Circuit principles and weak pigeonhole variants
... hold in the real world. Jeřábek’s result to some extent gives an upper bound on the theory required to prove a weak Kannan result, for once we know a hard string exists, if we can obtain a least such string, we can construct a fixed set which is hard for size-nk circuits. This kind of argument can ...
... hold in the real world. Jeřábek’s result to some extent gives an upper bound on the theory required to prove a weak Kannan result, for once we know a hard string exists, if we can obtain a least such string, we can construct a fixed set which is hard for size-nk circuits. This kind of argument can ...
PRESBURGER ARITHMETIC, RATIONAL GENERATING
... variables. We will restrict ourselves to counting functions such that gF (p) is finite for all p ∈ Nn . One could instead either include ∞ in the codomain of gF or restrict the domain of gF to where gF (p) is finite (the domain would itself be a Presburger set). A classic example is to take F (c, p) ...
... variables. We will restrict ourselves to counting functions such that gF (p) is finite for all p ∈ Nn . One could instead either include ∞ in the codomain of gF or restrict the domain of gF to where gF (p) is finite (the domain would itself be a Presburger set). A classic example is to take F (c, p) ...
Presburger arithmetic, rational generating functions, and quasi
... variables. We will restrict ourselves to counting functions such that gF (p) is finite for all p ∈ Nn . One could instead either include ∞ in the codomain of gF or restrict the domain of gF to where gF (p) is finite (the domain would itself be a Presburger set). A classic example is to take F (c, p) ...
... variables. We will restrict ourselves to counting functions such that gF (p) is finite for all p ∈ Nn . One could instead either include ∞ in the codomain of gF or restrict the domain of gF to where gF (p) is finite (the domain would itself be a Presburger set). A classic example is to take F (c, p) ...
EXHAUSTIBLE SETS IN HIGHER-TYPE
... is whether there are infinite examples. Intuitively, there can be none: how could one possibly check infinitely many cases in finite time? This intuition is correct when K is a set of natural numbers: it is a theorem that, in this case, K is exhaustible if and only if it is finite. This can be prove ...
... is whether there are infinite examples. Intuitively, there can be none: how could one possibly check infinitely many cases in finite time? This intuition is correct when K is a set of natural numbers: it is a theorem that, in this case, K is exhaustible if and only if it is finite. This can be prove ...
EXHAUSTIBLE SETS IN HIGHER
... is whether there are infinite examples. Intuitively, there can be none: how could one possibly check infinitely many cases in finite time? This intuition is correct when K is a set of natural numbers: it is a theorem that, in this case, K is exhaustible if and only if it is finite. This can be prove ...
... is whether there are infinite examples. Intuitively, there can be none: how could one possibly check infinitely many cases in finite time? This intuition is correct when K is a set of natural numbers: it is a theorem that, in this case, K is exhaustible if and only if it is finite. This can be prove ...
QUASI-MINIMAL DEGREES FOR DEGREE SPECTRA 1
... Let ϕ be a partial enumeration of A and e, x ∈ N. Then the modeling relation “|=” is defined as follows: (i) ϕ |= Fe (x) ⇐⇒ (∃v)(hv, xi ∈ We & (∀u ∈ Dv )(∃i)(1 ≤ i ≤ k & (u = h0, i, xu1 , . . . , xuri i & xu1 , . . . , xuri ∈ dom(ϕ) & (ϕ(xu1 ), . . . , ϕ(xuri )) ∈ Ri )∨ (u = h1, i, xu1 , . . . , xur ...
... Let ϕ be a partial enumeration of A and e, x ∈ N. Then the modeling relation “|=” is defined as follows: (i) ϕ |= Fe (x) ⇐⇒ (∃v)(hv, xi ∈ We & (∀u ∈ Dv )(∃i)(1 ≤ i ≤ k & (u = h0, i, xu1 , . . . , xuri i & xu1 , . . . , xuri ∈ dom(ϕ) & (ϕ(xu1 ), . . . , ϕ(xuri )) ∈ Ri )∨ (u = h1, i, xu1 , . . . , xur ...
210ch2 - Dr. Djamel Bouchaffra
... • We know that if a b then f(g(a)) f(g(b)) since the composition is injective. • Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point. • Hence, g(a) g(b) It follows that g must be an injection. However, f need not be an i ...
... • We know that if a b then f(g(a)) f(g(b)) since the composition is injective. • Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point. • Hence, g(a) g(b) It follows that g must be an injection. However, f need not be an i ...
Lecture 8: Back-and-forth - to go back my main page.
... (b) Th(M ) = Th(N ) and SSy(M ) = SSy(N ). Proof. The implication (a) ⇒ (b) is trivial. For the converse, suppose (b) holds. We carry out a back-and-forth argument to find an isomorphism M → N . By recursion, we will define (rm )m∈N in M and (sm )m∈N in N such that f : rm 7→ sm is an isomorphism M → ...
