Solving a PSPACE-complete problem by a linear interval-valued computation

1. Solving a PSPACE-complete problem by a linear interval-valued computation Benedek Nagy, Sándor Vályi University of Debrecen, Faculty of Informatics Hungary

2. Introduction • Definitions (interval-value, operations on them, computations and decidability) • Motivation • Complexity results • Proof I. • Variations on the theme • Proof II. (if time available)

3. Definitions • Benedek Nagy: An interval-valued computing device, in: “Computability in Europe 2005: New computational para­digms”, ILLC Publications • Interval-value: a subset of [0,1), union of subintervals of form [a,b). Also are representable by characteristic function. • The set of allowed interval-values: V. • One distinguished interval-value: FIRSTHALF.

4. Example. An interval-value FIRSTHALF

5. Operations on V: • Boolean operations, for subsets: + union, - complementation, • Shifts (L,R), [the above are analogous to the operators on finite bytes], • Product (*) [Nagy’s dissertation, many-valued logic].

6. Shift to the right If no such component: does nothing

7. Example for productoperator • Product

8. Interval-valued computation sequence A sequence of operator applications to already constructed interval-values, starting from FIRSTHALF. • That is, <S(0), S(1),…, S(n)>, (n  IN), • where S(0) = FIRSTHALF, • for each i<n+1: S(i+1) = op(S(k), S(j)), • (op  {+,-,L,R,*}, k<i,j<i) • (functional style, formalising the original paper!)

9. The value ||L|| of an interval-valued computation sequence: • ||FIRSTHALF||= [0,1/2). • The meaning of the operators are described informally by the previous figures (only in this talk, in paper: formal definitions).

10. Decidability • A language L is decidable by an interval-valued computation iff there is an algorithm A that for each input word w constructs an appropriate computation sequence with last element A(w) such that • w L if and only if ||A(w)|| is nonempty. • Further, we consider in this case -L also decidable. (Testing emptiness is also allowed then.)

11. Complexity • A language L is decidable by a linear interval-valued computation iff there is a positive constant c and there is a logarithmic space algorithm A, that for each input word w constructs an appropriate computation sequence with last element S(n) such that • w L if and only if || S(n)|| is nonempty, moreover, n does not exceed c|w|.

12. Theorem. (question: referee of first paper) There is a PSPACE-complete problem that is decidable by a linear interval-valued computation.  • We prove it for one of the basic examples for PSPACE-problems, namely, the problem of validity of a quantified Boolean formula. (QSAT) • In the paper submitted to this Conference we have given a formal proof. In this talk illustration only.

13. Illustration for the proof.

14. Illustration, Part 2.

15. Restriction on i-v computations • A language L is said to be decidable by a restricted polynomial size interval-valued computationiff • There exists a polynom P and a logarithmic space algorithm A with the following properties. • For each input word w, A constructs a computation sequence with last element S(n) such that • w L if and only if || S(n)|| is nonempty, • n does not exceed P(|w|), further, • the left side of the product applications is always • FIRSTHALF.

16. Theorem 2. (after submitting) • PSPACE coincides with the class of languages decidable by a restricted polynomial size interval-valued computation. In a technical report we have proved this formally. We outline this proof, at the end of this talk. If we release the condition on the left size of the operators, then we obtain that the class of languages decidable by a polynomial interval-valued computation is included in EXPSPACE. (equality?)

18. Motivation • Analogous computation. New version, compared to B—S—S, • Fits into computation over algebraic structures, • Analogous data representation by waves (1 dimension, this paper) is legitimate just as by images (2 dimension, Naughton and Woods, CiE 2005) [also in Theoretical C.S.], • Infinite Turing machine computations with dense memory organisation .

19. Motivation for the operators • Similarity to operators in digital computers • Turing completeness • Mathematically: The set of the allowed values equipped with the operators is a Boolean algebra with operators

20. The structure of interval-values with these operators • First-order theory of this structure, its metamathematical properties. • The closed terms of this structure (involving no variables, only the constant FIRSTHALF): equality, inclusion is decidable in exponential space. • What if arbitrary terms? • Monadic second-order theory of this structure: For possible description of the class of languages decidable by an arbitrary interval-valued computation. • Combination with the forthcoming variations:

21. Variations on the theme, further work: • The allowed basic interval-values can be varied: • -- Finite unions of subintervals • -- Arbitrary subsets • -- Unions of shifts of [0,1/2n] • All the questions remain interesting. • Other operators should also be developed, retaining Turing completeness. Esp. for the second variation!

22. Another natural ideas are the following ones: • What happens if not an algorithm is responsible for producing different computation sequences for different input words, but an interval-valued algorithm is a pair of a fixed computation sequence and a “digital-analog con­verter” which takes the input word and produces an interval-value – and the answer is taking by an “analog-digital converter”. • Imperative paradigm for i-valued computation.

23. Work to do, not mentioned yet • To show Turing completeness in a mathematical manner, e.g. give a problem in EXPSPACE\PSPACE solvable by a non-restricted polynomial i-v computation, • Investigate connections with interval temporal logic.