In computer graphics, people speak of screen's resolution as the screen's dimensions: "This screen's resolution is 1280x1024 pixels."

This colloquial usage is wrong, however (just like some people used to speak of bauds when they meant bps). The resolution measures screen's "density" - how big the pixels are, physically.

Screen resolution is usually measured as dots per inch - that is, pixels per inch. For example, this screen in front of me has a screen dimensions of 1280x1024 pixels, and physical size of 361x271 millimeters. This makes the screen resolution 90x96 dpi.

Typical computer screens have resolutions ranging from 75x75 to 100x100 dpi.

(node your homework)

Resolution is a method to prove logical statements. Actually it doesn't prove a statement, but it proves the unrealizability of it. But that's not a problem. If you want to prove something you just negate it, and prove the unrealizability of this formula.

Resolution works for propositional logic, predicate Logic and logical programs in the form of Horn clauses (that's why Prolog uses it).

propositional logic

To simplify the resolution algorithm, the formula should be in clause form (which is like the disjunctive normal form, but the disjunctions are lists and the whole formula is a list of lists). The algorithm works the following way:

  • Choose a clause C1, which contains an atom p
  • Choose a clause C2, which contains an atom not(p)
  • Generate a new clause C by deleting all p from C1 and all not(p) from C2 and combining both disjunctively
  • Do this until you can't generate any new clauses or till you generated an empty clause

If you generated an empty clause, the formula is unrealizable. If you negated the formula before you used the algorithm, you proved the formula. If you finished the algorithm without generating an empty clause, you either proved that it is not unrealizable or that it is wrong (if you negated it in the beginning).

Example: Your formula is: [ [a], [b], [c], [not a, not b, e], [not c, not e] ]

The disjunctions are numbered from 1 to 5:

res(1,4) = [not b, e] 6
res(2,6) = [e] 7
res(3,5) = [not e] 8
res(6,8) = [] 9
You just proved that the formula is unrealizable.

predicate logic

The algorithm works very similar. Again you have to have a formula in clause form (which is a bit more difficult here, as you have to convert the formula in prenex normal form, then into skolem normal form and finally into the disjunctive normal form). The algorithm again has the same rule for generating new clauses, but with the extension of the use of unification for the finding of equal atoms:

  • Choose a clause C1, which contains an atom p(a1,...,an)
  • Choose a clause C2, which contains an atom not(p(b1,...,bn)
  • p(a1,...,an) and (p(b1,...,bn) can be unified with an unificator U
  • Generate a new clause (C)U by deleting all p from C1 and all not(p) from C2 and combining both disjunctively
  • Do this until you can't generate any new clauses or till you generated an empty clause
  • But using only this rule is not enough for predicate logic, so there is another one:

  • you have a clause C which contains p(a1,...,an) and p(b1,...,bn)
  • or not(p(a1,...,an)) and not(p(b1,...,bn))
  • p(a1,...,an) and p(b1,...,bn) can be unified with an unificator U
  • generate a new clause D1 without p(a1,...,an) and p(b1,...,bn)
  • Horn clauses

    A logical program which contains only Horn clauses is a definite program. It can be proven by using SLD-resolution (linear resolution with selection function on definite clauses). The algorithm:

    you have: <- B1 and ... and Bm

    which is called a definite goal (your question to the program) and

    A <- A1 and ... and An

    which is a clause from your program. If a Bi and A can be unified with a unificator U, you can write:

    <- (B1 and ... and Bi-1 and A1 and ... and An and ... Bm)U

    Repeat this until you either get the empty clause(you've proven B1 and ... and Bm) or you can't do more resolutions (you've proven that B1 and ... and Bm is wrong). The problem with the resolution of definite programs is, that it does not neccessarily stop.

    Hello, Ed. I don't know if you'll ever find this message, or how old you'll be when you do. I'm half-hoping you never find it at all. This whole business seems... well, sickening would be the word.

    My name is Eduardo Frank MacPherson. I am you, from another timeline.

