Defining Causality Is . . . Defining Causality: . . . Algorithmic . . . The Corresponding . . . Towards a How to Define Space- . . . Towards a Working . . . Better Understanding Discussion and . . . of Space-Time Causality: This Definition Is . . . Space-Time Causality . . . Kolmogorov Complexity and Home Page Causality as a Matter Title Page ◭◭ ◮◮ of Degree ◭ ◮ Vladik Kreinovich 1 and Andres Ortiz 2 Page 1 of 14 1 Department of Computer Science Go Back 2 Departments of Mathematical Sciences and Physics University of Texas at El Paso, El Paso, TX 79968, USA Full Screen aortiz19@miners.utep.edu, vladik@utep.edu Close Quit
Defining Causality Is . . . Defining Causality: . . . 1. Defining Causality Is Important Algorithmic . . . • Causal relation e � e ′ between space-time events is one The Corresponding . . . of the fundamental notions of physics. How to Define Space- . . . Towards a Working . . . • In Newton’s physics , it was assumed that influences can Discussion and . . . propagate with an arbitrary speed: e = ( t, x ) � e ′ = ( t ′ , x ′ ) ⇔ t ≤ t ′ . This Definition Is . . . Space-Time Causality . . . • In Special Relativity , the speeds of all the processes are Home Page limited by the speed of light c : Title Page e = ( t, x ) � e ′ = ( t ′ , x ′ ) ⇔ c · ( t ′ − t ) ≥ d ( x, x ′ ) . ◭◭ ◮◮ • In the General Relativity Theory , the space-time is ◭ ◮ curved, so the causal relation � is even more complex. Page 2 of 14 • Different theories, in general, make different predic- Go Back tions about the causality � . Full Screen • So, to experimentally verify fundamental physical the- ories, we need to experimentally check whether e � e ′ . Close Quit
Defining Causality Is . . . Defining Causality: . . . 2. Defining Causality: Challenge Algorithmic . . . • Intuitively, e � e ′ means that: The Corresponding . . . How to Define Space- . . . – what we do in the vicinity of e Towards a Working . . . – changes what we observe at e ′ . Discussion and . . . • If we have two (or more) copies of the Universe, then: This Definition Is . . . – in one copy, we perform some action at e , and Space-Time Causality . . . Home Page – we do not perform this action in the second copy. • If the resulting states differ, this would indicate e � e ′ : Title Page World 1 World 2 ◭◭ ◮◮ ✻ ✻ ◭ ◮ e ′ e ′ rain ∗ ∗ no rain Page 3 of 14 rain dance ∗ ∗ no rain dance e e Go Back Full Screen • Alas, in reality, we only observe one Universe, in which we either perform the action or we do not. Close Quit
Defining Causality Is . . . Defining Causality: . . . 3. Algorithmic Randomness and Kolmogorov Com- Algorithmic . . . plexity: A Brief Reminder The Corresponding . . . • If we flip a coin 1000 times and still get get all heads, How to Define Space- . . . common sense tells us that this coin is not fair. Towards a Working . . . Discussion and . . . • Similarly, if we repeatedly flip a fair coin, we cannot 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. ◭ ◮ • In contrast, the shortest way to print a truly random sequence is to actually print it bit-by-bit: printf (01. . . ). Page 4 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 Quit
Defining Causality Is . . . Defining Causality: . . . 4. The Corresponding Notion of Independence Algorithmic . . . • If y is independent on x , then knowing x does not help The Corresponding . . . us generate y . How to Define Space- . . . Towards a Working . . . • If y depends on x , then knowing x helps compute y ; Discussion and . . . example: This Definition Is . . . – knowing the locations and velocities 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 . ◭◭ ◮◮ • 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 Page 5 of 14 def K ( y | x ) = 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 Quit
