Introduction to Theoretical Computer Science, lesson 3
... • A proof that a formula is a tautology or that a conclusion Z logically follows from premises can be performed: a) In the direct way – for instance by a truth-value table (only in PL), by natural deduction, etc. b) In the indirect way: P1 ... Pn Z is a contradiction; hence the set of premises ...
... • A proof that a formula is a tautology or that a conclusion Z logically follows from premises can be performed: a) In the direct way – for instance by a truth-value table (only in PL), by natural deduction, etc. b) In the indirect way: P1 ... Pn Z is a contradiction; hence the set of premises ...
Properties of Independently Axiomatizable Bimodal Logics
... Let EL denote the lattice of extensions of a modal logic. We have defined an operation − ⊗ − : (EK)2 → EK2 . ⊗ is a -homomorphism in both arguments. There are certain easy properties of this map which are noteworthy. Fixing the second argument we can study the map − ⊗ M : EK → EK2 . This is a -h ...
... Let EL denote the lattice of extensions of a modal logic. We have defined an operation − ⊗ − : (EK)2 → EK2 . ⊗ is a -homomorphism in both arguments. There are certain easy properties of this map which are noteworthy. Fixing the second argument we can study the map − ⊗ M : EK → EK2 . This is a -h ...
Nonmonotonic Logic II: Nonmonotonic Modal Theories
... The first two of these follow by predicate calculus. The third follows because ~CANFLY(FRED) is not a member of the fixed point. In other words, M CANFLY(FRED) is in Astheory(fixed-point). So by the first proper axiom, CANFLY(FRED) is in the fixed point as well. Of course, I have not proven that thi ...
... The first two of these follow by predicate calculus. The third follows because ~CANFLY(FRED) is not a member of the fixed point. In other words, M CANFLY(FRED) is in Astheory(fixed-point). So by the first proper axiom, CANFLY(FRED) is in the fixed point as well. Of course, I have not proven that thi ...
Logic and Proof - Collaboratory for Advanced Computing and
... 1. Given two distinct points, one can draw one and only one line segment connecting these points. 2. Given two distinct points, one can draw one and only one circle centered at the first point and passing through the second one. 3. Any two right angles are equal. 4. Every line segment can ...
... 1. Given two distinct points, one can draw one and only one line segment connecting these points. 2. Given two distinct points, one can draw one and only one circle centered at the first point and passing through the second one. 3. Any two right angles are equal. 4. Every line segment can ...
Slide 1
... • The checking algorithm A would then verify that the tour really does visit all of the cities and really does have total length K. without seeking all possible K solutions through each of the vertices. Polynomial. • The TSP, therefore, also belongs to NP. • How could a problem fail to belong to NP? ...
... • The checking algorithm A would then verify that the tour really does visit all of the cities and really does have total length K. without seeking all possible K solutions through each of the vertices. Polynomial. • The TSP, therefore, also belongs to NP. • How could a problem fail to belong to NP? ...
PLATONISM IN MODERN MATHEMATICS A University Thesis
... logic is not math, but logic itself is an area of mathematical study, and in no way do they rely on each other. Finally, set theorists describe all mathematical objects as sets. Cantor developed this school with his studies in the infinite. When he showed that there are certain infinite sets that ca ...
... logic is not math, but logic itself is an area of mathematical study, and in no way do they rely on each other. Finally, set theorists describe all mathematical objects as sets. Cantor developed this school with his studies in the infinite. When he showed that there are certain infinite sets that ca ...
number theory and methods of proof
... If a statement and its converse are true then the implication () works both ways () i.e. if p q and q p then p q and we say that p is true if and only if q is true. If p q then we say that p is a sufficient condition for q. If q p then we say that p is a necessary condition for q. If p ...
... If a statement and its converse are true then the implication () works both ways () i.e. if p q and q p then p q and we say that p is true if and only if q is true. If p q then we say that p is a sufficient condition for q. If q p then we say that p is a necessary condition for q. If p ...
Review of logic and proofs
... 6. Let P (x) = “x is a math major,” Q(x) = “x is taking MATH 174,” and R(x, y) = “x lives with y.” Let a = Alice, b = Bob, c = Cat, and d = Derrick. Write the following sentences in the simplest English possible. (a) ∀x[P (x) → Q(x)] (b) ∃xy[P (x) ∧ ¬P (y) ∧ R(x, y)] (c) ∀x[P (x) ∧ ∃yR(x, y) → P (y ...
