mitochondria that perform infinitely often the fusion process. 20
- 22. ω-regular path queries (cont.) ∙ Use of ω-regular expressions that correspond to the regular expressions used in regular path queries. ∙ These expressions can be translated into equivalent Büchi automata. ∙ E.g. the query asking for the mitochondria that performs infinitely often the fusion process could have the following form: SELECT * WHERE { < mitochondrion > [(performs)|(isNeighbour∗ /performs)]ω < mitochondrion >. } 21
- 23. the corresponding automaton s0 s1 a b a b Figure: A Büchi automaton A describing the ω-regular expression underneath the simplest example of mitochondrial fusion ([(a)|(b∗ a)]ω ). Symbol Explanation a performs b isNeighbour 22
- 24. our proposed definition Definition The associated automaton is an ω-automaton under the Büchi acceptance condition. It is a tuple (QS ∪ O, Σp, δ, q0, F, Acc), where: 1. QS ∪ O∈ {S ∪ O} is the finite set of states, where S is the set of subjects and O the set of objects of the triples, 2. Σp is a finite set of input symbols (alphabet) where each symbol is a predicate from the Linked Data graph, 3. δ : QS ∪ O × Σp −→ QS ∪ O is the transition function, 4. q0 ∈ QS ∪ O is the initial state of the automaton, 5. F ⊆ QS ∪ O is the set of accepting states, and 6. Acc defines the acceptance condition, in the case of the associated automaton we propose it is the Büchi acceptance condition. 23
- 25. another example query ∙ An example query of protein folding/misfolding which represents the amyloid aggregation could have the following form: SELECT * WHERE { < protein > (folding|misfolding)∗ |[translocates|mitochondrion]ω } 24
- 26. our contribution ∙ Combination of particular queries with automata on infinite inputs. ∙ Proposal of a novel methodology regarding queries and their association with ω-automata. ∙ We demonstrate some characteristic queries on simple instances. ∙ We proposed a methodology on how specific queries of regular paths in SPARQL could be translated into inputs for ω-automata. ∙ Interpretation of the the eventual computability of these queries using the well-known Büchi acceptance condition. 25
- 28. membrane computing ∙ Known as P systems with several proposed variants. ∙ Evolution depicted through rewriting rules on multisets of the form u→v ∙ imitating natural chemical reactions. ∙ u, v are multisets of objects. ∙ The hierarchical status of membranes evolves by constantly creating and destroying membranes, by membrane division etc. ∙ Types of communication rules: ∙ symport rules (one-way passing through a membrane) ∙ antiport rules (two-way passing through a membrane) 27
- 29. examples ◦ Membranes create hierarchical structures. ◦ Each membrane contains objects and rules. ◦ Represented either by a Venn diagram or a tree. (a) Hierarchical nested membranes (b) With simple objects and rules 28
- 30. p systems evolution and computation ∙ Via purely non deterministic, parallel rules. ∙ Characteristics of membrane systems: the membrane structure, multisets of objects, and rules. ∙ They can be represented by a string of labelled matching parentheses. ∙ Use of rules =⇒ transitions among configurations. ∙ A sequence of transitions is interpreted as computation. ∙ Accepted computations are those which halt and a successful computation is associated with a result. 29
- 31. p automata ∙ Variants of P systems with automata-like behaviour. ∙ Computation starts from an initial configuration. ∙ Acceptance is defined by a set of final states. ∙ They define a computable set of configurations satisfying certain conditions. ∙ The set of accepted input sequences forms the accepted language. ∙ A configuration of a P automaton with n membranes is defined as a n-tuple of multisets of object in each membrane. ∙ A run of a P automaton is defined as a process of altering its configurations in each step. 30
