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Home , Causality (physics), Time

Defining Is . . .

Defining Causality: . . .

Algorithmic . . .

The Corresponding . . . Towards a How to Define - . . . Better Understanding Towards a Working . . . Discussion and . . . of Space- Causality: This Definition Is . . . Space-Time Causality . . . Kolmogorov Complexity and Home Page Causality as a Title Page of Degree JJ II J I 1 2 Vladik Kreinovich and Andres Ortiz Page1 of 14 1Department of Computer 2Departments of Mathematical and Go Back

University of Texas at El Paso, El Paso, TX 79968, USA Full Screen [email protected], [email protected] Close

Quit Defining Causality Is . . .

Defining Causality: . . . 1. Defining Causality Is Important Algorithmic . . . 0 • Causal relation e 4 e between space-time events is one The Corresponding . . . of the fundamental notions of physics. How to Define Space- . . . • In Newton’s physics, it was assumed that influences can Towards a Working . . . propagate with an arbitrary : Discussion and . . . This Definition Is . . . e = (t, x) e0 = (t0, x0) ⇔ t ≤ t0. 4 Space-Time Causality . . . • In , the of all the processes are Home Page

limited by the speed of c: Title Page e = (t, x) e0 = (t0, x0) ⇔ c · (t0 − t) ≥ d(x, x0). 4 JJ II • In the Theory, the space-time is J I curved, so the causal relation is even more complex. 4 Page2 of 14 • Different theories, in general, make different predic- Go Back tions about the causality 4. • So, to experimentally verify fundamental physical the- Full Screen 0 ories, we need to experimentally check whether e 4 e . Close

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Defining Causality: . . . 2. Defining Causality: Challenge Algorithmic . . . 0 • Intuitively, e 4 e means that: The Corresponding . . . – what we do in the vicinity of e How to Define Space- . . . Towards a Working . . . – changes what we observe at e0. Discussion and . . .

• If we have two (or more) copies of the , then: This Definition Is . . .

– in one copy, we perform some at e, and Space-Time Causality . . . – we do not perform this action in the copy. Home Page 0 Title Page • If the resulting states differ, this would indicate e 4 e : World 1 World 2 JJ II

∗6e0 e0∗6 no rain J I

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rain dance∗e e∗ no rain dance Go Back

• Alas, in , we only observe one Universe, in which Full Screen we either perform the action or we do not. Close

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Defining Causality: . . . 3. Algorithmic Randomness and Kolmogorov Com- plexity: A Brief Reminder Algorithmic . . . The Corresponding . . .

• If we flip a coin 1000 and still get get all heads, How to Define Space- . . . common sense tells us that this coin is not fair. Towards a Working . . . • Similarly, if we repeatedly flip a fair coin, we cannot Discussion and . . . expect a periodic sequence 0101 ... 01 (500 times). This Definition Is . . . Space-Time Causality . . .

• Traditional probability theory does not distinguish be- Home Page tween random and non-random sequences. Title Page • Kolmogorov, Solomonoff, Chaitin: a sequence 0 ... 0 isn’t random since it can be printed by a short program. JJ II • In contrast, the shortest way to print a truly random J I sequence is to actually print it bit-by-bit: printf(01. . . ). Page4 of 14

• Let an integer C > 0 be fixed. We say that a string x Go Back

is random if K(x) ≥ len(x) − C, where Full Screen def K(x) = min{len(p): p generates x}. Close

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Defining Causality: . . . 4. The Corresponding of Independence Algorithmic . . . • If y is independent on x, then knowing x does not help The Corresponding . . . us generate y. How to Define Space- . . . • If y depends on x, then knowing x helps compute y; Towards a Working . . . example: Discussion and . . . This Definition Is . . .

