X - NUS School of Computing

... quantified statements. Some occur, through the presence of the word a or an. Others occur in cases where the general context of a sentence supplies part of its meaning. For example, in an algebra course in which the letter x is always used to indicate a real number, the predicate If x > 2 then x2 > ...

... quantified statements. Some occur, through the presence of the word a or an. Others occur in cases where the general context of a sentence supplies part of its meaning. For example, in an algebra course in which the letter x is always used to indicate a real number, the predicate If x > 2 then x2 > ...

... quantified statements. Some occur, through the presence of the word a or an. Others occur in cases where the general context of a sentence supplies part of its meaning. For example, in an algebra course in which the letter x is always used to indicate a real number, the predicate If x > 2 then x2 > ...

A Unified View of Induction Reasoning for First-Order Logic

... are distinguished, for which the elements of E are i) (vectors of) terms, and ii) (first-order) formulas. The first class includes the conventional induction methods, based on induction schemas. They explicitly define the induction hypotheses which are linked to the induction conclusions in order to ...

... are distinguished, for which the elements of E are i) (vectors of) terms, and ii) (first-order) formulas. The first class includes the conventional induction methods, based on induction schemas. They explicitly define the induction hypotheses which are linked to the induction conclusions in order to ...

Problems on Discrete Mathematics1 (Part I)

... correct. But, in general, we are not able to do so because the domain is usually an infinite set, and even worse, the domain can be uncountable, e.g., real numbers. To overcome this problem, we divide the domain into several categories and make sure that those categories cover the domain. Then we ex ...

... correct. But, in general, we are not able to do so because the domain is usually an infinite set, and even worse, the domain can be uncountable, e.g., real numbers. To overcome this problem, we divide the domain into several categories and make sure that those categories cover the domain. Then we ex ...

Deductive Databases with Universally Quantified Conditions

... a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. ...

... a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. ...

Carnap and Quine on the analytic-synthetic - Philsci

... Quine’s position are in fact quite close,1 and it becomes hard to understand why Quine’s victory has been so crushing. In this paper I will focus on one aspect of the debate, namely analyticity as truth by virtue of meaning, or rather truth by virtue of the rules of a chosen linguistic framework. Th ...

... Quine’s position are in fact quite close,1 and it becomes hard to understand why Quine’s victory has been so crushing. In this paper I will focus on one aspect of the debate, namely analyticity as truth by virtue of meaning, or rather truth by virtue of the rules of a chosen linguistic framework. Th ...

doc

... (a) Explain what it means to say that a sentence in the predicate calculus is semantically true. Also state what it means for a sequent to be semantically valid in predicate logic. A sentence in the predicate calculus is semantically true iff the sentence is true in very interpretation of the predic ...

... (a) Explain what it means to say that a sentence in the predicate calculus is semantically true. Also state what it means for a sequent to be semantically valid in predicate logic. A sentence in the predicate calculus is semantically true iff the sentence is true in very interpretation of the predic ...

Recursive Predicates And Quantifiers

... recursive schema that the given function symbols should not appear on the left will prevent any ambiguity being introduced by the interaction under Rl and R2 of equations in the combined system which were formerly in separate systems. Thus the definition can be considered as effected in a single gen ...

... recursive schema that the given function symbols should not appear on the left will prevent any ambiguity being introduced by the interaction under Rl and R2 of equations in the combined system which were formerly in separate systems. Thus the definition can be considered as effected in a single gen ...

Predicate Logic

... the set of real numbers: R False. 3 is a counterexample. the set of positive integers not exceeding 4: {1, 2, 3, 4} False. 3 is a counterexample. Also note that here ∀P (x) is P (1) ∧ P (2) ∧ P (3) ∧ P (4), so its enough to observe that P (3) is false. the set of real numbers in the interval [10, 39 ...

... the set of real numbers: R False. 3 is a counterexample. the set of positive integers not exceeding 4: {1, 2, 3, 4} False. 3 is a counterexample. Also note that here ∀P (x) is P (1) ∧ P (2) ∧ P (3) ∧ P (4), so its enough to observe that P (3) is false. the set of real numbers in the interval [10, 39 ...

