- Please DO NOT use any AI tools like Chat-GPT to generate codes/answers as that'd a violation.

1. Resolution strategies a. (5) Is the following refutation of 1-5 - a unit proof? - a unit preference proof? - a set of support proof? (say why (not)) - an ordered proof? (say why (not)) 1. B(x,y) C(x) 2. B(x,y) C(y) 3. B(D,E) 4. C(F) 5. B(r,s) B(r,u) Denial of conclusion Refutation: r[3,5a] 6. B(D,u) r[6,2a] 7. C(y) r[7,4] 8. [] b. (5) answer the same questions for the refutation of 1-6 below. (Using a unit clause of form = for paramodulating also is a possible step in a unit proof or unit preference proof.)

1. L(R,K) 2. L(R,y) K=y 3. C(K) 4. L(x,y) L(y,x) 5. L(J,R) 6. C(x) Denial of conclusion

Refutation: r[4a,5] 7. L(R,J) r[7,2a] 8. K=J p[8L,3] 9. C(J) r[6,9) 10. []

3. (6) A frequent paramodulation error students make is to do equality substitutions *within* the clause containing the equality (while at the same time making the substitution in the other clause). The *only* substitutions we make in the clause containing the equality are those dictated by the *unification* of one side of the equality with a term in the other clause. Here's practice in doing it correctly: Find all paramodulants of P(f(x)) A=f(x), (f(g(B))=A) P(f(B))

4. (A simplified example of planning using resolution & Green's method) We use At(x,y,s) to express that x is at y *in situation* s; and similarly In(x,y,s) to express that x is in y *in situation* s. Constant B will refer to Bob, C to Bob's car, W to his work location, and S is our name for a certain situation in Bob's world. You can think of a "situation" as the state of the world at a brief moment in time; actions (represented as functions on situations) change that situation to a new one at a later moment. The 'get-in-car' function represents the action of Bob getting in his car, and this maps any given situation s to a new one, namely, get-in-car(s). Similarly the 'drive-to-work' function represents the action of Bob driving to work, which changes the starting situation to one where Bob is at work (W).

a. (6) Given: At(B,C,S), s.At(B,C,s) => In(B,C,get-in-car(s)) s.In(B,C,s) => At(B,W,drive-to-work(s))

Use resolution to prove that there is a situation in which Bob is at work. b. (6) Use the proof in (a) to answer the question "In what situation is Bob at work?", by attaching Cordell Green's answer literal and carrying it through the proof, with the appropriate substitutions.

c. (3) Ituitively interpret the answer obtained, keeping in mind the explanations above of what the functions mean. 5. a. (6) The simplest example of Natural Logic (NLog) inference that we saw in class was "Every dog barks" |- "Every poodle makes noise". Provide a quick resolution derivation of the conclusion *without* assuming falsity of that conclusion, i.e., a forward proof. (Though resolution isn't complete for forward inference, it's still pretty powerful.) You may assume any facts that are used in making the NLog inference.

b. (6) An NLog inference not readily expressible in standard FOL is this: "Governor Greg prevented a million would-be voters from casting ballots" |- "A governor prevented many would-be voters from voting"; |- "Many would-be voters did not vote". Outline the properties of "prevent", both in terms of what it implicates and the polarity of its complement, that enable these inferences; also mention any generalization/specialization relations that need to be used (Careful: The polarity properties here are different from what you might guess at first sight.)

c. (3) Represent "Many voters did not vote" in Episodic Logic (EL). Do this by analogy with the examples in the slides. Include a constant for the episode of many voters not voting.

6. Specialist inference about time ``````````````````````````````` If you look at the "Time Specialist" page in the "Specialist Inference" notes/slides, you see the numbering from 1 to 6 on the central (blue) time chain. Suppose now that we want to add a new time-point node into the chain, located after node 4 but before node 5 (in the non-strict sense of "before" and "after", i.e., allowing for equality). a. (4) Suggest a simple *constant-time* way to make such insertions, without losing the benefit of constant-time lookup of before/after relations on the chain. b. (4) Can this method be used for an unlimited number of insertions, or is there a limitation due to the realities of information storage in a computer -- other than that the total memory capacity of any computer is finite?

