CS 399: Constructive Logic Final Exam (Sample Solution) Name Instructions
... Unequality: We will say x and y are unequal if ¬(x =N y). In the next two problems you will show that x 6= y if and only if x and y are unequal. The proof terms for =N are given on page 57 of the Pfenning notes. (B) Prove that ∀x ∈ nat. ∀y ∈ nat.¬(x =N y) ⊃ x 6= y. Give your ...
... Unequality: We will say x and y are unequal if ¬(x =N y). In the next two problems you will show that x 6= y if and only if x and y are unequal. The proof terms for =N are given on page 57 of the Pfenning notes. (B) Prove that ∀x ∈ nat. ∀y ∈ nat.¬(x =N y) ⊃ x 6= y. Give your ...
Section.8.3
... Example (functions). ƒ(x) = y iff ƒ(x, y) is true iff (x, y) ƒ. So a higher-order logic allows sets to be quantified or to be elements of other sets. Classifying Logics by Order The order of a predicate is 1 if its arguments are terms. Otherwise the order is n + 1 where n is the maximum order of t ...
... Example (functions). ƒ(x) = y iff ƒ(x, y) is true iff (x, y) ƒ. So a higher-order logic allows sets to be quantified or to be elements of other sets. Classifying Logics by Order The order of a predicate is 1 if its arguments are terms. Otherwise the order is n + 1 where n is the maximum order of t ...
L-spaces and the P
... (T, S)-preserving if the following holds: T is an Aronszajn tree, S ⊆ ω1 , and for every λ > (2|P |+ℵ1 )+ and countable N ≺ H(λ, ∈) such that P, T, S ∈ N and δ = N ∩ ω1 6∈ S, and every p ∈ N ∩ P there is some q ≥ p (bigger conditions are stronger) such that (1) q is (N, P ) generic; and (2) for ever ...
... (T, S)-preserving if the following holds: T is an Aronszajn tree, S ⊆ ω1 , and for every λ > (2|P |+ℵ1 )+ and countable N ≺ H(λ, ∈) such that P, T, S ∈ N and δ = N ∩ ω1 6∈ S, and every p ∈ N ∩ P there is some q ≥ p (bigger conditions are stronger) such that (1) q is (N, P ) generic; and (2) for ever ...
MATHEMATICAL NOTIONS AND TERMINOLOGY
... What makes some problems computationally hard and others easy? • First, by understanding which aspect of the problem is at the root of the difficulty. • Second, you may be able to settle for less than a perfect solution to the problem. • Third, some problems are hard only in the worst case situation ...
... What makes some problems computationally hard and others easy? • First, by understanding which aspect of the problem is at the root of the difficulty. • Second, you may be able to settle for less than a perfect solution to the problem. • Third, some problems are hard only in the worst case situation ...
Modalities in the Realm of Questions: Axiomatizing Inquisitive
... Building on ideas from inquisitive semantics, the recently proposed framework of inquisitive epistemic logic (IEL) provides the tools to model and reason about scenarios in which agents do not only have information, but also entertain issues. This framework has been shown to allow for a generalizati ...
... Building on ideas from inquisitive semantics, the recently proposed framework of inquisitive epistemic logic (IEL) provides the tools to model and reason about scenarios in which agents do not only have information, but also entertain issues. This framework has been shown to allow for a generalizati ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... In program verification, an interpolating theorem prover often interacts tightly with various decision procedures. It is therefore advantageous for the interpolants computed by the prover to be expressible in simple logic fragments. Unfortunately, interpolation procedures for expressive first-order fr ...
... In program verification, an interpolating theorem prover often interacts tightly with various decision procedures. It is therefore advantageous for the interpolants computed by the prover to be expressible in simple logic fragments. Unfortunately, interpolation procedures for expressive first-order fr ...
Unit-1-B - WordPress.com
... Mathematical Reasoning We need mathematical reasoning to determine whether a mathematical argument is correct or incorrect Mathematical reasoning is important for artificial intelligence systems to reach a conclusion from knowledge and facts. We can use a proof to demonstrate that a particular stat ...
