Incompleteness in the finite domain
... terms. The “finite domain” in the title refers to the fact that the lengths of computations and lengths of proofs of instances of the problems that we consider are at most exponential, hence there is a finite bound on them. Perhaps, a more precise term would be “exponential domain”. In previous pres ...
... terms. The “finite domain” in the title refers to the fact that the lengths of computations and lengths of proofs of instances of the problems that we consider are at most exponential, hence there is a finite bound on them. Perhaps, a more precise term would be “exponential domain”. In previous pres ...
Abella: A System for Reasoning about Relational Specifications
... part of his doctoral work carried out at the University of Minnesota [19]. Kaustuv Chaudhuri and Yuting Wang have subsequently designed and implemented extensions to the system, resulting in an updated release. The various authors of this tutorial and several of their colleagues have been involved i ...
... part of his doctoral work carried out at the University of Minnesota [19]. Kaustuv Chaudhuri and Yuting Wang have subsequently designed and implemented extensions to the system, resulting in an updated release. The various authors of this tutorial and several of their colleagues have been involved i ...
proof terms for classical derivations
... Variables annotate assumptions, and the term constructors of pairing and λ-abstraction correspond to the introduction of conjunctions and conditionals respectively. Now the terms corresponding to the proofs bear the marks of the different proof behaviour. The first proof took an assumption p to p ∧ ...
... Variables annotate assumptions, and the term constructors of pairing and λ-abstraction correspond to the introduction of conjunctions and conditionals respectively. Now the terms corresponding to the proofs bear the marks of the different proof behaviour. The first proof took an assumption p to p ∧ ...
Chu Spaces - Stanford University
... It is convenient to view Chu spaces as organized either by rows or by columns. For the former, we define r̂ : A → (X → Σ) as r̂(a)(x) = r(a, x), and refer to the function r̂(a) : X → Σ as row a of A. Dually we define ř : X → (A → Σ) as ř(x)(a) = r(a, x) and call ř(x) : A → Σ column x of A. When r ...
... It is convenient to view Chu spaces as organized either by rows or by columns. For the former, we define r̂ : A → (X → Σ) as r̂(a)(x) = r(a, x), and refer to the function r̂(a) : X → Σ as row a of A. Dually we define ř : X → (A → Σ) as ř(x)(a) = r(a, x) and call ř(x) : A → Σ column x of A. When r ...
ppt - UBC Computer Science
... strategies for a given theorem, including o all those listed in the "pre-class" learning goals o logical equivalences, o propositional rules of inference o rules of inference on quantifiers i.e. be able to apply the strategies listed in the Guide to Proof Strategies reference sheet on the course web ...
... strategies for a given theorem, including o all those listed in the "pre-class" learning goals o logical equivalences, o propositional rules of inference o rules of inference on quantifiers i.e. be able to apply the strategies listed in the Guide to Proof Strategies reference sheet on the course web ...
George VOUTSADAKIS CATEGORICAL ABSTRACT ALGEBRAIC
... Around the end of the nineteenth and the beginning of the last century, several authors considered the process of coordinatization of various axiomatically defined abstract geometries. Among them were the affine plane geometry, affine plane geometry with the Desargues and Pappus properties, projecti ...
... Around the end of the nineteenth and the beginning of the last century, several authors considered the process of coordinatization of various axiomatically defined abstract geometries. Among them were the affine plane geometry, affine plane geometry with the Desargues and Pappus properties, projecti ...
Sample pages 2 PDF
... Example We can write ‘If it is raining then I am drenched’ as ‘It is raining’ ⇒ ‘I am drenched’. If it is raining, but I am inside and thus not wet, this proposition is false. If it is raining and I am having a shower, then it is true. If it is not raining, then it is true whether or not I am wet. A ...
... Example We can write ‘If it is raining then I am drenched’ as ‘It is raining’ ⇒ ‘I am drenched’. If it is raining, but I am inside and thus not wet, this proposition is false. If it is raining and I am having a shower, then it is true. If it is not raining, then it is true whether or not I am wet. A ...