- 32. rules used in membrane computing ... ... b) a) c) exo (a,in)aa(b,in)ba (a,out) cab c→a bbbba (a,out) caa c→bb cca =⇒ =⇒ =⇒ 31
- 33. a case study ∙ A biological model of mitochondrial fusion by Alexiou et al, expressed in BioAmbient calculus. ∙ Cell is divided into hierarchically nested ambients. ∙ 3 proteins are required (Mfn1, Mfn2 and OPA1) for the successful fusion. ∙ Fusion can occur: ∙ by the merging of two membrane-bounded segments. ∙ when segments may enter or exit one another. ∙ Synchronized capabilities that can alter ambients’ state are: entry, exit, or merge of other compartments. 32
- 34. our approach ∙ Every ambient ≡ membrane subsystem. ∙ Hierarchical structure of ambients ≡ membrane-like segments. ∙ Biomolecular rules from Bioambient calculus to P automata rewriting rules. ∙ Actions altering ambients’ state (entry, exit, or merge). Initial configuration: [[[[[[]AO1[]K]PM1M2]RM1M2]GM1M2]OMOM1M2 [[[[[]BO1]PO1]RO1]GO1]IMOM1M2]skin/cell Final configuration: [[]PM1M2[]RM1M2[]GM1M2[]OMOM1M2[]PO1[]RO1[]GO1[]IMOM1M2[]K[]AO1[]BO1]skin/cell 33
- 35. the production of the the protein mfn1-mfn2 Initial config. consecutive use of appropriate rule −−−−−−−−−−−−−−−−−−→ final config. and halt. ∙ Initial configuration: [[[[]PM1M2]RM1M2]GM1M2]OMOM1M2 ∙ Final configuration: []PM1M2[]RM1M2[]GM1M2[]OMOM1M2 ∙ Halting configuration through consecutive exo operations. [[[[]PM1M2]RM1M2]GM1M2]OMOM1M2 exo −−→ [[[]PM1M2]RM1M2]GM1M2[]OMOM1M2 exo −−→ [[]PM1M2]RM1M2[]GM1M2[]OMOM1M2 exo −−→ []PM1M2[]RM1M2[]GM1M2[]OMOM1M2 ∙ Similarly for the rest of the model. 34
- 36. definition i Definition A generic P system (of degree m, m ≥ 1) with the characteristics described above can be defined as a construct Π=(V, T, C, H, µ, w1, ..., wm, (R1, ..., Rm), (H1, ..., Hm) i0) , where 1. V is an alphabet and its elements are called objects. 2. T ⊆ V is the output alphabet. 3. C ⊆ V, C ∩ T = ⊘ are catalysts. 4. H is the set {pino, exo, mate, drip} of membrane handling rules. 5. µ is a membrane structure consisting of m membranes, with the membranes and the regions labeled in a one-to-one way with elements of a given set H. 6. wi, 1 ≤ i ≤ m, are strings representing multisets over V associated with the regions 1,2, ... ,m of µ. 7. Ri , 1 ≤ i ≤ m, are finite sets of evolution rules over the alphabet set V associated with the regions 1,2, ... , m of µ. These object evolution rules have the form u → v. 8. Hi , 1 ≤ i ≤ m, are finite sets of membrane handling rules rules over the set H associated with the regions 1,2, ... , m of µ. 9. i0 is a number between 1 and m and defines the initial configuration of each region of the P system. 35
- 37. definition ii Definition Formally, a one-way P automaton with n membranes (n ≥ 1) and antiport rules is a construct Π=(V, µ, P1, ..., Pn, c0, F), where: 1. V is a finite alphabet of objects, 2. µ is the underlying membrane structure of the automaton with n membranes, 3. Pi is a finite set of antiport rules for membrane i with 1≤i≤n without promoters/inhibitors, where each antiport rule is of the form (a, out; b, in) with a, b being multisets consisting of elements of the set V, 4. c0 is the initial configuration of Π , and 5. F is the set of accepting configurations of Π. 36
- 38. schematic view Figure: The step by step process through consecutive exo operations. 37
- 40. overall Inherent compartmentalization, easy extensibility and direct intuitive appearance for biologists. Suitable in cases when few number of objects are involved or slow reactions. Need for deeper understanding of mitochondrial fusion ◦ Connections with neurodegenerative diseases and malfunctions. Probability theory and stochasticity (many biological functions are of stochastic nature). Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 Drp1 Mff Fis1 39