– knowing the locations and x of a mechan- Space-Time Causality . . .

ical system at time t Home Page – helps compute the locations and velocities y at time Title Page t + ∆t. JJ II • Let an integer C > 0 be fixed. We say that a string y is independent of x if K(y | x) ≥ K(y) − C, where J I

Page5 of 14 K(y | x) def= min{len(p): p(x) generates y}. Go Back • We say that a string y is dependent on the string x if Full Screen

K(y | x) < K(y) − C. Close

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Defining Causality: . . . 5. How to Define Space-Time Causality: First Seem- ing Reasonable Algorithmic . . . The Corresponding . . . 0 • At first glance, we can check whether e 4 e as follows: How to Define Space- . . . – First, we perform observations and Towards a Working . . . in the vicinity of the e, and get the x. Discussion and . . . This Definition Is . . . – We also perform measurements and observations in the vicinity of the event e0, and produce x0. Space-Time Causality . . . Home Page – If x0 depends on x, i.e., if K(x0 | x)  K(x0), then we claim that e can casually influence e0. Title Page 0 JJ II • If e 4 e , then indeed knowing what happened at e can 0 help us predict what is happening at e . J I • However, the inverse is not necessarily true. Page6 of 14 • We may have x ≈ x0 because both e and e0 are influ- Go Back 00 enced by the same event e . Full Screen 0 00 • Example: both e and e receive the same signal from e . Close

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Defining Causality: . . . 6. Towards a Working Definition of Causality Algorithmic . . . • According to , the Universe is quantum The Corresponding . . . in ; for microscopic measurements: How to Define Space- . . . – we cannot predict the exact results, Towards a Working . . . Discussion and . . . – we can only predict probabilities of different out- comes; the actual observations are truly random. This Definition Is . . . Space-Time Causality . . .

• For each space-time event e: Home Page

– we can up such a random-producing experiment Title Page in the small vicinity of e, and JJ II – generate a random sequence re. J I • This random sequence re can affect results. Page7 of 14 • So, if we know the random sequence re, it may help us predict future observations. Go Back 0 0 Full Screen • Thus, if e 4 e , then for some observations x performed 0 0 0 in the small vicinity of e , we have K(x | re)  K(x ). Close

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Defining Causality: . . . 7. Discussion and Resulting Definition Algorithmic . . .

0 The Corresponding . . . • Reminder: when e 4 e , then the random sequence re can affect the measurement results at e0: How to Define Space- . . . 0 0 Towards a Working . . . K(x | re)  K(x ). Discussion and . . . 0 • If e 46 e , then the random sequence re cannot affect This Definition Is . . . 0 0 0 the measurement results at e : K(x | re) ≈ K(x ). Space-Time Causality . . . Home Page • So, we arrive at the following semi-formal definition: Title Page – For a space-time event e, let re denote a random sequence generated in the small vicinity of e. JJ II 0 0 – We say e 4 e if for some observations x performed J I 0 in the small vicinity of e , we have Page8 of 14 0 0 K(x | re)  K(x ). Go Back • Our definition follows the of casuality as mark Full Screen

transmission, with the random sequence as a mark. Close

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Defining Causality: . . . 8. This Definition Is Consistent with Physical In- tuition Algorithmic . . . The Corresponding . . . 0 • If e 4 e , then we can send all the bits of re from e How to Define Space- . . . 0 to e . Towards a Working . . . • The signal x0 received in the vicinity of e0 will thus be Discussion and . . . This Definition Is . . . identical to re. Space-Time Causality . . . • Thus, generating x0 based on r does not require any e Home Page 0 computations at all: K(x | re) = 0. Title Page 0 • Since the sequence x = re is random, we have K(x0) ≥ len(x0) − C. JJ II J I 0 0 • When re = x is sufficiently long (len(x ) > 2C), we have K(x0) ≥ len(x0) − C > 2C − C = C, hence Page9 of 14 0 0 0 0 Go Back 0 = K(x | re) < K(x ) − C and K(x | re)  K(x ). • So, our definition is indeed in accordance with the Full Screen physical intuition. Close

Quit Defining Causality Is . . .

