Transact-SQL User-Defined Functions
... Any omission or misuse (of any kind) of service marks or trademarks should not be regarded as intent to infringe on the property of others. The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. This book is sold as ...
... Any omission or misuse (of any kind) of service marks or trademarks should not be regarded as intent to infringe on the property of others. The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. This book is sold as ...
LPF and MPLω — A Logical Comparison of VDM SL and COLD-K
... Another approach has been followed in LPF, the logic underlying VDM SL. Here the possibility of undefinedness is extended to the formulae by adding a truth value N (neithertrue-nor-false), so terms and formulae are in this respect treated on an equal footing. This makes LPF a non-classical logic wit ...
... Another approach has been followed in LPF, the logic underlying VDM SL. Here the possibility of undefinedness is extended to the formulae by adding a truth value N (neithertrue-nor-false), so terms and formulae are in this respect treated on an equal footing. This makes LPF a non-classical logic wit ...
Foundations of Computation - Department of Mathematics and
... the logical combination of the truth values of the two propositions “I wanted to leave” and “I left.” Or consider the proposition “I wanted to leave but I did not leave.” Here, the word “but” has the same logical meaning as the word “and,” but the connotation is very different. So, in mathematical l ...
... the logical combination of the truth values of the two propositions “I wanted to leave” and “I left.” Or consider the proposition “I wanted to leave but I did not leave.” Here, the word “but” has the same logical meaning as the word “and,” but the connotation is very different. So, in mathematical l ...
lecture notes in logic - UCLA Department of Mathematics
... 4A. Tarski and Gödel (First Incompleteness Theorem). . . . . . . . . . . 139 4B. Numeralwise representability in Q . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4C. Rosser, more Gödel and Löb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4D. Computability and undec ...
... 4A. Tarski and Gödel (First Incompleteness Theorem). . . . . . . . . . . 139 4B. Numeralwise representability in Q . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4C. Rosser, more Gödel and Löb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4D. Computability and undec ...
REGULAR COST FUNCTIONS, PART I: LOGIC AND ALGEBRA
... define functions. For achieving this, the automata uses an extra mechanism involving the asymptotic behaviors of counters for deciding whether an infinite word should be accepted or not. This extra mechanism has no equivalent in distance automata, and is in some sense ‘orthogonal’ to the machinery i ...
... define functions. For achieving this, the automata uses an extra mechanism involving the asymptotic behaviors of counters for deciding whether an infinite word should be accepted or not. This extra mechanism has no equivalent in distance automata, and is in some sense ‘orthogonal’ to the machinery i ...
Full abstraction for PCF - Department of Computer Science, Oxford
... the natural numbers. The interpretation of each type of Finitary PCF in the fully abstract model is a finite poset. A natural question is whether these finite posets can be effectively presented. Suppose that we have a category of sequential domains as described in the previous paragraph, yielding a ...
... the natural numbers. The interpretation of each type of Finitary PCF in the fully abstract model is a finite poset. A natural question is whether these finite posets can be effectively presented. Suppose that we have a category of sequential domains as described in the previous paragraph, yielding a ...
Proof Pearl: Defining Functions over Finite Sets
... Alternative 2 above resembles the inductive definition of fold. Whichever alternative is chosen, we should only prove enough results about cardinality to allow the definition of fold : many lemmas about cardinality are instances of more general lemmas about set summation and can be obtained easily onc ...
... Alternative 2 above resembles the inductive definition of fold. Whichever alternative is chosen, we should only prove enough results about cardinality to allow the definition of fold : many lemmas about cardinality are instances of more general lemmas about set summation and can be obtained easily onc ...
Incompleteness in the finite domain
... sentences have proofs in T of polynomial length [13].1 It turned out that the answer to his question is yes [28], but it is still possible, and seems very plausible, that for natural variations of this question there are no polynomial length proofs. Namely, this should be true if we ask about the le ...
... sentences have proofs in T of polynomial length [13].1 It turned out that the answer to his question is yes [28], but it is still possible, and seems very plausible, that for natural variations of this question there are no polynomial length proofs. Namely, this should be true if we ask about the le ...
Barwise: Infinitary Logic and Admissible Sets
... We say that two structures M and N , of arbitrary cardinality, are potentially isomorphic if there is a back-and-forth family for M, N . It is obvious that isomorphic structures are potentially isomorphic. In the other direction, potentially isomorphic structures are very similar to each other, but ...
... We say that two structures M and N , of arbitrary cardinality, are potentially isomorphic if there is a back-and-forth family for M, N . It is obvious that isomorphic structures are potentially isomorphic. In the other direction, potentially isomorphic structures are very similar to each other, but ...
Proof Pearl: Defining Functions Over Finite Sets
... Alternative 2 above resembles the inductive definition of fold. Whichever alternative is chosen, we should only prove enough results about cardinality to allow the definition of fold : many lemmas about cardinality are instances of more general lemmas about set summation and can be obtained easily o ...
... Alternative 2 above resembles the inductive definition of fold. Whichever alternative is chosen, we should only prove enough results about cardinality to allow the definition of fold : many lemmas about cardinality are instances of more general lemmas about set summation and can be obtained easily o ...
How complicated is the set of stable models of a recursive logic
... described by a logic program?”. The first result in this direction is the classical result implicit in ([Smullyan, 1961]) and, more recently, reproved by Andreka and Nemeti ([Andreka and Nemeti, 1978]). This can be summarized as follows: the least model of a recursive Horn program is a recursively e ...
... described by a logic program?”. The first result in this direction is the classical result implicit in ([Smullyan, 1961]) and, more recently, reproved by Andreka and Nemeti ([Andreka and Nemeti, 1978]). This can be summarized as follows: the least model of a recursive Horn program is a recursively e ...