- 41. our contribution ∙ We proposed a novel, original method to describe actual biomolecular models using membrane automata. ∙ We proposed a novel variant of P automata, ∙ Combination of membrane automata with process calculi. ∙ We present the advantages and disadvantages of this methodology, ∙ Both P systems and P automata are formal tools, with enhanced power and efficiency =⇒ could shed light to the problem of modeling complex biological processes. 40
- 43. moore’s law ∙ “The number of transistors incorporated in a chip will approximately double every 24 months.” ∙ 8086 (1978) was 16-bit, had 29.000 transistors, and integration technology of 3.2 μm. ∙ Intel(Haswell-E) had integration technology of 22nm and contained 2.6 billion transistors. ∙ Skylake is constructed with integration technology of 14nm. ∙ Moore’s law is about to reach its physical limits. 42
- 44. consequences of moore’s law ∙ Continuously decreasing size of the computing circuits. ∙ Technological and physical limitations (limits of lithography in chip design). ∙ New technologies to overcome these barriers, with Quantum Computation being a possible candidate. ∙ Ability of these systems to operate at a microscopic level. ∙ Redesign and revisit of well-studied models and structures from classical computation. 43
- 45. basics of quantum computing ∙ QC considers the notion of computing as a natural, physical process. ∙ It must obey to the postulates of quantum mechanics. ∙ Bit ⇒ Qubit. ∙ It was initially discussed in the works of Richard Feynman in the early ’80s. 44
- 46. dirac symbolism bra-ket notation ∙ Introduced by Paul Dirac. ∙ State 0 is represented as ket |0⟩ and state 1 as ket |1⟩. ∙ Every ket corresponds to a vector in a Hilbert space. For example: |0⟩ = [ 1 0 ] , |1⟩ = [ 0 1 ] . (2) 45
- 47. kets και bras ∙ A qubit is in state |ψ⟩ described by: |ψ⟩ = c0 |0⟩ + c1 |1⟩ (3) where c0 and c1 are called probability amplitudes ∙ They are complex numbers for which |c0|2 + |c1|2 = 1. ∙ For every ket |ψ⟩ there is a bra ⟨ψ| which is: ⟨ψ| = c∗ 0 ⟨0| + c∗ 1 ⟨1| (4) where c∗ 0 and c∗ 1 are the complex conjugates of c0 and c1. 46
- 48. to quantum automata using dirac formalism ∙ Each state of the machine is a superposition of the basis states |ψi⟩. ∙ They have the form |ψ⟩=c1 |ψ1⟩ + c2 |ψ2⟩ + · · · + cn |ψn⟩, ∙ The probability of observing the state |ψ′ ⟩=c1 |ψ′ 1⟩ + c2 |ψ′ 2⟩ + · · · + cn |ψ′ n⟩ is p, with p = ∑ ψ′∈F |ci|2 (F is the set of accepting states). ∙ In a MO-automaton the projection matrix P is applied strictly once. ∙ In MM-automata, there are three disjoint sets of states: the Qa (accepting states), the Qr (rejecting states) and the Qn of neutral states. ∙ Measurement after reading each symbol. 47
- 49. quantum automata variations ∙ Measure-many approach ∙ Measure-once approach ∙ There are regular languages not recognized by a quantum automaton. ∙ We have to blame the reversibility of the quantum system! ∙ But they are space-efficient. ∙ 2-way variants are more powerful. 48
- 50. automata and computation (standard and quantum) ∙ Finite automata ⇒ simple models of computation. ∙ Finite quantum automata ∙ A quantum system where each symbol represents the application of a unitary transformation. ∙ Proposed after the middle of the 1990s. ∙ They can be seen as a generalization of probabilistic finite automata. ∙ Transitions are weighted with a probability amplitude ⇒ vectors in a Hilbert space. ∙ Probability semantics under which automata accept or reject. 49
- 51. terminology needed for clarification ∙ Σ ⇒ the alphabet ∙ Σ∗ ⇒ the set of all finite strings over Σ ∙ If U is an n × n square matrix , ¯U is its conjugate, and U† its transpose and conjugate. ∙ Cn×n defines the set of all n × n complex matrices. ∙ A unitary operator (or matrix) U is an orthogonal matrix with complex entries that preserves the norms of vectors. ∙ Equivalently, a matrix U is unitary if it has an inverse and if ∥Uψ∥ = ∥ψ∥ for every vector ψ. ∙ Hn is an n-dimensional Hilbert space. 50