Defining Causality: . . . 9. Randomness Is a Matter of Degree Algorithmic . . . • Reminder: a sequence x is random if K(x) ≥ len(x)−C The Corresponding . . . for some C > 0. How to Define Space- . . .

Towards a Working . . . • For a given sequence x, its degree of randomness d(x) can be defined by the smallest integer C for which Discussion and . . . This Definition Is . . .

K(x) ≥ len(x) − C. Space-Time Causality . . .

Home Page • One can check that this smallest integer is equal to the difference d(x) = len(x) − K(x). Title Page • For random sequences, the degree d(x) is small. JJ II J I • For sequences which are not random, the degree d(x) is large. Page 10 of 14 • In general, the smaller the difference d(x), the more Go Back random is the sequence x. Full Screen

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Defining Causality: . . . 10. Space-Time Causality Is a Matter of Degree Algorithmic . . .

0 0 The Corresponding . . . • Our definition of causality is that K(x | re) < K(x ) − C for some large integer C. How to Define Space- . . .

Towards a Working . . . • The larger the integer C, the more confident we are that an event e can causally influence e0. Discussion and . . . This Definition Is . . .

• It is therefore reasonable to define a degree of causality Space-Time Causality . . . 0 c(e, e ) as the largest integer C for which Home Page

0 0 K(x | re) < K(x ) − C. Title Page

• One can check that this largest integer is equal to the JJ II 0 0 0 difference c(e, e ) = K(x ) − K(x | re) − 1. J I • The larger this difference c(e, e0), the more confident Page 11 of 14 0 we are that e can influence e . Go Back

• In other words, just like randomness turns out to be a Full Screen

matter of degree, causality is also a matter of degree. Close

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Defining Causality: . . . 11. Remaining Open Problems Algorithmic . . . • It is desirable to explore possible physical of The Corresponding . . . such “degrees of causality” c(e, e0). How to Define Space- . . .

Towards a Working . . . • Maybe this function c(e, e0) is related to relativistic metric – the amount of between e and e0? Discussion and . . . This Definition Is . . .

• Another open problem: the above definition works for Space-Time Causality . . .

objects in a small vicinity of one spatial location. Home Page

• In quantum physics, not all objects are localized in Title Page space-time. JJ II • We can have situations when the states of two spatially J I separated are entangled. Page 12 of 14 • It is desirable to extend our definition to such objects Go Back as well. Full Screen

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Quit Defining Causality Is . . .

Defining Causality: . . . 12. Conclusions Algorithmic . . . • We propose a new operationalist definition of causality The Corresponding . . . 0 0 e 4 e between space-time events e and e . How to Define Space- . . . • Namely, to check whether an event e can casually in- Towards a Working . . . fluence an event e0, we: Discussion and . . . This Definition Is . . .

– generate a truly random sequence re in the small Space-Time Causality . . .

vicinity of the event e, and Home Page

– perform observations in the small vicinity of the Title Page event e0. JJ II • If some observation results x0 (obtained near e0) depend 0 J I on re, then we claim that e 4 e . Page 13 of 14 • On the other hand, if all observation results x0 are in- 0 Go Back dependent on re, then we claim that e 46 e . • This new definition naturally leads to a conclusion that Full Screen

space-time causality is a matter of degree. Close

Quit 13. Acknowledgments Defining Causality Is . . . Defining Causality: . . . Algorithmic . . . This was supported in part: The Corresponding . . . How to Define Space- . . . • by the National Science Foundation grants HRD-0734825 Towards a Working . . . (Cyber-ShARE Center of Excellence) and DUE-0926721, Discussion and . . . This Definition Is . . . • by Grant 1 T36 GM078000-01 from the National Insti- Space-Time Causality . . . tutes of Health, and Home Page

• by a grant on F-transforms from the Office of Naval Title Page Research. JJ II The authors are thankful: J I

• to John Symons for valuable discussions, and Page 14 of 14

• to the anonymous referees for useful suggestions. Go Back

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