Module 31
... – If T ==>*G2 x, then x is in (L(G1))* – The two statements are equivalent because • x in L(G2) means that T ==>*G2 x ...
... – If T ==>*G2 x, then x is in (L(G1))* – The two statements are equivalent because • x in L(G2) means that T ==>*G2 x ...
Module 31
... – If T ==>*G2 x, then x is in (L(G1))* – The two statements are equivalent because • x in L(G2) means that T ==>*G2 x ...
... – If T ==>*G2 x, then x is in (L(G1))* – The two statements are equivalent because • x in L(G2) means that T ==>*G2 x ...
4. Divisibility and the Greatest Common Divisor Definition. Let a, b
... (a) There exist u, v ∈ Z such that gcd(a, b) = au + bv. Moreover, gcd(a, b) is the smallest positive integer representable in the form am + bn with m, n ∈ Z. (b) If c is any integer such that c | a and c | b, then c | gcd(a, b). Before proving this theorem, we give an illustration of part (a). Let a ...
... (a) There exist u, v ∈ Z such that gcd(a, b) = au + bv. Moreover, gcd(a, b) is the smallest positive integer representable in the form am + bn with m, n ∈ Z. (b) If c is any integer such that c | a and c | b, then c | gcd(a, b). Before proving this theorem, we give an illustration of part (a). Let a ...
A Horn Clause that Implies an Undecidable Set of Horn Clauses ⋆ 1
... the author of the theorem here, however following [22] we shall present a proof of it. Proof: Let H = (Q(t1 ) =⇒ Q(t2 )). Let Q(s1 ) be the (hypothetical) label of the premise of the derivation and Q(s2 ) be the one of the conclusion (s1 and s2 are ground terms). Following[22] we define H1 = H . Let ...
... the author of the theorem here, however following [22] we shall present a proof of it. Proof: Let H = (Q(t1 ) =⇒ Q(t2 )). Let Q(s1 ) be the (hypothetical) label of the premise of the derivation and Q(s2 ) be the one of the conclusion (s1 and s2 are ground terms). Following[22] we define H1 = H . Let ...
Math 318 Class notes
... g is called a left inverse of f if g f = id A ; g is called a right inverse of f if f g = id B . Proposition 3.11. f is injective iff it has a left inverse. Proof. ⇒ Write B = range( f ) and for b ∈ B0 , let a = g(b) be the unique element such that f ( a) = b. Next, let a ∈ A be arbitrary and define ...
... g is called a left inverse of f if g f = id A ; g is called a right inverse of f if f g = id B . Proposition 3.11. f is injective iff it has a left inverse. Proof. ⇒ Write B = range( f ) and for b ∈ B0 , let a = g(b) be the unique element such that f ( a) = b. Next, let a ∈ A be arbitrary and define ...
EXTRA CREDIT PROJECTS The following extra credit projects are
... the function and the truth value assignment as the object. Again, when one draws a table of values for this function, it has the form of a truth table as we discussed in the first weeks of class. A question arises here: is it possible that there are functions in Fn that don’t ...
... the function and the truth value assignment as the object. Again, when one draws a table of values for this function, it has the form of a truth table as we discussed in the first weeks of class. A question arises here: is it possible that there are functions in Fn that don’t ...
Recursion (Ch. 10)
... 'prints digits of n vertically starting with low-order digit' if n <10: # base case: one digit number print(n) else: # n has at least 2 digits print(n%10) # prints last digit of n reverse(n//10) # recursively print in reverse all but the last digit ...
... 'prints digits of n vertically starting with low-order digit' if n <10: # base case: one digit number print(n) else: # n has at least 2 digits print(n%10) # prints last digit of n reverse(n//10) # recursively print in reverse all but the last digit ...
Ultrasheaves
... and Butz [4]. Pitts uses the filter construction on coherent categories to prove completeness and interpolation results. Makkai’s topos of types is related to the prime filters in Pitts construction. The precise relation between the two toposes is considered by Butz, who uses filters to construct ge ...
... and Butz [4]. Pitts uses the filter construction on coherent categories to prove completeness and interpolation results. Makkai’s topos of types is related to the prime filters in Pitts construction. The precise relation between the two toposes is considered by Butz, who uses filters to construct ge ...
FORMALIZATION OF HILBERT`S GEOMETRY OF INCIDENCE AND
... whether Geometry is a logical consequence of explicitly stated axioms, or in other words, whether the axioms given to a reasoning machine will make the sequence of all theorems appear” (1902, 252–253). Hilbert himself states as the basic principle of his study of geometry, “to express every question ...
... whether Geometry is a logical consequence of explicitly stated axioms, or in other words, whether the axioms given to a reasoning machine will make the sequence of all theorems appear” (1902, 252–253). Hilbert himself states as the basic principle of his study of geometry, “to express every question ...
Recursion (Ch. 10)
... 'prints digits of n vertically starting with low-order digit' if n <10: # base case: one digit number print(n) else: # n has at least 2 digits print(n%10) # prints last digit of n reverse(n//10) # recursively print in reverse all but the last digit ...
... 'prints digits of n vertically starting with low-order digit' if n <10: # base case: one digit number print(n) else: # n has at least 2 digits print(n%10) # prints last digit of n reverse(n//10) # recursively print in reverse all but the last digit ...
Recurrent points and hyperarithmetic sets
... There is considerable freedom in the choice of the integers nt,i , and a specific recursive choice is given in Long Delays. To prove our main theorem any recursive choice that guarantees the truth of the various lemmata in §1 of Long Delays will do: as before, when we are discussing the shift functi ...