- 52. quantum computation states and formalism ∙ Two types of quantum states: pure and mixed states. ∙ A pure state is a state represented by a single ket vector |ψ⟩ in a Hilbert space over complex numbers. ∙ A mixed state is a statistical distribution of pure states (usually described with density matrices). ∙ The evolution of a quantum system is described by unitary transformations. ∙ The states of an n-level quantum system are self-adjoint positive mappings of Hn with unit trace. ∙ An observable of a quantum system is a self-adjoint mapping Hn → Hn. ∙ Each state qi ∈ Q with |Q| = n can be represented by a vector ei = (0, . . . , 1, . . . , 0). 51
- 53. quantum computation applying matrices, observables, and projection ∙ Each of the states is a superposition of the form n∑ i=1 ciei. ∙ n is the number of states ∙ ci ∈ C are the coefficients with |c1|2 + |c2|2 + · · · + |cn|2 = 1 ∙ ei denotes the (pure) basis state corresponding to i. ∙ Each symbol σi ∈ Σ a unitary matrix/operator Uσi and each observable O an Hermitian matrix O. ∙ The possible outcomes of a measurement are the eigenvalues of the observable. ∙ Transition from one state to another is achieved through the application of a unitary operator Uσi . ∙ The probability of obtaining a result p is ∥πPi∥, where π is the current state (or a superposition) and Pi is the projection matrix of the measured basis state. ∙ The state after the measurement collapses to the πPi / ∥πPi∥. 52
- 55. to quantum ω-automata definition ◦ A simple periodic, one-way quantum ω-automaton is a tuple (Q, Σ, Uδ, q0, π0, F, P, Acc) where: 1. Q is a finite set of states, 2. Σ is the input alphabet, 3. Ua is the n × n unitary matrix that describes the transitions among the states for each symbol a ∈ Σ, 4. q0 ∈ Q is the initial (pure) state, 5. π0 is the initial vector, 6. F ∈ Q is the set of final states, 7. P is the set [P0, P1, . . . , Pn] of the projection matrices of states, and 8. Acc is an acceptance condition. 54
- 56. their functionality explanation ∙ It starts at its initial pure state q0, i.e. the state vector of the system is the π0. ∙ Transitions among the states are expressed with complex amplitude. ∙ Acc defines the acceptance condition. Periodic quantum acceptance condition It defines that infinitely often the measurement of the quantum system finds with some probability the automaton in one of the final states. Almost-sure periodic quantum acceptance condition It defines that infinitely often the measurement of the quantum system finds the automaton in one of the final states with probability 1. 55
- 57. periodic quantum automaton ∙ A simple m-periodic, 1-way quantum ω-automaton with periodic measurements is a tuple (Q, Σ, Uδ, q0, m, π0, F, P, Acc) where: 1. Q is a finite set of states, 2. Σ is the input alphabet, 3. Uα : Q × Σ −→ C[0,1] is the n × n unitary matrix that describes the transitions among the states for each symbol a ∈ Σ, 4. q0 ∈ Q is the (pure) initial state, 5. m ∈ N defines the measurement period, 6. π0 is the vector of the initial pure state q0, 7. F ∈ Q is the set of final states, 8. P is the set [P0, P1, . . . , Pn] of the projection matrices of states, and 9. Acc is the almost-sure periodic quantum acceptance condition. 56
- 58. transitions on quantum periodic automata ∙ The transition matrix for every symbol has the form of: Uϕ = i( cos(ϕ) sin(ϕ) − sin(ϕ) cos(ϕ) ) ∙ ϕ defines the period (if m is the period of the transition, then ϕ = π/m). ∙ Counter-clockwise rotation. ∙ We can reverse the rotation by transposing the Uϕ. ∙ Then we have UT ϕ = i ( cos(ϕ) − sin(ϕ) sin(ϕ) cos(ϕ) ) . ∙ Both return the system to its initial state after the same period. 57
