Satisfiability.nb

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Cell[TextData[StyleBox["Solving problems with propositional logic",
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 "Mechanical computers are really nothing more complicated than very high \
speed machines that perform elementary logical operations on zeros and ones. \
Higher level concepts and their operations such as integers and their \
arithmetic are built up from these basic logical operations implemented in \
the computer hardware, and these concepts are then used to build higher \
abstraction structures and superstructures. But deep down, all computation \
boils down to zeros and ones and the elementary operations to use them to \
produce more zeros and ones.\nA fundamental result from the theory of ",
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can be mechanistically converted to a set of propositional variables and a \
corresponding set of logical ",
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these clauses all together are ",
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then allow us to reconstruct the solution of the original problem from the \
truth values of the propositions of the corresponding satisfiability problem. \
With modern satisfiability solvers, it is often actually easier and even more \
efficient to solve problems by converting them to propositional logic instead \
of trying to implement the backtracking algorithm and think up to the \
dependencies inside the problem that can be used to guide this search, since \
we can just let the satisfiability solver implicitly discover these \
dependencies.\n",
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every time a new instance is printed in the daily newspaper. Here is one \
possible sudoku found online, described on that web page as being quite \
difficult for human solvers. "
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 "An ",
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 "-by-",
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 " propositions to express the claims of the form \[OpenCurlyDoubleQuote]The \
tile in coordinates (",
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propositional logic variables. The rules of Sudoku then enforce a bunch of \
logical clauses to express ",
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Furthermore, we must also remember to write ",
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giving its tiles any values at all, which would be within the rules  of the \
previous constraintsm but not in their spirit. A mindless program trying to \
solve a problem has no concept of what we finger-quotes \
\[OpenCurlyDoubleQuote]intended\[CloseCurlyDoubleQuote] when defining that \
problem, so the constraint clauses must really explicitly say everything that \
is allowed and not allowed. (An artificial intelligence may always find \
surprising lateral thinking solutions for any optimization problems that we \
give it.)\nWe start by defining the tiles as list of pairs of the form (x, \
y), followed by a utility function ",
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 "Two tiles are neighbours if they reside either in the same row, in the same \
column, or inside the same 3-by-3 square called a ",
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 "Unlike predicate logic and higher order logics all the way up to our \
natural language, propositional logic does not allow us to express ",
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computational problems are encoded as sets of propositional logic formulas, \
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variables, so we cannot just use the letters ",
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in the introductory logic textbooks. Instead, most satisfiability solvers \
simply number these propositions with positive integers 1, 2, 3, ...\nWith \
the fully symbolic nature of ",
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backwards to compute a pre-image that produces the result image when \
convolved forward with that same kernel. And solving sets of equations is the \
daily bread and butter of ",
 StyleBox["Mathematica",
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 ". Again, the kernels out to be squares of odd length and their weights \
normally should add up to one, but this is not a law of nature, and the \
reader might want to experiment with deconvolution using  kernels of \
different shapes and sum to experience the glitchy results that deconvolution \
can create, especially when combined with color space conversions followed by \
the use of different kernels for each separated component of that colour \
space. However, a balanced deconvolution whose weights sum up to one, as the \
simple 3-by-3 kernel used below, smooths out the image and makes it look more \
natural than the original lifeless computer-generated image."
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interactions.\nThe Conway\[CloseCurlyQuote]s Game of Life automaton proceeds \
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used in this context merely to mean the binary states 0 and 1. For \
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even though technically this is incorrect, since many initial states will \
eventually produce living tiles arbitrary far from the initial position. Same \
way as we did when solving Sudoku, we begin by defining a slew of \
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 "So far using propositional logic to compute the next step of Conway\
\[CloseCurlyQuote]s Game of Life has not been anything special, since unlike \
Sudoku that needs recursion to try out the possible values for each tile and ",
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more possible values remaining because its twenty neighbours have already \
used up all the values from 1 to 9, the evolution from the initial state of \
Conway\[CloseCurlyQuote]s Game of Life is rather trivial to compute forward \
any number of time steps with nested loop that loop over the time and the \
game board, which makes this problem easy enough to serve as an exercise in a \
first programming course. However, the propositional logic approach allows us \
to do something that would not be anywhere as trivial to solve using \
imperative programming; execute the rules ",
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were applied to it!\nIn all programming languages, computational statements \
are intentionally designed so that the state of the system at time ",
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 " + 1 is easy to compute from the given the state of the system at the \
previous time ",
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logic formulas that constrain the values of tiles between consecutive time \
steps are perfectly symmetric, so the previous satisfiability solver can be \
used, without any modification whatsoever to its structure, to solve the \
values of tiles at time ",
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of the game are perfectly deterministic in the forward direction so that the \
initial state of the game fully determines all of its future states, even one \
googolplex steps ahead unimaginably far in the future, the same determinism \
does not hold when time moves in the reverse directions. The exact same state \
of Conway\[CloseCurlyQuote]s Game of Life at time ",
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number of solutions, typically exponentially many. For example, any \
individual tile inside some 3-by-3 area that is empty at time ",
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tell which one of these result-equivalent states actually took place. On the \
other hand, it is also perfectly possible that no such previous state exists \
at all, because the current state is logically impossible to produce from any \
state using the rules of Conway\[CloseCurlyQuote]s Game of Life! (A trivial \
example of such an impossible state would be one whose every tile is alive.)\n\
This introduction of exponential nondeterminism is inherent and unavoidable \
in all universal computation any time (heh) that we attempt to reverse the \
arrow of time to find out which particular input would produce the desired \
output. Since Conway\[CloseCurlyQuote]s Game of Life is computationally \
universal and Turing-complete, the same phenomenon will inevitably occur in \
our attempt to reverse that game. If all computations were mechanistically \
reversible, any encryption algorithm could be executed in reverse to work as \
its own decryption algorithm. More generally, any algorithm that checks \
whether the offered solution of some problem is a valid one could be run \
backwards to work as an algorithm that finds a solution to that particular \
problem! Countless other similar goodies, equivalents of having a Christmas \
every day from the computational point of view, would rain on humanity like \
endless manna from heavens.\nThe state of each tile at time ",
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neighbourhood. The same simplicity does not hold in backwards direction. \
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time ",
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every tile indirectly depends on every other tile at the same time! The \
satisfiability solver then has an exponentially harder row to hoe when it \
tries to build up a working global combination of tile values that \
simultaneously satisfies the given constraints. When time flow forward, the \
value of each time can be computed in complete isolation from the values of \
the tiles for the rest of the board, which allows this task in principle to \
be parallelized without limit so that each tile could be computed with a \
separate processor for blazingly fast generation of future states.\nHowever, \
the logic of setting up the problem using our previous functions is exactly \
the same backwards as it was forward. The only difference is that the clauses \
now define how the known current state at time 1 depends on the unknown \
previous state at time 0, instead of how the unknown next state at time 2 \
deterministically depends on the known current state at time 1. "
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