... (b) Th(M ) = Th(N ) and SSy(M ) = SSy(N ). Proof. The implication (a) ⇒ (b) is trivial. For the converse, suppose (b) holds. We carry out a back-and-forth argument to find an isomorphism M → N . By recursion, we will define (rm )m∈N in M and (sm )m∈N in N such that f : rm 7→ sm is an isomorphism M → ...
First-order possibility models and finitary
... normal modal logics extending K with standard axioms (such as D, T, 4, B, and 5), Holliday [Hol14] is able to give such a completeness proof. On the other hand, there are normal modal logics which do not admit such a finitary completeness proof.1 In the case of first-order logics, it is not quite so ...
... normal modal logics extending K with standard axioms (such as D, T, 4, B, and 5), Holliday [Hol14] is able to give such a completeness proof. On the other hand, there are normal modal logics which do not admit such a finitary completeness proof.1 In the case of first-order logics, it is not quite so ...
Sense and denotation as algorithm and value
... truth: check to see if at any time Niarchos and Onassis were married to two sisters. On this account, we can define a faithful translation (at the sentence level) as an arbitrary syntactic transformation of sentences from one language to another which preserves referential intension, and that would ...
... truth: check to see if at any time Niarchos and Onassis were married to two sisters. On this account, we can define a faithful translation (at the sentence level) as an arbitrary syntactic transformation of sentences from one language to another which preserves referential intension, and that would ...
Linearizing some recursive logic programs
... A Datalog program P is a finite set of function-free Horn clauses, called rules, of the form: Q(X1 , . . . , Xn ) ←− Q1 (Y1,1 , . . . , Y1,n1 ), . . . , Qp (Yp,1 , . . . , Yp,np ) where X1 , . . . , Xn are variables, the Yi,j ’s are either variables or constants, Q is an intensional predicate, the Q ...
... A Datalog program P is a finite set of function-free Horn clauses, called rules, of the form: Q(X1 , . . . , Xn ) ←− Q1 (Y1,1 , . . . , Y1,n1 ), . . . , Qp (Yp,1 , . . . , Yp,np ) where X1 , . . . , Xn are variables, the Yi,j ’s are either variables or constants, Q is an intensional predicate, the Q ...
Review - UT Computer Science
... respect to particular interpretations of interest. One example is Presburger arithmetic, in which the universe is the natural numbers and there is a single function, plus, whose properties are axiomatized. There are other theories that are incomplete because we have not yet added enough axioms. But ...
... respect to particular interpretations of interest. One example is Presburger arithmetic, in which the universe is the natural numbers and there is a single function, plus, whose properties are axiomatized. There are other theories that are incomplete because we have not yet added enough axioms. But ...
Discrete Maths - Department of Computing | Imperial College London
... elements of one set to another. Haskell functions can be viewed as mathematical functions, although they also have the additional property that they are computable. [In fact, the idea of a computable function can be expressed precisely. For example, Turing machines describe the computable functions, ...
... elements of one set to another. Haskell functions can be viewed as mathematical functions, although they also have the additional property that they are computable. [In fact, the idea of a computable function can be expressed precisely. For example, Turing machines describe the computable functions, ...
The First Incompleteness Theorem
... A formalized theory T built in some formalized language L has a determinate set of axioms and a deductive system for deriving theorems from the axioms. For a properly formalized theory, it is required that it is effectively decidable whether a purported proof is indeed a correctly constructed T -pro ...
... A formalized theory T built in some formalized language L has a determinate set of axioms and a deductive system for deriving theorems from the axioms. For a properly formalized theory, it is required that it is effectively decidable whether a purported proof is indeed a correctly constructed T -pro ...
AppA - txstateprojects
... • In theoretical computer science, automata theory is the study of abstract machines and problems which they are able to solve. It is closely related to formal language theory as the automata are often classified by the class of formal languages they are able to recognize. – An abstract machine, als ...
... • In theoretical computer science, automata theory is the study of abstract machines and problems which they are able to solve. It is closely related to formal language theory as the automata are often classified by the class of formal languages they are able to recognize. – An abstract machine, als ...
The Formulae-as-Classes Interpretation of Constructive Set Theory
... (see [16]) to discover a simple formalism that relates to Bishop’s constructive mathematics as classical Zermelo-Fraenkel Set Theory with the axiom of choice relates to classical Cantorian mathematics. CST provides a standard set theoretical framework for the development of constructive mathematics ...
... (see [16]) to discover a simple formalism that relates to Bishop’s constructive mathematics as classical Zermelo-Fraenkel Set Theory with the axiom of choice relates to classical Cantorian mathematics. CST provides a standard set theoretical framework for the development of constructive mathematics ...
LOWNESS NOTIONS, MEASURE AND DOMINATION
... Dobrinen and Simpson observed that WKL0 0 Gδ -REG and asked whether any (or all) of Gδ -REG, Gδ -ε or POS implied ACA0 . They suggested finding simpler recursion theoretic equivalences of a.e. domination and u.a.e. domination to help answer this question. At that time, it was known that A is complet ...
