Theories.Axioms,Rules of Inference
... What do axioms do for us? That is where a logic comes in, with rules of inference, which allow us to derive theorems from axioms and other theorems. This is the alternate characterization of theorems, instead of saying a theorem is a valid(true in all possible assignments to free variables) formula ...
... What do axioms do for us? That is where a logic comes in, with rules of inference, which allow us to derive theorems from axioms and other theorems. This is the alternate characterization of theorems, instead of saying a theorem is a valid(true in all possible assignments to free variables) formula ...
Homework 2
... i.e., Yt < K for some K > 0 and all t, and that Yt → Xτ a.s. as t → ∞. (c) Show that it follows that E(Xτ ) = 1 (this is almost the optional stopping theorem, except that we have not required that τ < ∞!) The rest is easy. What is the mean and variance of τ ? (You don’t have to give a rigorous argum ...
... i.e., Yt < K for some K > 0 and all t, and that Yt → Xτ a.s. as t → ∞. (c) Show that it follows that E(Xτ ) = 1 (this is almost the optional stopping theorem, except that we have not required that τ < ∞!) The rest is easy. What is the mean and variance of τ ? (You don’t have to give a rigorous argum ...
And this is just one theorem prover!
... Output: meaning of the answer • If the theorem prover says “yes” to a formula, what does that tell us? – Soudness: theorem prover says yes implies formula is correct – Subject to bugs in the Trusted Computing Base (TCB) – Broad defn of TCB: part the system that must be correct in order to ensure th ...
... Output: meaning of the answer • If the theorem prover says “yes” to a formula, what does that tell us? – Soudness: theorem prover says yes implies formula is correct – Subject to bugs in the Trusted Computing Base (TCB) – Broad defn of TCB: part the system that must be correct in order to ensure th ...
And this is just one theorem prover!
... Output: meaning of the answer • If the theorem prover says “yes” to a formula, what does that tell us? – Soudness: theorem prover says yes implies formula is correct – Subject to bugs in the Trusted Computing Base (TCB) – Broad defn of TCB: part the system that must be correct in order to ensure th ...
... Output: meaning of the answer • If the theorem prover says “yes” to a formula, what does that tell us? – Soudness: theorem prover says yes implies formula is correct – Subject to bugs in the Trusted Computing Base (TCB) – Broad defn of TCB: part the system that must be correct in order to ensure th ...
Ramsey`s theorem for colors from a metric space
... compact but not metric, and where the statement of the theorem fails. Let K be any compact space where not every sequence has a convergent subsequence. For instance, K can be an uncountable product of unit intervals. Take some sequence a1 , a2 , . . . in K which does not have any convergent subseque ...
... compact but not metric, and where the statement of the theorem fails. Let K be any compact space where not every sequence has a convergent subsequence. For instance, K can be an uncountable product of unit intervals. Take some sequence a1 , a2 , . . . in K which does not have any convergent subseque ...
The Coase theorem states that private parties can find efficient
... Imagine a farm and a ranch next to each other. The rancher's cows occasionally wander over to the farm and damage the farmer's crops. The farmer has an incentive to bargain with the rancher to find a more efficient solution. If it is more efficient to prevent cattle trampling a farmer's field by fe ...
... Imagine a farm and a ranch next to each other. The rancher's cows occasionally wander over to the farm and damage the farmer's crops. The farmer has an incentive to bargain with the rancher to find a more efficient solution. If it is more efficient to prevent cattle trampling a farmer's field by fe ...
remarks on the topologies for minkowski space-time
... (1) The causal order can be deduced from the considered topology, and (2) The group of homeomorphisms of the considered topology is generated by the inhomogeneous Lorentz group and dilations. Some examples of such topologies are constructed in [6], [3] and [4], using some particular properties of R ...
... (1) The causal order can be deduced from the considered topology, and (2) The group of homeomorphisms of the considered topology is generated by the inhomogeneous Lorentz group and dilations. Some examples of such topologies are constructed in [6], [3] and [4], using some particular properties of R ...
2010-10
... • Do the data provide evidence of discrimination? • Alternative explanations based on classical economics • Additional variables: percent unemployed in the subject percent non-academic jobs in the subject median non-academic salary in the subject • Which model(s) are most useful ? ...
... • Do the data provide evidence of discrimination? • Alternative explanations based on classical economics • Additional variables: percent unemployed in the subject percent non-academic jobs in the subject median non-academic salary in the subject • Which model(s) are most useful ? ...
Rewriting Predicate Logic Statements
... We now play with Arrow’s Impossibility Theorem because it’s a fascinating proof. But.. Poundstone would remind us that there are systems not subject to this theorem! ...
... We now play with Arrow’s Impossibility Theorem because it’s a fascinating proof. But.. Poundstone would remind us that there are systems not subject to this theorem! ...
On the Uniqueness of the Decomposition of Manifolds, Polyhedra
... Let M3, N3 be connected oriented Seifert fibred 3-manifolds. If M3 M3 ≈ N3 N3 then M3 ≈ N3 unless M3 and N3 are lens spaces with isomorphic fundamental groups. ...
... Let M3, N3 be connected oriented Seifert fibred 3-manifolds. If M3 M3 ≈ N3 N3 then M3 ≈ N3 unless M3 and N3 are lens spaces with isomorphic fundamental groups. ...
Chapter 1
... assuming a fixed number of countries & equal population size Purpose of MD social choice: find attractive rule to judge whether one situation X is better or worse than another, say Y But what is attractive? introduce axioms: create simple imaginary situations X and Y in which it is (relatively) ...
