Designing, Understanding, 1st Approach: Need . . . Interval - PowerPoint PPT Presentation

Main Problem: . . . Our Main Claim: Use . . . Why Logic? 1st Approach: . . . Designing, Understanding, 1st Approach: Need . . . Interval Computations and Analyzing Proof Mining 2nd Approach: . . . Potential Use of . . . Unconventional Potential Use of . . . Explicit Use of . . . Computation: Unconventional . . . Title Page The Important Role of Logic ◭◭ ◮◮ and Constructive ◭ ◮ Mathematics Page 1 of 22 Go Back Grigori Mints Stanford University, gmints@stanford.edu Full Screen Close Vladik Kreinovich University of Texas at El Paso, vladik@utep.edu Quit

Our Main Claim: Use . . . Why Logic? 1. Main Problem: Reminder and Possible Approaches 1st Approach: . . . • Problem: computations are often too slow. 1st Approach: Need . . . Interval Computations • Traditional approaches: Proof Mining • design faster super-computers ( hardware ); 2nd Approach: . . . • design faster algorithms . Potential Use of . . . • Limitations of the traditional approaches: Potential Use of . . . • re hardware: we use the same physical processes as Explicit Use of . . . before; Unconventional . . . Title Page • re algorithms: we solve the same exact problem as before – while often, data are imprecise . ◭◭ ◮◮ • Possible new approaches: ◭ ◮ • hardware: use unconventional physical (and biolog- Page 2 of 22 ical) processes; Go Back • algorithms: perform computations only up to accu- Full Screen racy that matches the input accuracy. Close Quit

Our Main Claim: Use . . . Why Logic? 2. Our Main Claim: Use Logic (and Constructive Math- 1st Approach: . . . ematics) 1st Approach: Need . . . • Alternative approaches (reminder): Interval Computations Proof Mining • use unconventional physical (and biological) pro- 2nd Approach: . . . cesses; Potential Use of . . . • perform computations only up to accuracy that Potential Use of . . . matches the input accuracy. Explicit Use of . . . • Our claim: for these approaches to succeed, it is crucial Unconventional . . . to further develop: Title Page • the corresponding tools of mathematical logic , and ◭◭ ◮◮ • the related methods of constructive mathematics . ◭ ◮ Page 3 of 22 Go Back Full Screen Close Quit

Our Main Claim: Use . . . Why Logic? 3. Why Logic? 1st Approach: . . . • Claim: logic is useful on all the stages of solving a 1st Approach: Need . . . problem : Interval Computations • we specify a problem; Proof Mining 2nd Approach: . . . • we design and implement an algorithm ; Potential Use of . . . • we verify the corresponding program. Potential Use of . . . • Specifying a problem: Explicit Use of . . . • sometimes, the problem is to solve an equation; Unconventional . . . • in general, a proper formulation requires quantifiers Title Page etc. (stable control, etc.) – i.e., logic. ◭◭ ◮◮ • Designing an algorithm: logic programming transforms ◭ ◮ a logical specification into an algorithm. Page 4 of 22 • Program verification: logic helps in reasoning about Go Back programs; e.g., pre-condition implies post-condition. Full Screen • Proof assistant programs (based on logic) help to prove. Close Quit

Our Main Claim: Use . . . Why Logic? 4. 1st Approach: Computations with Limited Accu- racy 1st Approach: . . . 1st Approach: Need . . . • Objective: compute f ( x ) with accuracy ε > 0. Interval Computations Proof Mining • Idea: compute x only with accuracy δ > 0 for which 2nd Approach: . . . d ( x, x ′ ) ≤ δ implies d ( f ( x ) , f ( x ′ )) ≤ ε. Potential Use of . . . • Constructive math: a number is r n s.t. d ( r n , x ) ≤ 2 − n ; Potential Use of . . . a constructive function is a pair of: Explicit Use of . . . Unconventional . . . • an algorithm f : X → Y and Title Page • an algorithm ε → δ . ◭◭ ◮◮ • Status: this is developed only for a few problems. ◭ ◮ • Research Direction I.1: Page 5 of 22 • develop general constructive mathematics techniques , Go Back • with a special emphasis on problems requiring in- Full Screen tensive computations (e.g., large-scale PDE) . Close Quit