Defining Causality Is . . . Defining Causality: . . . 5. How to Define Space-Time Causality: First Seem- Algorithmic . . . ing Reasonable Idea The Corresponding . . . • At first glance, we can check whether e � e ′ as follows: How to Define Space- . . . Towards a Working . . . – First, we perform observations and measurements Discussion and . . . in the vicinity of the event e , and get the results x . This Definition Is . . . – We also perform measurements and observations in Space-Time Causality . . . the vicinity of the event e ′ , and produce x ′ . Home Page – If x ′ depends on x , i.e., if K ( x ′ | x ) ≪ K ( x ′ ), then we claim that e can casually influence e ′ . Title Page ◭◭ ◮◮ • If e � e ′ , then indeed knowing what happened at e can help us predict what is happening at e ′ . ◭ ◮ • However, the inverse is not necessarily true. Page 6 of 14 • We may have x ≈ x ′ because both e and e ′ are influ- Go Back enced by the same past event e ′′ . Full Screen • Example: both e and e ′ receive the same signal from e ′′ . Close Quit
Defining Causality Is . . . Defining Causality: . . . 6. Towards a Working Definition of Causality Algorithmic . . . • According to modern physics, the Universe is quantum The Corresponding . . . in nature; for microscopic measurements: How to Define Space- . . . Towards a Working . . . – we cannot predict the exact measurement results, Discussion and . . . – we can only predict probabilities of different out- This Definition Is . . . comes; the actual observations are truly random. Space-Time Causality . . . • For each space-time event e : Home Page – we can set up such a random-producing experiment Title Page in the small vicinity of e , and ◭◭ ◮◮ – generate a random sequence r e . ◭ ◮ • This random sequence r e can affect future results. Page 7 of 14 • So, if we know the random sequence r e , it may help us Go Back predict future observations. • Thus, if e � e ′ , then for some observations x ′ performed Full Screen in the small vicinity of e ′ , we have K ( x ′ | r e ) ≪ K ( x ′ ). Close Quit
Defining Causality Is . . . Defining Causality: . . . 7. Discussion and Resulting Definition Algorithmic . . . • Reminder: when e � e ′ , then the random sequence r e The Corresponding . . . can affect the measurement results at e ′ : How to Define Space- . . . K ( x ′ | r e ) ≪ K ( x ′ ) . Towards a Working . . . Discussion and . . . • If e � � e ′ , then the random sequence r e cannot affect This Definition Is . . . the measurement results at e ′ : K ( x ′ | r e ) ≈ 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 r e denote a random ◭◭ ◮◮ sequence generated in the small vicinity of e . – We say e � e ′ if for some observations x ′ performed ◭ ◮ in the small vicinity of e ′ , we have Page 8 of 14 K ( x ′ | r e ) ≪ K ( x ′ ) . Go Back • Our definition follows the ideas of casuality as mark Full Screen transmission , with the random sequence as a mark. Close Quit
Defining Causality Is . . . Defining Causality: . . . 8. This Definition Is Consistent with Physical In- Algorithmic . . . tuition The Corresponding . . . • If e � e ′ , then we can send all the bits of r e from e How to Define Space- . . . to e ′ . Towards a Working . . . • The signal x ′ received in the vicinity of e ′ will thus be Discussion and . . . identical to r e . This Definition Is . . . • Thus, generating x ′ based on r e does not require any Space-Time Causality . . . Home Page computations at all: K ( x ′ | r e ) = 0. Title Page • Since the sequence x ′ = r e is random, we have ◭◭ ◮◮ K ( x ′ ) ≥ len( x ′ ) − C. ◭ ◮ • When r e = x ′ is sufficiently long (len( x ′ ) > 2 C ), we Page 9 of 14 have K ( x ′ ) ≥ len( x ′ ) − C > 2 C − C = C, hence 0 = K ( x ′ | r e ) < K ( x ′ ) − C and K ( x ′ | r e ) ≪ K ( x ′ ) . Go Back Full Screen • So, our definition is indeed in accordance with the physical intuition. Close Quit
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