Lecture 09

... Example: Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps. Solution: Let P(n) be the proposition that postage of n cents can be formed using 4-cent and 5-cent stamps. – BASIS STEP: P(12), P(13), P(14), and P(15) hold. • P(12) uses three 4-cent ...

... Example: Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps. Solution: Let P(n) be the proposition that postage of n cents can be formed using 4-cent and 5-cent stamps. – BASIS STEP: P(12), P(13), P(14), and P(15) hold. • P(12) uses three 4-cent ...

Principle of Mathematical Induction

... used to prove that a proposition is true for all natural numbers 1,2,3,… , although there are many variations of the basic method. The method is particularly important in discrete mathematics, and one often sees theorems proven by induction in areas like computer science. The technique is so intuiti ...

... used to prove that a proposition is true for all natural numbers 1,2,3,… , although there are many variations of the basic method. The method is particularly important in discrete mathematics, and one often sees theorems proven by induction in areas like computer science. The technique is so intuiti ...

A Conditional Logical Framework *

... Conditional Logical Framework LFK is the same exploited in [HLL07] for the General Logical Framework GLF. However, there is an important difference between the two frameworks in the definition of predicates. On one hand, predicates in [HLL07] are used both to determine whether β-reduction fires and ...

... Conditional Logical Framework LFK is the same exploited in [HLL07] for the General Logical Framework GLF. However, there is an important difference between the two frameworks in the definition of predicates. On one hand, predicates in [HLL07] are used both to determine whether β-reduction fires and ...

Beyond Quantifier-Free Interpolation in Extensions of Presburger

... uninterpreted functions (UF), this allows us to encode the theory of extensional arrays (AR), using uninterpreted function symbols for read and write operations. Our interpolation procedure extracts an interpolant directly from a proof of A ⇒ C. Starting from a sound and complete proof system based ...

... uninterpreted functions (UF), this allows us to encode the theory of extensional arrays (AR), using uninterpreted function symbols for read and write operations. Our interpolation procedure extracts an interpolant directly from a proof of A ⇒ C. Starting from a sound and complete proof system based ...

Complete Sequent Calculi for Induction and Infinite Descent

... • This least prefixed point can be approached via a sequence ...

... • This least prefixed point can be approached via a sequence ...

vmcai - of Philipp Ruemmer

... calculus, the proof rules are extended by labelled formulae and annotations that reduce, at the root of a closed proof, to interpolants. In earlier work, we presented a similar procedure for quantifier-free Presburger arithmetic [2]. In program verification, an interpolating theorem prover often int ...

... calculus, the proof rules are extended by labelled formulae and annotations that reduce, at the root of a closed proof, to interpolants. In earlier work, we presented a similar procedure for quantifier-free Presburger arithmetic [2]. In program verification, an interpolating theorem prover often int ...

Contents 1 The Natural Numbers

... be found in the first two chapters of Stoll’s book. We shall use the result stated by the theorem, however. Theorem 1. Suppose that A is a given set with a fixed element a0 selected in A. Suppose further that we have functions hn : A → A, one for each n ∈ N. Then, there exists a unique function k : ...

... be found in the first two chapters of Stoll’s book. We shall use the result stated by the theorem, however. Theorem 1. Suppose that A is a given set with a fixed element a0 selected in A. Suppose further that we have functions hn : A → A, one for each n ∈ N. Then, there exists a unique function k : ...

Day00a-Induction-proofs - Rose

... • How do we actually construct a proof by strong induction? To show that p(n) is true for all n n0 : – Step 0: Believe in the "magic." • You will show that it's not really magic at all. But you have to believe. • If, when you are in the middle of an induction proof, you begin to doubt whether the ...

... • How do we actually construct a proof by strong induction? To show that p(n) is true for all n n0 : – Step 0: Believe in the "magic." • You will show that it's not really magic at all. But you have to believe. • If, when you are in the middle of an induction proof, you begin to doubt whether the ...

YABLO WITHOUT GODEL

... or is true for. It is to be read: For all x and y, the formula ‘ϕ(x, y)’ is satisfied by x and y iff ϕ(x, y). An instance in English would be the following sentence: ‘is bigger than’ is satisfied by objects x and y iff x is bigger than y. The variables x and y are fixed; the first two variables in a ...