    This isn't the history module, that comes later, so I'll be blunt. Our Earth has fallen. We were thrust into interstellar war and we lost. Badly. So badly that right now, over a decade after the end of hostilities, we still don't have a clear idea of the full extent of the damage. Your job is to change the course of that war. We are giving you all the information you need. Unfortunately the system we're using to do this is complicated and difficult and... less untested than untestable. If you've found a way to read this now, you should be able to access the rest of the data in the same way. Otherwise... information may only be able to surface gradually, as flashes of inspiration, or dreams, or feelings which I'm told are something like deja vu. You will have the information you need at the time you need it. Whether you will be able to use it is down to you. We can't guide your actions from here, or alter your personality. We know you're going to be smart, hard-working and scientifically-inclined. We're hoping you'll also be tough, and... well, if you figure out a word which is like "patriotic" but applies to your planet instead of your country, then we hope you'll be something like that, too.

    To be honest, a lot of people are secretly hoping you'll turn out to be the next Alexander the Great.

    That's what I don't like about this. What we've given you is a greater burden than anybody ever carried, and we didn't ask your permission. Certainly, I volunteered for the project - because my life is the only life I'm remotely happy messing with. But you and I, I have realised, are two different people. I took away your life and replaced it with a mission.

    For that, I'm sorry.

    Good luck.

    Ed MacPherson
    Mount Kerrig Research Facility, Nevada, USA
    2041.09.16

    "I don't understand," I say. "What does that mean?"

    "In their universe," says Ed, slowly and patiently, "the Eridanians won the war. They EMPed the entire planet Earth back into the stone age and neutron-bombed what was left. We fought back, but our technology was a century too slow to keep up with them. Because it was under a mountain, the Kerrig facility - a time travel research base in their universe just like ours - was one of the few places which was shielded sufficiently to have any working electrical technology after the attacks finally ceased.

    "They deconstructed downed Eridanian drones and took scientific and tactical information from the few intact data structures they could scavenge. They constructed entire libraries - all the information about the enemy they could find, along with ready-made guides for the design and manufacture of weapons, nanoassemblers, unbelievably strong metal alloys... even spaceships. They put all the information they could fit into a tiny data package. And they asked for a volunteer. Turns out, in that universe, I was still pretty good at physics; I was one of the ones at Kerrig at the end of it all. I volunteered to be the subject. My mind was scanned and we moulded the data to fit it.

    "We took a piece of highly sophisticated medical equipment - somewhere in Japan-plus-ten, someone is drawing schematics for it, right now - and hooked it together with our time machine and projected the data backwards in time, to me.

    "And the irony is - we got it as right as we possibly could have. But we didn't fully understand time travel. We had no guiding lights. We thought that by sending someone back in time to stop the war we could change history. It would all have been so perfect. To disappear and be replaced by a better world. I don't want to guess how many times we repeated the experiment when that didn't happen. Or what we did when we realised we were going to have to see our war through to the bitter end, alone."

    "You can... remember everything now? All that data is there?"

    "Yes," says Ed.

    And there is a moment of silence.

    "So... what happens now? How do we stop the war?"

    "Don't you get it? We knew why the Eridanians attacked us. We knew it would be impossible to coexist with them. Even if we could find a way to talk with them, the damage has already been done. The level of radio output coming from Earth-minus-ten of 1998 is already high enough to permanently brain-damage these creatures. That's not to mention the fact that in eleven years' time, the signals from Earth-plus-ten of 2008 begin to arrive, more than doubling the volume in an instant. Even instantly ceasing all radio activity on Earth right now wouldn't save them. No, the war is unavoidable. The plan, all along, was to win the war for Humanity. I'm supposed to wipe them out. It's them or us."

    "There has to be a third option."

    "The only third option would be to block the signals somehow. But if I knew how to do that, it would have been part of the plan to start with. You'd need godlike powers. You'd need to... to build a wall across trillions of kilometres of space, or something. A wall, or a force-field, or a... chasm. Wait a minute." Ed turns to the robot avatar of the Hotel Infinity, whom I had forgotten was even there. "The energy virus," he says.

    "The last piece of the puzzle," says the robot.

    "Can you do it?"

    "It's already been done."