... There is considerable freedom in the choice of the integers nt,i , and a specific recursive choice is given in Long Delays. To prove our main theorem any recursive choice that guarantees the truth of the various lemmata in §1 of Long Delays will do: as before, when we are discussing the shift functi ...
An Introduction to Löb`s Theorem in MIRI Research
... One (anachronistic) way of stating Gödel’s key insight is that you can use computer programs to search for proofs, and you can prove statements about computer programs. If we think about any conjecture in mathematics that can be stated in terms of arithmetic, you can write a rather simple program t ...
... One (anachronistic) way of stating Gödel’s key insight is that you can use computer programs to search for proofs, and you can prove statements about computer programs. If we think about any conjecture in mathematics that can be stated in terms of arithmetic, you can write a rather simple program t ...
PPT
... In Classical Logic, which is what we’ve been discussing, the goal is to formalize theories. In Intuitionistic Logic, theorems are viewed as programs. They give explicit evidence that a claim is true. ...
... In Classical Logic, which is what we’ve been discussing, the goal is to formalize theories. In Intuitionistic Logic, theorems are viewed as programs. They give explicit evidence that a claim is true. ...
classden
... the model theory of a program must needs be ‘dynamic’. But these differences, important as they are, should not blind us to the fact that the two forms of semantics are essentially one. We shall model the difference between ‘static’ and ‘dynamic’ semantics as a difference in type here. While static ...
... the model theory of a program must needs be ‘dynamic’. But these differences, important as they are, should not blind us to the fact that the two forms of semantics are essentially one. We shall model the difference between ‘static’ and ‘dynamic’ semantics as a difference in type here. While static ...
A Calculus for Belnap`s Logic in Which Each Proof Consists of Two
... This is the notion of entailment considered in Belnap [5, 6], but not that of Arieli & Avron [1], who use a single-barrelled notion. The two notions of entailment are coextensional on sets of formulas based on classical connectives only, but not on formulas based on a functionally complete set of co ...
... This is the notion of entailment considered in Belnap [5, 6], but not that of Arieli & Avron [1], who use a single-barrelled notion. The two notions of entailment are coextensional on sets of formulas based on classical connectives only, but not on formulas based on a functionally complete set of co ...
On Action Logic
... omitted. Also, (K1), (K2) can be replaced with (C1), (C2) [22], which is not true for Kleene algebras ((C1), (C2) hold in right-handed Kleene algebras, and not every right-handed Kleene algebra is a Kleene algebra [10]). An action algebra is *-continuous (as a Kleene algebra) iff a∗ =l.u.b.{an : n ...
... omitted. Also, (K1), (K2) can be replaced with (C1), (C2) [22], which is not true for Kleene algebras ((C1), (C2) hold in right-handed Kleene algebras, and not every right-handed Kleene algebra is a Kleene algebra [10]). An action algebra is *-continuous (as a Kleene algebra) iff a∗ =l.u.b.{an : n ...
Lecture notes from 5860
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
Slide 1
... A conditional is vacuously true if its hypothesis is false. A conditional is trivially true if its conclusion is true. Proof Techniques: We’ll give sample proofs about numbers. Here are some definitions. ...
... A conditional is vacuously true if its hypothesis is false. A conditional is trivially true if its conclusion is true. Proof Techniques: We’ll give sample proofs about numbers. Here are some definitions. ...
Discrete Mathematics (2009 Spring) Induction and Recursion
... So, according to the inductive inference rule, the property is proved. ...
... So, according to the inductive inference rule, the property is proved. ...
AppA - txstateprojects
... objects to objects in the world). We say a theory is sound w.r.t. w iff every theorem in the theory corresponds to a fact that is true in w. We say a theory is complete w.r.t. w iff every fact that is true in w corresponds to a theorem in the theory. ...
... objects to objects in the world). We say a theory is sound w.r.t. w iff every theorem in the theory corresponds to a fact that is true in w. We say a theory is complete w.r.t. w iff every fact that is true in w corresponds to a theorem in the theory. ...
PDF
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
Discrete Mathematics, Chapter 5: Induction and Recursion
... n ∈ N, we complete these steps: Basis Step: Show that P(0) is true. (In the instantiation of the formula for well-founded induction this is the only case where there are no R-“smaller” elements y .) Inductive Step: Show that P(k ) → P(k + 1) is true for all k ∈ N. To complete the inductive step, we ...
... n ∈ N, we complete these steps: Basis Step: Show that P(0) is true. (In the instantiation of the formula for well-founded induction this is the only case where there are no R-“smaller” elements y .) Inductive Step: Show that P(k ) → P(k + 1) is true for all k ∈ N. To complete the inductive step, we ...
P - Department of Computer Science
... • Completeness Theorem: there exists some set of inference rules R such that, given any set of axioms A and a sentence c, there is a proof of c, starting with A and applying the rules in R, iff c is entailed by A. • Incompleteness Theorem: any theory that is derived from a decidable set of axioms an ...
... • Completeness Theorem: there exists some set of inference rules R such that, given any set of axioms A and a sentence c, there is a proof of c, starting with A and applying the rules in R, iff c is entailed by A. • Incompleteness Theorem: any theory that is derived from a decidable set of axioms an ...
Redundancies in the Hilbert-Bernays derivability conditions for
... they are closed under cut. It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with foundations, for it treats logics with quantifiers, and in that case one can dispose entirely of the first and second derivability conditions. re ...
... they are closed under cut. It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with foundations, for it treats logics with quantifiers, and in that case one can dispose entirely of the first and second derivability conditions. re ...