... or is true for. It is to be read: For all x and y, the formula ‘ϕ(x, y)’ is satisfied by x and y iff ϕ(x, y). An instance in English would be the following sentence: ‘is bigger than’ is satisfied by objects x and y iff x is bigger than y. The variables x and y are fixed; the first two variables in a ...

Mathematical Induction

... Basis: The sum of the first 0 natural numbers is indeed 0. Inductive step: Assume the sum of the first k natural numbers is k(k-1)/2 (inductive hypothesis). We want to show that then the same is true for k+1 instead of k, that is, the sum of the first k+1 natural numbers is (k+1)((k+1)-1)/2, i.e. it ...

... Basis: The sum of the first 0 natural numbers is indeed 0. Inductive step: Assume the sum of the first k natural numbers is k(k-1)/2 (inductive hypothesis). We want to show that then the same is true for k+1 instead of k, that is, the sum of the first k+1 natural numbers is (k+1)((k+1)-1)/2, i.e. it ...

Chapter 5 Predicate Logic

... The only difference is that the quantifiers range over a whole set of values from D, not just a pair of values. Law 4 says that a universal quantifier can be raised out of a disjunction. This is a logical consequence, not a logical equivalence. Thus, if we know that ((∀x)G(x) ∨ (∀x)H(x)), then we kn ...

... The only difference is that the quantifiers range over a whole set of values from D, not just a pair of values. Law 4 says that a universal quantifier can be raised out of a disjunction. This is a logical consequence, not a logical equivalence. Thus, if we know that ((∀x)G(x) ∨ (∀x)H(x)), then we kn ...

Problems on Discrete Mathematics1

... We use Dx , Dy to denote the domains of x and y, respectively. Note that Dx and Dy do not have to be the same. In the above example, P (3, 2) is the proposition 3 ≥ 22 with truth value F . Similarly, Q(Boo, dog) is a proposition with truth value T if there is a dog named Boo. Note: Any proposition i ...

... We use Dx , Dy to denote the domains of x and y, respectively. Note that Dx and Dy do not have to be the same. In the above example, P (3, 2) is the proposition 3 ≥ 22 with truth value F . Similarly, Q(Boo, dog) is a proposition with truth value T if there is a dog named Boo. Note: Any proposition i ...

Building explicit induction schemas for cyclic induction reasoning

... of induction hypotheses representing ‘not yet proved’ formulas. The induction hypotheses can be defined before their use, by explicit induction schemas that can be directly embedded in inference systems using explicit induction rules. On the other hand, the induction hypotheses can also be defined b ...

... of induction hypotheses representing ‘not yet proved’ formulas. The induction hypotheses can be defined before their use, by explicit induction schemas that can be directly embedded in inference systems using explicit induction rules. On the other hand, the induction hypotheses can also be defined b ...

Specification Predicates with Explicit Dependency Information

... easy to read for humans. They are indispensable for the specification of inherently recursive properties such as reachability. Especially in first-order program logics there is no other alternative to specify properties recursively. Such state-dependent predicate or function symbols, which are somet ...

... easy to read for humans. They are indispensable for the specification of inherently recursive properties such as reachability. Especially in first-order program logics there is no other alternative to specify properties recursively. Such state-dependent predicate or function symbols, which are somet ...

High Level Verification of Control Intensive Systems

... It is usually the case that different predicates are not independent. We describe efficient methods to compute constraints between predicates, which are added as invariants to the abstract model to make it more accurate. Another issue that we address in this paper is the following: Current predicate ...

... It is usually the case that different predicates are not independent. We describe efficient methods to compute constraints between predicates, which are added as invariants to the abstract model to make it more accurate. Another issue that we address in this paper is the following: Current predicate ...

Yablo`s paradox

... The rest of the argument is as before. Construing the argument in this way, we do not have to talk of satisfaction. There is therefore no predicate involved, and a fortiori no fixed-point predicate. We therefore have a paradox without circularity.6 Such a suggestion would be disingenuous, though. As ...

... The rest of the argument is as before. Construing the argument in this way, we do not have to talk of satisfaction. There is therefore no predicate involved, and a fortiori no fixed-point predicate. We therefore have a paradox without circularity.6 Such a suggestion would be disingenuous, though. As ...