    *

    We request a window, some way to see the event as its light - or absence thereof - reaches us. In response, the Hotel Infinity instantly teleports us to an invisible force-shielded bubble of air on the exterior of its hull.

    Standing in it, entirely comfortable, but apparently directly exposed to hard vacuum, is one of the most unnerving experiences imaginable.

    The exterior of the ship is a burnt orange-red, and has the approximate texture of a field of rocks strewn with crushed circuit boards and radio equipment. Its hull (shell?) is vaguely conical but very irregular, and stretches for five hundred metres ahead of us and another five hundred behind. To the left and right it curves precipitously downwards - it feels like I could step off into space any time I wanted.

    Dominating the "sky" above us are two of the ship's eight wings - enormous red bat-like structures which ring the fatter end of the ship. These have apparently evolved as a means of dumping heat into space, a process vital to survival in the hot near-sun habitat from which the Hotel Infinity's race hail.

    We rotate away from Epsilon Eridani and the orange glare is replaced with a glittering star field.

    "I was given the technology to beat the waves of robot drones that the Eridanians sent to us," says Ed. "But when the Eridanians realised they were losing, they changed their strategy, something nobody in the other timeline could have accurately predicted. They conceded that they would never shut us up, and instead they began working on this. You see, there's more than one kind of energy virus.

    "They weren't trying to cut an infinitely expanding sphere out of the universe. That was a mistake. They were trying to create a spherical shell; a virus which eats up spacetime in two dimensions, curving around slowly as it expands, then meets itself as it forms a complete sphere and stops completely. They just hit the wrong combination first. And the Andromedans, in their infinite wisdom, know the right combination.

    "It'll block out every piece of incoming electromagnetic radiation in this layer of the universe - light, radio, microwaves, ultraviolet. Light will still be able to escape, though, so the star won't be hidden. And because the higher layers aren't affected, we can still pass through the barrier with the tunnel drive, which means we can leave... and come back."

    Eight minutes pass silently. Then, as we watch, our home star is blotted out, cut out of the sky as if by some cosmic holepunch. As time passes, the dark circle begins to expand, eclipsing more stars. With its edge advancing at the speed of light, it'll take about nineteen hours to enclose the whole solar system.

    Around us, although we can't tell, blissful radio darkness falls. "Well," says Ed, "that's my purpose in life fulfilled."


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    Res`o*lu"tion (-l?"sh?n), n. [F. r'esolution. L. resolutio a loosening, solution. See Resolve.]

    1.

    The act, operation, or process of resolving. Specifically: (a) The act of separating a compound into its elements or component parts. (b) The act of analyzing a complex notion, or solving a vexed question or difficult problem.

    The unraveling and resolution of the difficulties that are met with in the execution of the design are the end of an action. Dryden.

    2.

    The state of being relaxed; relaxation.

    [Obs.]

    3.

    The state of being resolved, settled, or determined; firmness; steadiness; constancy; determination.

    Be it with resolution then to fight. Shak.

    4.

    That which is resolved or determined; a settled purpose; determination. Specifically: A formal expression of the opinion or will of an official body or a public assembly, adopted by vote; as, a legislative resolution; the resolutions of a public meeting.

    5.

    The state of being resolved or firm in opinion or thought; conviction; assurance.

    [Obs.]

    Little resolution and certainty there is as touching the islands of Mauritania. Holland.

    6. Math.

    The act or process of solving; solution; as, the resolution of an equation or problem.

    7. Med.

    A breaking up, disappearance; or termination, as of a fever, a tumor, or the like.

    8. Mus.

    The passing of a dissonant into a consonant chord by the rising or falling of the note which makes the discord.

    Joint resolution. See under Joint, a. -- Resolution of a forcemotion Mech., the separation of a single force or motion into two or more which have different directions, and, taken together, are an equivalent for the single one; -- the opposite of composition of a force. -- Resolution of a nebula Astron., the exhibition of it to the eye by a telescope of such power as to show it to be composed of small stars.

    Syn. -- Decision; analysis; separation; disentanglement; dissolution; resolvedness; resoluteness; firmness; constancy; perseverance; steadfastness; fortitude; boldness; purpose; resolve. See Decision.

     

    © Webster 1913.

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