... Dobrinen and Simpson observed that WKL0 0 Gδ -REG and asked whether any (or all) of Gδ -REG, Gδ -ε or POS implied ACA0 . They suggested finding simpler recursion theoretic equivalences of a.e. domination and u.a.e. domination to help answer this question. At that time, it was known that A is complet ...
Lowness notions, measure and domination
... Dobrinen and Simpson observed that WKL0 0 Gδ -REG and asked whether any (or all) of Gδ -REG, Gδ -ε or POS implied ACA0 . They suggested finding simpler recursion theoretic equivalences of a.e. domination and u.a.e. domination to help answer this question. At that time, it was known that A is complet ...
... Dobrinen and Simpson observed that WKL0 0 Gδ -REG and asked whether any (or all) of Gδ -REG, Gδ -ε or POS implied ACA0 . They suggested finding simpler recursion theoretic equivalences of a.e. domination and u.a.e. domination to help answer this question. At that time, it was known that A is complet ...
Basic Set Theory
... When performing set theoretic computations, you should declare the domain in which you are working. In set theory this is done by declaring a universal set. Definition 2.8 The universal set, at least for a given collection of set theoretic computations, is the set of all possible objects. If we decl ...
... When performing set theoretic computations, you should declare the domain in which you are working. In set theory this is done by declaring a universal set. Definition 2.8 The universal set, at least for a given collection of set theoretic computations, is the set of all possible objects. If we decl ...
Translating the Hypergame Paradox - UvA-DARE
... hypergame argument. In both cases X = N; if (u~)~~N is a sequence that contains all real numbers, for each n fix a decimal representation of ur&, and call u~,~ the mth digit of Us. Then define the binary relation R on iV as mRn iff u~,~ = 1. Up to this point, the diagonal and the hypergame proofs fo ...
... hypergame argument. In both cases X = N; if (u~)~~N is a sequence that contains all real numbers, for each n fix a decimal representation of ur&, and call u~,~ the mth digit of Us. Then define the binary relation R on iV as mRn iff u~,~ = 1. Up to this point, the diagonal and the hypergame proofs fo ...
ordinal logics and the characterization of informal concepts of proof
... such, is familiar from the need for the derivability conditions in the second incompleteness theorem. Thus, if P(m, n) is a proof predicate for a consistent system ($), ri denotes the negation of n, and Con 8 denotes (m)->P(m, r 0= V) thenP x (m,n),i.e.P(m,n) &(p)[p < m->—^P(p,n)] is also a proof pr ...
... such, is familiar from the need for the derivability conditions in the second incompleteness theorem. Thus, if P(m, n) is a proof predicate for a consistent system ($), ri denotes the negation of n, and Con 8 denotes (m)->P(m, r 0= V) thenP x (m,n),i.e.P(m,n) &(p)[p < m->—^P(p,n)] is also a proof pr ...
On the Characteristic Functions of Fuzzy Systems 1 Introduction 2
... be named canonical partition of the interval I. The points ek play a role in considerations only regarding the derivability of Sugeno systems and regarding Mamdani systems; for sake of generality we use them here too. One could define the canonical partition without the points ek and introduce them o ...
... be named canonical partition of the interval I. The points ek play a role in considerations only regarding the derivability of Sugeno systems and regarding Mamdani systems; for sake of generality we use them here too. One could define the canonical partition without the points ek and introduce them o ...
Appendix A Sets, Relations and Functions
... This chapter explains the basics of formal set notation, and gives an introduction to relations and functions. The chapter ends with a short account of the principle of proof by mathematical induction. ...
... This chapter explains the basics of formal set notation, and gives an introduction to relations and functions. The chapter ends with a short account of the principle of proof by mathematical induction. ...
Lecture notes from 5860
... of the statements that can be derived by means of the formal rules (or, what amounts to the same, how they are understood) because understanding a language, even a formal one, is not merely to understand its rules as rules of symbol manipulation. Believing that is the mistake of formalism.” The firs ...
... of the statements that can be derived by means of the formal rules (or, what amounts to the same, how they are understood) because understanding a language, even a formal one, is not merely to understand its rules as rules of symbol manipulation. Believing that is the mistake of formalism.” The firs ...
PDF
... We can imagine a simpler principle if we work in a sub-logic in which we do not keep track of all the evidence. This kind of sub-logic is called classical logic, and we will examine it more later in the course. {Least Number Principle:} ∃x : N.A(x) ⇒ ∃y : N.(A(y)&∀z : N.z < y ⇒∼ A(z)). There are man ...
... We can imagine a simpler principle if we work in a sub-logic in which we do not keep track of all the evidence. This kind of sub-logic is called classical logic, and we will examine it more later in the course. {Least Number Principle:} ∃x : N.A(x) ⇒ ∃y : N.(A(y)&∀z : N.z < y ⇒∼ A(z)). There are man ...