Mathematical Logic
... Example. (P → Q) → R ∧ ∀xR0 (x) has as s.p.p.’s the whole formula, R ∧ ∀xR0 (x), R, ∀xR0 (x), R0 (t). The positive subformulas are the s.p.p.’s and in addition P ; the negative subformulas are P → Q, Q. 2. Natural Deduction We introduce Gentzen’s system of natural deduction. To allow a direct corres ...
... Example. (P → Q) → R ∧ ∀xR0 (x) has as s.p.p.’s the whole formula, R ∧ ∀xR0 (x), R, ∀xR0 (x), R0 (t). The positive subformulas are the s.p.p.’s and in addition P ; the negative subformulas are P → Q, Q. 2. Natural Deduction We introduce Gentzen’s system of natural deduction. To allow a direct corres ...
Lecture Slides
... strategies for a given theorem, including o all those listed in the "pre-class" learning goals o logical equivalences, o propositional rules of inference o rules of inference on quantifiers i.e. be able to apply the strategies listed in the Guide to Proof Strategies reference sheet on the course web ...
... strategies for a given theorem, including o all those listed in the "pre-class" learning goals o logical equivalences, o propositional rules of inference o rules of inference on quantifiers i.e. be able to apply the strategies listed in the Guide to Proof Strategies reference sheet on the course web ...
Prime Implicates and Prime Implicants: From Propositional to Modal
... languages, one naturally wonders whether these notions can be suitably generalized to these more expressive logics. We recall that in propositional logic the prime implicates of a formula are defined to be its logically strongest clausal consequences. The restriction to clauses is made in order to r ...
... languages, one naturally wonders whether these notions can be suitably generalized to these more expressive logics. We recall that in propositional logic the prime implicates of a formula are defined to be its logically strongest clausal consequences. The restriction to clauses is made in order to r ...
Full Text - Institute for Logic, Language and Computation
... natural language and not in mathematics. But GoodmanÕs goal is also to show the difficulties of the program of the empiricist logicians (especially Carnap): reducing all scientific language to a formal one. The problem is that the notion of disposition necessitates an analysis of counterfactual cond ...
... natural language and not in mathematics. But GoodmanÕs goal is also to show the difficulties of the program of the empiricist logicians (especially Carnap): reducing all scientific language to a formal one. The problem is that the notion of disposition necessitates an analysis of counterfactual cond ...
full text (.pdf)
... From a practical standpoint, many simple program manipulations such as loop unwinding and basic safety analysis do not require the full power of PDL, but can be carried out in a purely equational subsystem using the axioms of Kleene algebra. However, tests are an essential ingredient for modeling r ...
... From a practical standpoint, many simple program manipulations such as loop unwinding and basic safety analysis do not require the full power of PDL, but can be carried out in a purely equational subsystem using the axioms of Kleene algebra. However, tests are an essential ingredient for modeling r ...
Annals of Pure and Applied Logic Commutative integral bounded
... (see, for instance, [2,17]). The next lemma shows that for residuated lattices, (1.11) implies (1.12) without assuming the distributivity of the underlying lattice. Lemma 1.6. A stonean residuated lattice A satisfies Eq. (1.12). Proof. Observe first that by (1.5), for any pair of elements of a resid ...
... (see, for instance, [2,17]). The next lemma shows that for residuated lattices, (1.11) implies (1.12) without assuming the distributivity of the underlying lattice. Lemma 1.6. A stonean residuated lattice A satisfies Eq. (1.12). Proof. Observe first that by (1.5), for any pair of elements of a resid ...
An Institution-Independent Generalization of Tarski`s Elementary
... “theory” only the set E, and not the whole pair (Σ, E). A presentation morphism ϕ : (Σ, E) → (Σ 0 , E 0 ) is a signature morphism ϕ : Σ → Σ 0 such that ϕ(E) ⊆ E 0• . A presentation morphism between theories is called theory morphism. For a presentation (Σ, E), we let Mod(Σ, E) denote the category of ...