7. Specialist reasoning about 3-D models ````````````````````````````````````` People have powerful abilities for visualizing 3-D objects in space (and in motion!), which general AI systems will need as well. Let's consider a very simple case, that of the relationship between two rect- angular blocks A, B (prisms with 6 rectangular faces); for simplicity we assume that both have their edges aligned with the three Cartesian coordinate directions. Thus we can describe the blocks themselves and their locations in space by specifying the coordinate intervals each of them occupies in 3 dimensions. For example, one of them, A, might occuply the intervals ((X1,X2), (Y1,Y2), (Z1,Z2)) and the other, B, ((X1',X2'), (Y1',Y2'), (Z1',Z2')). a. (4) Specify the conditions on these parameters for the blocks to be "separated", i.e., not intersecting or touching. b. (4) Specify the conditions on these parameters for the blocks to be "touching", in the sense of sharing a finite area of some face (while the blocks are otherwise nonintersecting). c. (4) Now suppose that the second "block", B, is actually a box (with walls so thin that we can neglect their thickness). Specify the conditions for the first block, A, to *fit* inside the box B, and the conditions for it to *be* inside the box (possibly "floating" inside it). d. (4) Make up 2 clauses each of which contains predicates such as Touching, Separated, or Inside, where you can obtain a resolvent (in the generalized sense) based on clashing predicates; one of the clashing predicates should contain a variable which also occurs in another literal of that clause. (NB: This would be an efficient reasoning method, since the clash-detection is constant-time.)

8. Specialist reasoning about colors ````````````````````````````````` (Here a preamble is in order, but the problems are easy.) Have a look at the color specialist in the "Specialist Inference" notes/slides. The colored figure is far from perfectly shaded -- the top of the cylinder should also be colored, the transitions should be gradual, etc. You can imagine creating this color space by mixing paints, except that instead of viscous paint, you're mixing sand-like tiny pellets, which come in the following colors: - all the rainbow hues from red to orange to yellow to green to blue to purple (actually, an artist could do with just 3, namely red, yellow, and blue, mixing other hues from these); they are arrayed circumferentially on the cylinder, with the purest, undiluted hues showing at the topmost circumference; - black; - white. Then when most of the pellets are black, you have the subcylinder labeled "black" at the top center. When most of the pellets are white, you have the "white" disk comprising the bottom of the cylinder. The proportion of black pellets decreases as you move radially outward from the cylinder axis, and the proportion of white pellets decreases as you move straight upward .

The intriguing aspect of this is that it provides a very simple "subjective geometry" of our color perception. As noted in class, there are 11 main English color terms that we use to "carve up" the subjective color space (viz., red, orange, yellow, green, blue, purple, brown, pink, gray, black, and white; it's similar in other languages, though some lump together two "adjacent" colors, such as blue and green, into a single color term). In the cylindrical color space, each term corresponds simply to a coordinate-bounded region, much like the blocks in question 3! Let's assume the ranges of the 3 dimensions are hue: 0- 360 (starting where "red" begins and ending after a full circuit where "purple" ends; i.e., where reddish purple transitions into purplish red); dilution: 0 - 1 (starting at the top of the cylinder and reaching 1 at the bottom, where the cylinder has a final "white" disk); purity: 0 - 1 (starting at the axis of the cylinder and extending radially to the cylinder surface, where the value is 1) So, for example, we might have these characterizations of "blue" and of "black": blue: hue: 240 - 300 black: hue: 0 - 360 dilution: 0 - 0.8 dilution: 0 - 0.5 purity: 0.3 - 1 purity: 0 - 0.3

Note that some pairs of colors could be considered "strictly incompatible", in the sense that you can't get an "in-between" color; e.g., you can't get a reddish green (a uniform color that is both "sort of red" and "sort of green"), or a bluish orange, or a yellowish purple, or a blackish yellow, etc. Others are weakly incompatible; e.g., while you wouldn't consider an object you judge to be "clearly orange" as also being red, you can well imagine an object that is reddish-orange (i.e., sort of red, and also sort of orange), i.e., in the immediate vicinity of the surface where red transitions to orange.

a. (4) Using the color cylinder, suggest (in mathematical form) how we could determine predicate incompatibility (and hence "color-predicate resolution") in constant time for strictly incompatible predicates such as Red vs Green, Orange vs Blue, Pink vs Yellow, Blue vs Brown, etc.; b. (4) Give an example of generalized resolution based on such predicates, using 2 clauses where at least one has 2 literals, and one of the clashing color literals has a variable argument. c. (4) Of course we know many color terms other than the 11 basic ones, such as "beige", "maroon", "lime-green", etc. It turns out that these also correspond to simple coordinate-bounded regions in the cylindrical space, in some cases intersecting two basic colors (e.g., chartreuse), in others lying within the bounds of a basic color (e.g., burgundy -- a red in the upper, darker (further inward) portion of the red region). The latter case gives rise to further "weak" incompatibilities, dependent on negation. Give an example, and say why the incompatibility seems weak.

- Please answer all the questions/parts as all are part of one single assignment and I cannot ask them in parts.