Our Main Claim: Use . . . Why Logic? 5. 1st Approach: Need for Logic 1st Approach: . . . • Decomposition: solutions to complex problems usually 1st Approach: Need . . . come from combining solutions to subproblems. Interval Computations Proof Mining • Logic is needed for combination: e.g., stable robust con- trol means it is stable ∀ possible parameter values. 2nd Approach: . . . Potential Use of . . . • Constructive logic: we want to preserve constructions: Potential Use of . . . • ∃ x P ( x ) should mean that we can construct such x ; Explicit Use of . . . • ∀ x ∃ y P ( x, y ) ↔ ∃ algorithm ϕ : X → Y s.t. P ( x, ϕ ( x )). Unconventional . . . Title Page • It was originated by Kolmorogov: e.g., A ∨¬ A is false. ◭◭ ◮◮ • Fact: constructive logic is used in constructive math. ◭ ◮ • Research Direction I.2: Page 6 of 22 • develop general constructive logic techniques , Go Back • with a special emphasis on problems requiring in- Full Screen tensive computations. Close Quit

Our Main Claim: Use . . . Why Logic? 6. Interval Computations 1st Approach: . . . • Constructive math: algorithms work for all accuracies. 1st Approach: Need . . . Interval Computations • In practice: we only need one given accuracy. Proof Mining • Solution: interval computations (a.k.a. applied con- 2nd Approach: . . . structive mathematics). Potential Use of . . . • Idea: if we know an estimate � x w/accuracy ∆, then Potential Use of . . . Explicit Use of . . . x ∈ x = [ � x − ∆ , � x + ∆] . Unconventional . . . Title Page • Traditional approach: we also know probability distri- def bution for ∆ x = � x − x (usually Gaussian). ◭◭ ◮◮ • Where it comes from: calibration using standard MI. ◭ ◮ Page 7 of 22 • Problem: calibration is not possible in: Go Back • fundamental science – no better Measuring Instr. (MI); • manufacturing – too expensive to calibrate. Full Screen Close Quit

Our Main Claim: Use . . . Why Logic? 7. Interval Computations (cont-d) 1st Approach: . . . x 1 1st Approach: Need . . . ✲ Interval Computations x 2 y = f ( x 1 , . . . , x n ) f ✲ Proof Mining · · · ✲ 2nd Approach: . . . x n ✲ Potential Use of . . . Potential Use of . . . • Given: algorithm y = f ( x 1 , . . . , x n ) and x i = [ x i , x i ]. Explicit Use of . . . • Compute: the corresponding range of y : Unconventional . . . Title Page y = [ y, y ] = { f ( x 1 , . . . , x n ) | x 1 ∈ [ x 1 , x 1 ] , . . . , x n ∈ [ x n , x n ] } . ◭◭ ◮◮ • Fact: this problem is NP-hard even for quadratic f . ◭ ◮ • Challenge: find a good approximation Y ⊇ y . Page 8 of 22 • Applications: spaceflights, super-colliders, robotics, chem- Go Back ical engineering, nuclear safety, etc. Full Screen • Modal logic is efficiently used: � ( y ∈ y ), ♦ ( x = � x ). Close Quit

Our Main Claim: Use . . . Why Logic? 8. Proof Mining 1st Approach: . . . • Historically: first existence proofs were direct (constr.). 1st Approach: Need . . . Interval Computations • Currently: many proofs are indirect – they prove ∃ x P ( x ) Proof Mining without constructing such x . 2nd Approach: . . . • Meta-results: sometimes, we can extract (“mine”) a Potential Use of . . . constructive proof from a non-constructive one. Potential Use of . . . • Example (Kohlenbach): uniqueness → computability. Explicit Use of . . . Unconventional . . . • Idea: to find x s.t. f ( x ) = 0, compute min B ε ( x i ) | f ( x ) | Title Page w/increasing accuracy for x i from ε -nets, ε = 2 − 1 , 2 − 2 , . . . ◭◭ ◮◮ • Research Direction I.4: ◭ ◮ • further develop proof mining, Page 9 of 22 • with a special emphasis on its use to develop algo- rithms for realistic large-scale problems. Go Back Full Screen Close Quit

Our Main Claim: Use . . . Why Logic? 9. 2nd Approach: Unconventional Computations 1st Approach: . . . • Main idea: use non-standard physical processes to speed 1st Approach: Need . . . up computations. Interval Computations Proof Mining • Most well-known example: quantum computing: • search in an un-sorted array of size n in time √ n 2nd Approach: . . . Potential Use of . . . (Grover); Potential Use of . . . • factoring large integers in polynomial time (Shor). Explicit Use of . . . • Limitation: the only provable speed-up is polynomial. Unconventional . . . Title Page • Other schemes: can potentially lead to exponential ◭◭ ◮◮ speed-up. ◭ ◮ • In other words: we can potentially solve NP-hard prob- lems in polynomial time. Page 10 of 22 • Simplest example: acausal processes – compute and Go Back send the result back in time. Full Screen Close Quit