... “theory” only the set E, and not the whole pair (Σ, E). A presentation morphism ϕ : (Σ, E) → (Σ 0 , E 0 ) is a signature morphism ϕ : Σ → Σ 0 such that ϕ(E) ⊆ E 0• . A presentation morphism between theories is called theory morphism. For a presentation (Σ, E), we let Mod(Σ, E) denote the category of ...
JUXTAPOSITION - Brown University
... §1. Introduction. Methods of combining logics are of great interest.1 Formal systems that result from the combination of multiple logical systems into a single system have applications in mathematics, linguistics, and computer science. For example, there are many applications for logics with multipl ...
... §1. Introduction. Methods of combining logics are of great interest.1 Formal systems that result from the combination of multiple logical systems into a single system have applications in mathematics, linguistics, and computer science. For example, there are many applications for logics with multipl ...
INDEX SETS FOR n-DECIDABLE STRUCTURES CATEGORICAL
... they showed that the index set of computable categorical structures is Π11 -complete. Combining the methods from [2] and from [10], Bazhenov, Goncharov and Marchuk showed that also the index set of computable structures of algorithmic dimension n > 1 is Π11 complete [18]. On the other hand, the inde ...
... they showed that the index set of computable categorical structures is Π11 -complete. Combining the methods from [2] and from [10], Bazhenov, Goncharov and Marchuk showed that also the index set of computable structures of algorithmic dimension n > 1 is Π11 complete [18]. On the other hand, the inde ...
Nominal Monoids
... nominal sets were used to prove independence of the axiom of choice, and other axioms. In Computer Science, they have been rediscovered by Gabbay and Pitts in [7], as an elegant formalism for modeling name binding. Since then, nominal sets have become a lively topic in semantics. They were also inde ...
... nominal sets were used to prove independence of the axiom of choice, and other axioms. In Computer Science, they have been rediscovered by Gabbay and Pitts in [7], as an elegant formalism for modeling name binding. Since then, nominal sets have become a lively topic in semantics. They were also inde ...
a PDF file of the textbook - U of L Class Index
... The first of these deductions is very famous, but the second one is lame. It may seem odd to even call it a deduction, because the two hypotheses have nothing at all to do with the conclusion, but, given our definition, it does count as a deduction. However, it is is a very poor one, so it cannot be ...
... The first of these deductions is very famous, but the second one is lame. It may seem odd to even call it a deduction, because the two hypotheses have nothing at all to do with the conclusion, but, given our definition, it does count as a deduction. However, it is is a very poor one, so it cannot be ...
Preferences and Unrestricted Rebut
... as well as in the argumentation version of Default Logic [3,8,17] and ASPIC+[12]. When it comes to reasoning with strict and defeasible rules, one of the landmark papers is the work of Caminada and Amgoud [2], in which a number of rationality postulates are specified: direct consistency, indirect co ...
... as well as in the argumentation version of Default Logic [3,8,17] and ASPIC+[12]. When it comes to reasoning with strict and defeasible rules, one of the landmark papers is the work of Caminada and Amgoud [2], in which a number of rationality postulates are specified: direct consistency, indirect co ...
On perturbations of continuous structures - HAL
... Abstract. We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately ω-saturated up to arbitrarily small ...
... Abstract. We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately ω-saturated up to arbitrarily small ...
The Premiss-Based Approach to Logical Aggregation Franz Dietrich & Philippe Mongin
... natural approach consists in singling out those propositions - to be called premisses - which are in some sense basic to the other judgments, aggregating them in some way, and then see what eventually results for the others. We will assume that if a formula p represents a premiss, a decision on p is ...
... natural approach consists in singling out those propositions - to be called premisses - which are in some sense basic to the other judgments, aggregating them in some way, and then see what eventually results for the others. We will assume that if a formula p represents a premiss, a decision on p is ...
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
... Abstract. We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately ω-saturated up to arbitrarily small ...
... Abstract. We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately ω-saturated up to arbitrarily small ...