A Taste of Categorical Logic — Tutorial Notes
... π [A] = {π(z) | z ∈ A} where π : X × Y → X is the first projection. Observe that this gives rise to ∃YX . Geometrically, if we draw X on the horizontal axis and Y on the vertical axis, A is a region on the graph. The image π [A] includes all x ∈ X such that the vertical line at x intersects A in at ...
... π [A] = {π(z) | z ∈ A} where π : X × Y → X is the first projection. Observe that this gives rise to ∃YX . Geometrically, if we draw X on the horizontal axis and Y on the vertical axis, A is a region on the graph. The image π [A] includes all x ∈ X such that the vertical line at x intersects A in at ...
Gödel Without (Too Many) Tears
... (i) an effectively formalized language L, (ii) an effectively decidable set of axioms, (iii) an effectively formalized proof-system in which we can deduce theorems from the axioms. But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effecti ...
... (i) an effectively formalized language L, (ii) an effectively decidable set of axioms, (iii) an effectively formalized proof-system in which we can deduce theorems from the axioms. But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effecti ...
you can this version here
... (i) an effectively formalized language L, (ii) an effectively decidable set of axioms, (iii) an effectively formalized proof-system in which we can deduce theorems from the axioms. But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effecti ...
... (i) an effectively formalized language L, (ii) an effectively decidable set of axioms, (iii) an effectively formalized proof-system in which we can deduce theorems from the axioms. But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effecti ...
Curry: A Tutorial Introduction
... Now it is time to make some remarks about the syntax of Curry (which is actually very similar to Haskell [16]). The names of functions and parameters usually start with a lowercase letter followed by letters, digits and underscores. The application of a function f to an expression e is denoted by ju ...
... Now it is time to make some remarks about the syntax of Curry (which is actually very similar to Haskell [16]). The names of functions and parameters usually start with a lowercase letter followed by letters, digits and underscores. The application of a function f to an expression e is denoted by ju ...
HONEST ELEMENTARY DEGREES AND DEGREES OF RELATIVE
... closure under bounded primitive recursion. These closure properties lead to useful alternative characterizations. To wit, the elementary functions are exactly the closure of 0, the successor function, the projection functions, the exponential function 2x , and the max function under composition and ...
... closure under bounded primitive recursion. These closure properties lead to useful alternative characterizations. To wit, the elementary functions are exactly the closure of 0, the successor function, the projection functions, the exponential function 2x , and the max function under composition and ...
Document
... Languages (cont’d.) • xy is the concatenation of the two strings x and y; this is the basic operation on strings – If x = ab and y = bab then xy = abbab and yx = babab – For every string x, x = x = x – |xy| = |x| + |y| ...
... Languages (cont’d.) • xy is the concatenation of the two strings x and y; this is the basic operation on strings – If x = ab and y = bab then xy = abbab and yx = babab – For every string x, x = x = x – |xy| = |x| + |y| ...
Document
... Languages (cont’d.) • xy is the concatenation of the two strings x and y; this is the basic operation on strings – If x = ab and y = bab then xy = abbab and yx = babab – For every string x, x = x = x – |xy| = |x| + |y| ...
... Languages (cont’d.) • xy is the concatenation of the two strings x and y; this is the basic operation on strings – If x = ab and y = bab then xy = abbab and yx = babab – For every string x, x = x = x – |xy| = |x| + |y| ...
Closed Sets of Higher
... that...”). We can then see the benefit of lambda notation in defining higherorder operations which mathematicians would otherwise have difficulty writing out in full. When we say that we are taking a less foundational perspective, we mean that we take for granted the existence of sets and functions ...
... that...”). We can then see the benefit of lambda notation in defining higherorder operations which mathematicians would otherwise have difficulty writing out in full. When we say that we are taking a less foundational perspective, we mean that we take for granted the existence of sets and functions ...
A Note on the Relation between Inflationary Fixpoints and Least
... Section 2) every monotone function has a unique least fixpoint which can also be obtained by an explicit induction process. As monotonicity of a function is in general undecidable, first-order formulas that are positive in X (and therefore monotone) are usually considered only. A similar but seeming ...
... Section 2) every monotone function has a unique least fixpoint which can also be obtained by an explicit induction process. As monotonicity of a function is in general undecidable, first-order formulas that are positive in X (and therefore monotone) are usually considered only. A similar but seeming ...
Recursion
... procedure itself. A procedure that goes through recursion is said to be 'recursive'. To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps that are to be taken based on a set of rules. The running of a procedure ...
... procedure itself. A procedure that goes through recursion is said to be 'recursive'. To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps that are to be taken based on a set of rules. The running of a procedure ...
Sets, Logic, Computation
... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...
... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...
On the structure of honest elementary degrees - FAU Math
... Q sum and bounded product, that is, respectively f (~x, y) = i
... Q sum and bounded product, that is, respectively f (~x, y) = i
Notes on Simply Typed Lambda Calculus
... 2. Is closed under composition: h(x) = f(g1 (x) . . . gn(x)). 3. Is closed under primitive recursion: h(x, 0) = f(x), h(x, n + 1) = g(x, n, f(x, n)). 4. Is closed under minimisation, whenever this gives a total function: if for each x there is n such that f(x, n) = 0, then take g(x) to be the least ...
... 2. Is closed under composition: h(x) = f(g1 (x) . . . gn(x)). 3. Is closed under primitive recursion: h(x, 0) = f(x), h(x, n + 1) = g(x, n, f(x, n)). 4. Is closed under minimisation, whenever this gives a total function: if for each x there is n such that f(x, n) = 0, then take g(x) to be the least ...
Sets, Logic, Computation
... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...
... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...