- 59. quantum periodic automata periodicity ∙ After m applications of the transition matrix U, the state of the system is Um |ψ⟩, where |ψ⟩ is the state of the system before the m transitions. ∙ But Um =im ( −1 0 0 −1 ) since Um =im ( cos(mϕ) sin(mϕ) − sin(mϕ) cos(mϕ) ) and ϕ = π/m. ∙ In 2m timesteps we obtain the U2m = ( 1 0 0 1 ) ∙ It is the same! ∙ Their difference is a phase of π, since ϕ = 2mπ/m = 2π. 58
- 60. transitions of the state vector 0 π/4 π/2 3π/4 π 5π/4 7π/4 3π/2 r = 1 ϕϕ ϕ ϕ Figure: The vector is in the initial state and for every phase transition with angle ϕ = π/4 it is rotated counter-clockwise. After m − 1 (=4) transitions the system is in the state that is symmetric to the initial. 59
- 61. a quantum game - captain picard vs q ∙ The “PQ Penny Flip” game was described by David A. Meyer in 1999. ∙ A game that showed the superiority of quantum strategies over the classical ones. ∙ A player has a dominant strategy, no matter what the other players chooses in every round. 60
- 62. quantum games and automata ∙ Association of dominant strategies of repeated quantum games with quantum automata that recognize infinite periodic inputs. ∙ Shown in the PQ-PENNY quantum game where the quantum strategy outplays the choice of a pure or mixed strategy with probability 1. ∙ therefore the associated quantum automaton accepts with probability 1. ∙ We proposed a novel game played on the evolution of an automaton, where players’ actions and strategies are also associated with periodic quantum automata. 61
- 63. our proposed game ∙ 2-player game over the evolution of a simple DFA. ∙ A game played over a 2-state automaton with an alphabet consisting of two symbols. ∙ q0 is the winning state for Player 1 and q1 the one for Player 2. q0 q1 a b a b Figure: The automaton that corresponds to Player 1. In the case of Player 2, the accepting state is the state q1. Table: Transition matrix of the automaton. q0 q1 q0 a b q1 a b 62
- 64. provider vs. measurer ∙ Player 1 chooses a symbol and runs it on the automaton until Player 2 decides to stop the procedure. ∙ Player 2 has no knowledge about either the read symbols or the current state. ∙ The payoff for each player depends on the current state and the number of bs. Table: The game’s payoff matrix. The state in the columns denotes the automaton’s state after reading the final symbol when Player 2 stops the procedure. |b| ≤ |a| |b| > |a| q0 (1,1) (2,0) q1 (0,2) (1,1) 63
- 65. strategies on this game ∙ Player 1 chooses one of the two symbols not knowing when Player 2 is about to stop the procedure ∙ Every reading of a b symbol gets his win in stake. ∙ If he insists on reading only the symbol a, he guarantees a (1,1) result for himself. ∙ (1,1) is the Nash point for the deterministic version. ∙ In the quantum version we observe a different behaviour. ∙ Player 2 doesn’t actually stop the evolution but rather, he measures the current state (thus the name “measurer”). ∙ Player 1 still chooses the symbols in each timestep. 64
- 66. the quantum version ∙ The automaton is on a superposition of states. ∙ Player 1 associates a quantum operator, similar to those of periodic quantum automata, to the symbol a. ∙ He associates b with a specific matrix that actually does not alter the quantum state. ∙ E.g. he chooses the matrix U1 = i ( cos(ϕ) sin(ϕ) − sin(ϕ) cos(ϕ) ) with ϕ = π/m, m = 2 for the a and the U2 = i ( −1 0 0 −1 ) with ϕ = π/m, m = 1 for the b. ∙ Player 1 chooses the for the first 2 inputs the symbol a and then he consequently applies the matrix U2. ∙ This offers strategies that can be described as inputs in quantum periodic automata ∙ E.g. for w = aabbbbbbbbbbbb . . . bbbb we have the payoff (2,0). 65