... Mathematical Reasoning We need mathematical reasoning to determine whether a mathematical argument is correct or incorrect Mathematical reasoning is important for artificial intelligence systems to reach a conclusion from knowledge and facts. We can use a proof to demonstrate that a particular stat ...
On The Expressive Power of Three-Valued and Four
... consequence relation are de ned on FOUR as in every many-valued logic. The algebraic structure of FOUR has been generalized by Ginsberg [Gi88] to the general concept of a bilattice. He proposed Bilattices as a basis for a general framework for many applications. Bilattices were further investigated ...
... consequence relation are de ned on FOUR as in every many-valued logic. The algebraic structure of FOUR has been generalized by Ginsberg [Gi88] to the general concept of a bilattice. He proposed Bilattices as a basis for a general framework for many applications. Bilattices were further investigated ...
First-order possibility models and finitary
... FIRST-ORDER POSSIBILITY MODELS AND FINITARY COMPLETENESS PROOFS ...
... FIRST-ORDER POSSIBILITY MODELS AND FINITARY COMPLETENESS PROOFS ...
Probability Captures the Logic of Scientific
... Turning now to the choice of λ, Carnap (1980, pp. 111–119) considered the rate of learning from experience that different values of λ induce and came to the conclusion that λ should be about 1 or 2. Another consideration is this: So far as we know a priori, the statistical probability (roughly, long ...
... Turning now to the choice of λ, Carnap (1980, pp. 111–119) considered the rate of learning from experience that different values of λ induce and came to the conclusion that λ should be about 1 or 2. Another consideration is this: So far as we know a priori, the statistical probability (roughly, long ...
BEYOND ω-REGULAR LANGUAGES The notion of ω
... A deterministic max-automaton is defined like a BS-automaton, with the following differences: a) it is deterministic; b) it has an additional counter operation c := max(d, e); and c) its acceptance condition is a boolean (not necessarily positive) combination of Bconditions. The max operation looks ...
... A deterministic max-automaton is defined like a BS-automaton, with the following differences: a) it is deterministic; b) it has an additional counter operation c := max(d, e); and c) its acceptance condition is a boolean (not necessarily positive) combination of Bconditions. The max operation looks ...
The Complete Proof Theory of Hybrid Systems
... The suitability of an axiomatization can still be established by showing completeness relative to a fragment [Coo78, HMP77]. This relative completeness, in which we assume we were able to prove valid formulas in a fragment and prove that we can then prove all others, also tells us how subproblems ar ...
... The suitability of an axiomatization can still be established by showing completeness relative to a fragment [Coo78, HMP77]. This relative completeness, in which we assume we were able to prove valid formulas in a fragment and prove that we can then prove all others, also tells us how subproblems ar ...
God, the Devil, and Gödel
... well). However, in such a case what is usually alleged to have been disproved by Gödel is either that p or that q. It therefore requires not only the meta-mathematical result, but also considerable philosophical argument to establish the desired conclusion. I wish in this paper to examine in detail ...
... well). However, in such a case what is usually alleged to have been disproved by Gödel is either that p or that q. It therefore requires not only the meta-mathematical result, but also considerable philosophical argument to establish the desired conclusion. I wish in this paper to examine in detail ...
Lectures on Laws of Supply and Demand, Simple and Compound
... In the case above we will analyze it and show it is always true due to its structure.(You can see this for this simple example just by thinking about it.)In fact it is what is called in logic a tautology. We will let letters A, B or C represent single propositions and we will now investigate the tru ...
... In the case above we will analyze it and show it is always true due to its structure.(You can see this for this simple example just by thinking about it.)In fact it is what is called in logic a tautology. We will let letters A, B or C represent single propositions and we will now investigate the tru ...
CS 486: Applied Logic 8 Compactness (Lindenbaum`s Theorem)
... The key idea of Lindenbaum’s proof of this fact is to add formulas to the set S whenever they are consistent with the resulting set. Obviously this is an infinite process. But since there are only denumerably many formulas, we can give a recursive description of a method for construction a maximally ...
... The key idea of Lindenbaum’s proof of this fact is to add formulas to the set S whenever they are consistent with the resulting set. Obviously this is an infinite process. But since there are only denumerably many formulas, we can give a recursive description of a method for construction a maximally ...
