1. Here is a model M: Domain: {1, 2, 3, 4, 5} P : {1, 3, 5}, Q : {2, 4, 5}, R : ∅, S : {3, 4} a : 3, b : 4 Is the proposition ∀x((P x ∧ Qx) → Sx) ∨ (Rb ↔ P a) true or false in M? Explain.
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1. Here is a model M:
Domain: {1, 2, 3, 4, 5}
P : {1, 3, 5}, Q : {2, 4, 5}, R : ∅, S : {3, 4}
a : 3, b : 4
Is the proposition ∀x((P x ∧ Qx) → Sx) ∨ (Rb ↔ P a) true or false in M? Explain.
Step by step
Solved in 2 steps
- Consider a fully-connected artificial neural network with one hidden layer, i.e., a multilayer perceptron (MLP), which has 5 inputs, 3 neurons in the hidden layer, and 1 output neuron. The relation between the output y and the inputs x = [x1, . . . , x5] is given by y(x) = f (w, φ(x)), where φ(x) = [φ1(x), φ2(x), φ3(x)] 1. Draw the diagram that shows the inputs, nuerons, connections, correspond-ing weight parameters, and activation functions. 2. Explain the relation y(x) = f (w, φ(x)): write the explicit relation, explainthe role of functions f and φ(x), and state examples of functions.Consider a Diffie-Hellman scheme with a common prime q = 17 and a primitive root α = 3. a) If user A has a private key XA=4, what is A’s public key, YA? b) A sends YA to B. If B has a private key XB=6, what is the shared secret key, K that B can calculate and share with A? c) If B computes YB and sends it to A, what is the shared secret Key, K computed by A?In this question, we will explore the semantic properties of propositional Horn clauses. For any set of clauses S, define Is to be the interpretation that satisfies an atom p if and only if S = p. Show that if S is a set of positive Horn clauses, then Is |= S. Give an example of a set of clauses S where Is \|= S. Suppose that S is a set of positive Horn clauses and that c is a negative Horn clause. Show that if Is \|= c then SU{c} is unsatisfiable. Suppose that S is a set of positive Horn clauses and that T is a set of negative ones. Using part (c), show that if SU{c} is satisfiable for every c E T, then SUT is satisfiable also. In the propositional case, the normal Prolog interpreter can be thought of as taking a set of positive Horn clauses S (the program) and a single negative clause c (the query) and determining whether or not SU{c} is satisfiable. Use part (d) to conclude that Prolog can be used to test the satisfiability of an arbitrary set of Horn Clauses.
- Consider the following knowledge base in first order logic: p(X) ← q(X) ∧ r(X, Y ) q(X) ← s(X) ∧ r(Y, Y ) s(a) r(a, a) What is the result of the query p(a) using a backward chaining inference algorithm? Show the intermediary queries.Consider the following knowledge base Prove that Q is true with: 1. P → Q 2. L ∧ M → P 3. B ∧ L → M 4. A ∧ P → L 5. A ∧ B → L 6. A 7. B Forward-Chaining Backward-Chaining Resolution Prove t → s: 1. p → q 2. [q ∧ r] → s 3. [t ∧ u] → r 4. u → w 5. t → y 6. y → u 7. r → p 8. p → m i. Express in clause form ii. Forward-Chaining iii. Backward-Chaining iv. ResolutionGiven a set of n positive integers, C = {c1,c2, ..., cn} and a positive integer K, is there a subset of C whose elements sum to K? A dynamic program for solving this problem uses a 2-dimensional Boolean table T, with n rows and k + 1 columns. T[i,j] 1≤ i ≤ n, 0 ≤ j ≤ K, is TRUE if and only if there is a subset of C = {c1,c2, ..., ci} whose elements sum to j. Which of the following is valid for 2 ≤ i ≤ n, ci ≤ j ≤ K? a) T[i, j] = ( T[i − 1, j] or T[i, j − ci]) b) T[i, j] = ( T[i − 1, j] and T[i, j − ci ]) c) T[i, j] = ( T[i − 1, j] or T[i − 1, j − ci ]) d) T[i, j] = ( T[i − 1, j] and T[i − 1, j − cj ]) In the above problem, which entry of the table T, if TRUE, implies that there is a subset whose elements sum to K? a) T[1, K + 1] b) T[n, K] c) T[n, 0] d) T[n, K + 1]
