This topic contains 7 replies, has 1 voice, and was last updated by 
josh October 21, 2022 at 7:45 pm.
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joshFor example, consider bowls of soup – there are ranges of actions that work & ranges that fail.
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joshWhen I imagine solutions in my head, I think that the engineering implementations involve working with systems of multidimensional inequalities that include lots of symbolic variables. Picking a system that can do that well or building one would probably be part of my solution.
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joshThe way that computer algebra systems have been implemented so far is not very theoretical. Any given system offers a variety of “expression” types that include distinct sets of variables, and any given expression type may be “simplified” or possibly “solved” in some cases using tactical algorithms that are suited to that type. See, for example The expression tactics that Wolfram/Mathematica has implemented.
Can we make a tighter theoretical & software generalization?
Say expressions are either descriptions of quantities (with an implicit equality to an unknown), equalities or inequalities, and various tactics can be used to describe the potentially satisfying sets of variable assignments – i.e. is the generalized subspace null or this or that? What is the role of new proofs in expanding the library tactics? Available as “certified”? How about approximate & probabilistically approximate evaluations of the subspaces?
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joshA plan for a robot could be based around changes to the locations & states of some systems of objects. The timeline probably involves symbolic temporal variables for times and durations of transitions. Often the static states of repose & the dynamic states of transition are analyzed by different types of descriptions at different levels of details – consider, for example a step that involves “take engine X apart, lubricate key gears, and put it back together” prior to proceeding to use it in some way.
The desire to mix algebraic unknowns with “hard” mechanics and soft statistical generalizations seems motivated by real world practicalities and the need for hierarchical abstraction.
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joshWe can think about it in this: the transition from “Control Theory” and “Applied Math” to Industrial Control & Real World Design/Planning involves cobbling together models/constraints/partial descriptions of systems, organized around some set of goal objectives, using miscellaneous available knowledge, databases, & tables that may or may not be part of the solution boundaries. There is lots of value in creating software solutions that can compose, analyze, & work with “correct” and “smart” plans/control regimes while flagging incorrect and inefficient versions. These activities involve finding bridges between hard & soft descriptions of systems.
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joshIn general we want to say, regarding the product space of variables in this system:
Does the solution set exist or is it null?
If it exists, is the solution unique or a set of possibilities?
Either way, what is an example solution point & what is the simplest description of the feasible space of solutions?
What do we mean by simplest? This is a design concept, but probably there is a penalty for length & a penalty for using “complex” operators. So we may say that integral & differential operators, for example, have a high complexity penalty while equality & inequality operators in single variable equations have low complexity. Non-linear multiple variable inequality? Too complex,unles what we started out was worse. The quadratic formula is somewhat good, but not as great as just listing the solution set.-
joshIn many complex problems, the exact solution isn’t easy to compute with but there are approximate solutions to describing the feasible region that are in a happy form for further computation – e.g. lists of N-dim polyhedra. Again, we should be able to parameterize how we penalize approximation along with simplicity in the sense of good for compute engines vs. good for reading.
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