... 6. Let P (x) = “x is a math major,” Q(x) = “x is taking MATH 174,” and R(x, y) = “x lives with y.” Let a = Alice, b = Bob, c = Cat, and d = Derrick. Write the following sentences in the simplest English possible. (a) ∀x[P (x) → Q(x)] (b) ∃xy[P (x) ∧ ¬P (y) ∧ R(x, y)] (c) ∀x[P (x) ∧ ∃yR(x, y) → P (y ...
Inference Tasks and Computational Semantics
... • Proof theory is the syntactic approach to logic. • It attempts to define collections of rules and/or axioms that enable us to generate new formulas from old • That is, it attempts to pin down the notion of inference syntactically. • P |- Q versus P |= Q ...
... • Proof theory is the syntactic approach to logic. • It attempts to define collections of rules and/or axioms that enable us to generate new formulas from old • That is, it attempts to pin down the notion of inference syntactically. • P |- Q versus P |= Q ...
An application of results by Hardy, Ramanujan and Karamata
... The deeper reason for this can be described briefly as follows. Usual analytic number theory can be formalized within a logical framework RCA0 (see, for example, Simpson (1985) for a definition) which has only primitive recursive functions as provably total recursive functions by a standard result i ...
... The deeper reason for this can be described briefly as follows. Usual analytic number theory can be formalized within a logical framework RCA0 (see, for example, Simpson (1985) for a definition) which has only primitive recursive functions as provably total recursive functions by a standard result i ...
Infinity
... Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square. This being granted we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and ...
... Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square. This being granted we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and ...
The Closed World Assumption
... We view our program as a logical theory expressing knowledge about the world. In several situations, it is convenient to assume that the program contains complete information about certain kinds of logical statements. We can then make additional inferences about the world based on the assumed comple ...
... We view our program as a logical theory expressing knowledge about the world. In several situations, it is convenient to assume that the program contains complete information about certain kinds of logical statements. We can then make additional inferences about the world based on the assumed comple ...
Asymptotic densities in logic and type theory
... provable formulas in this language is surpassingly hight. Notice also that the classical tautologies in this language coincide with the intuitionistic ones. Theorem. 12. (see [11] page 592) For k = 1 the asymptotic density of the set of intuitionisticaly provable formulas T1→ exists and is: ...
... provable formulas in this language is surpassingly hight. Notice also that the classical tautologies in this language coincide with the intuitionistic ones. Theorem. 12. (see [11] page 592) For k = 1 the asymptotic density of the set of intuitionisticaly provable formulas T1→ exists and is: ...
Lecturecise 19 Proofs and Resolution Compactness for
... Thus, we see that the inductively proved statement holds even in this case. What the infinite formula D breaks is the second part, which, from the existence of interpretations that agree on an arbitrarily long finite prefix derives an interpretation for infinitely many variables. Indeed, this part e ...
... Thus, we see that the inductively proved statement holds even in this case. What the infinite formula D breaks is the second part, which, from the existence of interpretations that agree on an arbitrarily long finite prefix derives an interpretation for infinitely many variables. Indeed, this part e ...
Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory
... This paper presents a comparison of two axiomatic set theories over two non-classical logics. In particular, it suggests an interpretation of lattice-valued set theory as defined in [16] by S. Titani in fuzzy set theory as defined in [11] by authors of this paper. There are many different conception ...
... This paper presents a comparison of two axiomatic set theories over two non-classical logics. In particular, it suggests an interpretation of lattice-valued set theory as defined in [16] by S. Titani in fuzzy set theory as defined in [11] by authors of this paper. There are many different conception ...
cantor`s theory of transfinite integers
... of modern meta-mathematics during last decades confirms the validity of Cohen's pessimism. So, it is obviously that new ways are necessary here. One of such new ways, - a NON-metamathematical and NON-mathematical-logic way based on a so-called scientific cognitive computer visualization technique, - ...
... of modern meta-mathematics during last decades confirms the validity of Cohen's pessimism. So, it is obviously that new ways are necessary here. One of such new ways, - a NON-metamathematical and NON-mathematical-logic way based on a so-called scientific cognitive computer visualization technique, - ...