- Given a set of n positive integers, C = {c1,c2, ..., cn} and a positive integer K, is there a subset of C whose elements sum to K? A dynamic program for solving this problem uses a 2-dimensional Boolean table T, with n rows and k + 1 columns. T[i,j] 1≤ i ≤ n, 0 ≤ j ≤ K, is TRUE if and only if there is a subset of C = {c1,c2, ..., ci} whose elements sum to j. Which of the following is valid for 2 ≤ i ≤ n, ci ≤ j ≤ K? a) ?[?, ?] = ( ?[? − 1, ?] ?? ?[?, ? − ?? ]) b) ?[?, ?] = ( ?[? − 1, ?] ??? ?[?, ? − ?? ]) c) ?[?, ?] = ( ?[? − 1, ?] ?? ?[? − 1, ? − ?? ]) d) ?[?, ?] = ( ?[? − 1, ?] ??? ?[? − 1, ? − ?? ]) In the above problem, which entry of the table T, if TRUE, implies that there is a subset whose elements sum to K? a) ?[1, ? + 1] b) ?[?, ?] c) ?[?, 0] d) ?[?, ? + 1]For each pair of atomic sentences, give the most general unifier if it exists: 1. P(N, M, z), P(x, y, N). 2. Q(x, y, M), Q(N, M, x). 3. Knows(y, y), Knows(Father(x), x).link(a,b). link(a,c). link(b,c). link(b,d). link(c,d). link(d,e). link(d,f). link(e,f). link(f,g). Using The above Formulate the appropriate Prolog predicate "path(X,Y,N)" which is true if (and only if) there is a path of length "N" from node "X" to node "Y". For example, there is a path of length 2 from "a" to "d": "a->b->d", but also "a->c->d", and so "path(a,d,2)" gives "true" (two times). There is also a path of length 3 from "a" to "d": "a->b->c->d". Test this predicate out on the above network to verify whether or not it is working correctly. Once this is working correctly, note now, that e.g., "path(a,e,N)." will give multiple answers:
- Consider the wffs:φ1 ≡ p1 → (p2 → (p3 → p4))φ2 ≡ (p1 ∧ p2 ∧ p3) → p4(a) Technically speaking, neither φ1 nor φ2 is well-formed since neither is allowed by the formal syntaxof propositional logic. Correct them. Note, however, that we will freely make such trivial ’errors’ throughout this semester (as do most such courses).(b) Use truth tables (in the form defined in this course) to show that φ1 ↔ φ2.(c) After internalizing an intuitive understanding of this equality, propose an extension of it to n atoms.(d) State the number of rows in a truth table for proving the extensionConsider the wffs:φ1 ≡ p1 → (p2 → (p3 → p4))φ2 ≡ (p1 ∧ p2 ∧ p3) → p4(a) Technically speaking, neither φ1 nor φ2 is well-formed since neither is allowed by the formal syntaxof propositional logic. Correct them. Note, however, that we will freely make such trivial ’errors’throughout this semester (as do most such courses).(b) Use truth tables (in the form defined in this course) to show that φ1 ↔ φ2.(c) After internalizing an intuitive understanding of this equality, propose an extension of it to natoms.(d) State the number of rows in a truth table for proving the extension.Suppose propositional sentences and knowledge-base here. We have a knowledge-base called KB. KBPrime is defined as an union of (1) KB and (2) the negation of a propositional predicate P. Answer true/false to the following questions. Note that "|=" is used to denote entailment, and "|-" is used to denote derivation through resolution. a) If KB |= P, then KB |- P. That is, if KB entails P, then KB derives (using resolution) P. b) If KB |- P, then KB |= P. c) If KBPrime |- [], then KBPrime is not satisfiable. d) If KBPrime is not satisfiable, then KBPrime |- [].