... assuming a fixed number of countries & equal population size Purpose of MD social choice: find attractive rule to judge whether one situation X is better or worse than another, say Y But what is attractive? introduce axioms: create simple imaginary situations X and Y in which it is (relatively) ...
How to Solve an Equation: a topological approach
... • There are points where f ′ (x) = 0. First Condition: The map f must not have any Critical Points. ...
... • There are points where f ′ (x) = 0. First Condition: The map f must not have any Critical Points. ...
Mechanized foundations of finite group theory
... Four Color theorem [5]. This considerable formal development was joined with the drafting of a syntactic extension to the Coq theorem prover, aimed at making the proof-writing process shorter, that our work expands and refines. On the theoretic level, we also develop formal building blocks for finit ...
... Four Color theorem [5]. This considerable formal development was joined with the drafting of a syntactic extension to the Coq theorem prover, aimed at making the proof-writing process shorter, that our work expands and refines. On the theoretic level, we also develop formal building blocks for finit ...
A + B + C
... In POS standard form, every variable in the domain must appear in each sum term of the expression. You can expand a nonstandard POS expression to standard form by adding the product of the missing variable and its complement and applying rule 12, which states that (A + B)(A + C) = A + BC. Convert X ...
... In POS standard form, every variable in the domain must appear in each sum term of the expression. You can expand a nonstandard POS expression to standard form by adding the product of the missing variable and its complement and applying rule 12, which states that (A + B)(A + C) = A + BC. Convert X ...
Central Limit Theorem
... X1 , X2 , ..., Xn (not necessarily binary valued), as n → ∞, we have Zn → Z in the sense that ∀u ∈ R, Pr[Zn ≤ u] → Pr[Z ≤ u]. More specifically, for each > 0, there exists N ∈ N so that for every n > N and every u ∈ R, we have |Pr[Zn ≤ u] − Pr[Z ≤ u]| < . Definition 2. We use Z ∼ N (0, 1) to deno ...
... X1 , X2 , ..., Xn (not necessarily binary valued), as n → ∞, we have Zn → Z in the sense that ∀u ∈ R, Pr[Zn ≤ u] → Pr[Z ≤ u]. More specifically, for each > 0, there exists N ∈ N so that for every n > N and every u ∈ R, we have |Pr[Zn ≤ u] − Pr[Z ≤ u]| < . Definition 2. We use Z ∼ N (0, 1) to deno ...
Factoring Out the Impossibility of Logical Aggregation
... the results obtained is a striking impossibility theorem that abstractly generalizes the doctrinal paradox (Pauly and van Hees, 2006; see also Dietrich, 2006). This theorem states that a mapping de…ned on a universal domain is dictatorial - i.e., collapses the social judgment set into the set of a g ...
... the results obtained is a striking impossibility theorem that abstractly generalizes the doctrinal paradox (Pauly and van Hees, 2006; see also Dietrich, 2006). This theorem states that a mapping de…ned on a universal domain is dictatorial - i.e., collapses the social judgment set into the set of a g ...
Resources - CSE, IIT Bombay
... Case-2: B is An AnAn is a theorem (already proved) One is allowed to write A1, A2, A3,… An-1 |- (AnAn) i.e. |- (AnB) ...
... Case-2: B is An AnAn is a theorem (already proved) One is allowed to write A1, A2, A3,… An-1 |- (AnAn) i.e. |- (AnB) ...
Reinforcement Learning Leads to Risk Averse Behavior
... 1) In each period, the learner must choose one of N alternatives, each with a normally distributed reward, ri ,t . ...
... 1) In each period, the learner must choose one of N alternatives, each with a normally distributed reward, ri ,t . ...
Appendix 1: Utility Theory Much of the theory presented is based on
... developed in the book of Von Neumann and Morgenstern (1947). Further developments are given in Savage (1954), Blackwell and Girshick (1954) and Luce and Raiffa (1957). More recent descriptions may be found in Owen (1982) and Shubik (1984), and a more complete exposition of the theory may be found in ...
... developed in the book of Von Neumann and Morgenstern (1947). Further developments are given in Savage (1954), Blackwell and Girshick (1954) and Luce and Raiffa (1957). More recent descriptions may be found in Owen (1982) and Shubik (1984), and a more complete exposition of the theory may be found in ...
PDF
... In this entry, we will prove the substitution theorem for propositional logic based on the axiom system found here. Besides the deduction theorem, below are some additional results we will need to prove the theorem: 1. If ∆ ` A → B and Γ ` B → C, then ∆, Γ ` A → C. 2. ∆ ` A and ∆ ` B iff ∆ ` A ∧ B. ...
... In this entry, we will prove the substitution theorem for propositional logic based on the axiom system found here. Besides the deduction theorem, below are some additional results we will need to prove the theorem: 1. If ∆ ` A → B and Γ ` B → C, then ∆, Γ ` A → C. 2. ∆ ` A and ∆ ` B iff ∆ ` A ∧ B. ...
Document
... To illustrate these facts, consider three prizes z0 , z1 , and z2, where z2 ⊱ z1 ⊱ z0 . A lottery p can be depicted on a plane by taking p (z1) as the first coordinate (on the horizontal axis), and p (z2) as the second coordinate (on the vertical axis). p (z0) is 1 – p (z1) – p (z2). [See Figure 4 ...
... To illustrate these facts, consider three prizes z0 , z1 , and z2, where z2 ⊱ z1 ⊱ z0 . A lottery p can be depicted on a plane by taking p (z1) as the first coordinate (on the horizontal axis), and p (z2) as the second coordinate (on the vertical axis). p (z0) is 1 – p (z1) – p (z2). [See Figure 4 ...