- 67. overall ∙ Quantum automata with infinite computation are still unexplored. ∙ Different variants of machines, distinguished either by movement orientation or by the measurement mode. ∙ Need for models and verification processes for infinite QC. ∙ Useful in the verification of quantum systems and the design of quantum circuits. ∙ Space efficient for periodic ω-languages of the form (am b)ω . ∙ Connection with game theory and groups. ∙ Actions in such games form specific groups ∙ Consistency with the underlying quantum physics. 66
- 68. our contribution ∙ We proposed a novel definition regarding quantum computation with infinite horizon. ∙ It is a one of the first attempts that combine quantum computation and infinite inputs. ∙ We exploit the wave-like nature of quantum computing by presenting a computation scheme that accepts periodic languages of the form (am b)ω , where m is the periodicity. ∙ We illustrate this concept through examples and figures. ∙ Association of the proposed quantum automata with quantum games, their strategies, and group theory. 67
- 69. inspired by our work ∙ The organization of a related workshop. ∙ Natural, Unconventional, and Bio-inspired Algorithms and Computation Methods Workshop (NUBACoM 2016) in Sparta. ∙ The introduction of a new course in the curriculum of the Department of Informatics. ∙ “Introduction to Quantum and DNA Computing” ∙ Theses and PhD proposals. ∙ Establishment of a new research group in our department. ∙ Quantum and UnconventIonal CompuTing group ∙ Collaboration with the Bioinformatics and Human Electrophysiology Lab (BiHELab) ∙ The development of a novel quantum programming language called Qumin (in beta) by A. Singh. 68
- 70. potential applications ∙ Better and more efficient algorithms for querying Linked Data. ∙ Proper design and verification of unconventional means of computing. ∙ Design of new, original quantum algorithms. ∙ Universal quantum programming language(s) and architectures. ∙ Study on bio-inspired methods of performing actual computations, e.g. DNA sequences, nano-scale biological part etc. 69
- 71. further exploit of the results and our next steps ∙ Implementation of our theoretical results. ∙ Complexity issues and bounds. ∙ D-Wave ∙ IBM ∙ Use of quantum simulators. ∙ Further research based on our results. ∙ Already submitted works in progress. 70
- 72. publications Giannakis, K., and Andronikos, T. Membrane automata for modeling biomolecular processes. Natural Computing (2015), 1–13. Giannakis, K., and Andronikos, T. Mitochondrial fusion through membrane automata. In GeNeDis 2014. Springer, 2015, pp. 163–172. Giannakis, K., and Andronikos, T. Use of büchi automata and randomness for the description of biological processes. International Journal of Scientific World 3, 1 (2015), 113–123. Giannakis, K., Papalitsas, C., and Andronikos, T. Quantum automata for infinite periodic words. In Information, Intelligence, Systems and Applications, IISA 2015, The 6th International Conference on (2015), IEEE. Giannakis, K., Papalitsas, C., Kastampolidou, K., Singh, A., and Andronikos, T. Dominant strategies of quantum games on quantum periodic automata. Computation 3, 4 (2015), 586–599. Giannakis, K., Theocharopoulou, G., Papalitsas, C., Andronikos, T., and Vlamos, P. Associating ω-automata to path queries on webs of linked data. Engineering Applications of Artificial Intelligence 51 (2016), 115 – 123. Mining the Humanities: Technologies and Applications. Theocharopoulou, G., and Giannakis, K. Web mining to create semantic content: A case study for the environment. In Artificial Intelligence Applications and Innovations, L. Iliadis, I. Maglogiannis, H. Papadopoulos, K. Karatzas, and S. Sioutas, Eds., vol. 382 of IFIP Advances in Information and Communication Technology. Springer Berlin Heidelberg, 2012, pp. 411–420. Theocharopoulou, G., Giannakis, K., and Andronikos, T. The mechanism of splitting mitochondria in terms of membrane automata. In Signal Processing and Information Technology (ISSPIT), 2015 IEEE International Symposium on (2015), IEEE. 71
- 73. Thank you for your attention Any questions? 72