CHAPTER 5 SOME EXTENSIONAL SEMANTICS
... If T is the only designated value, the third value ⊥ corresponds to some notion of incomplete information, like undefined or unknown and is often denoted by the symbol U or I. If, on the other hand, ⊥ corresponds to inconsistent information, i.e. its meaning is something like known to be both true a ...
... If T is the only designated value, the third value ⊥ corresponds to some notion of incomplete information, like undefined or unknown and is often denoted by the symbol U or I. If, on the other hand, ⊥ corresponds to inconsistent information, i.e. its meaning is something like known to be both true a ...
Speaking Logic - SRI International
... pigeons and three holes. Write a propositional formula for checking that a given finite automaton hQ, Σ, q, F , δi with alphabet Σ, set of states S, initial state q, set of final states F , and transition function δ from hQ, Σi to Q accepts some string of length 5. Formalize the statement that a gra ...
... pigeons and three holes. Write a propositional formula for checking that a given finite automaton hQ, Σ, q, F , δi with alphabet Σ, set of states S, initial state q, set of final states F , and transition function δ from hQ, Σi to Q accepts some string of length 5. Formalize the statement that a gra ...
Propositional Logic: Why? soning Starts with George Boole around 1850
... This is the syntax of the symbolic languages of propositional logic 1. “If student takes Analysis of Algorithms, then it takes 1805” can be represented by (r → p) 2. “Student takes Analysis of Algorithms or Advanced Programming”: (q ∨ r) 3. “Student takes Analysis of Algorithms or Advanced Programm ...
... This is the syntax of the symbolic languages of propositional logic 1. “If student takes Analysis of Algorithms, then it takes 1805” can be represented by (r → p) 2. “Student takes Analysis of Algorithms or Advanced Programming”: (q ∨ r) 3. “Student takes Analysis of Algorithms or Advanced Programm ...
monadic second order logic
... All variables in S1S0 are set variables, Xj Atomic formulas are of the form X ⊆Y and succ (X,Y ) X ⊆Y is true if X is a subset of Y Succ ( X,Y ) is true if X and Y are singletons {x } and {y } respectively and y = x +1 ...
... All variables in S1S0 are set variables, Xj Atomic formulas are of the form X ⊆Y and succ (X,Y ) X ⊆Y is true if X is a subset of Y Succ ( X,Y ) is true if X and Y are singletons {x } and {y } respectively and y = x +1 ...
3.3 Inference
... We then used the definition of even numbers, and our previous parenthetic comment suggests that it was natural for us to use the definition symbolically. The definition tells us that if m is an even number, then there exists another integer i such that m = 2i. We combined this with the assumption that ...
... We then used the definition of even numbers, and our previous parenthetic comment suggests that it was natural for us to use the definition symbolically. The definition tells us that if m is an even number, then there exists another integer i such that m = 2i. We combined this with the assumption that ...
full text (.pdf)
... Automated deduction systems such as NuPrl [4, 6] and Mizar [16] have language support for accumulating results in libraries for later reference. However, the mechanisms for providing this support are typically not considered interesting enough to formalize in the underlying logic, although it is pos ...
... Automated deduction systems such as NuPrl [4, 6] and Mizar [16] have language support for accumulating results in libraries for later reference. However, the mechanisms for providing this support are typically not considered interesting enough to formalize in the underlying logic, although it is pos ...
Modus ponens
... While modus ponens is one of the most commonly used concepts in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution". Modus ponens allows one to el ...
... While modus ponens is one of the most commonly used concepts in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution". Modus ponens allows one to el ...
Partial Correctness Specification
... These specifications are ‘partial’ because for {P } C {Q} to be true it is not necessary for the execution of C to terminate when started in a state satisfying P It is only required that if the execution terminates, then Q holds {X = 1} WHILE T DO X := X {Y = 2} – this specification is true! ...
... These specifications are ‘partial’ because for {P } C {Q} to be true it is not necessary for the execution of C to terminate when started in a state satisfying P It is only required that if the execution terminates, then Q holds {X = 1} WHILE T DO X := X {Y = 2} – this specification is true! ...
Deep Inference - Department of Computer Science
... Deep inference could succinctly be described as an extreme form of linear logic [12]. It is a methodology for designing proof formalisms that generalise Gentzen formalisms, i.e. the sequent calculus and natural deduction [11]. In a sense, deep inference is obtained by applying some of the main conce ...
... Deep inference could succinctly be described as an extreme form of linear logic [12]. It is a methodology for designing proof formalisms that generalise Gentzen formalisms, i.e. the sequent calculus and natural deduction [11]. In a sense, deep inference is obtained by applying some of the main conce ...
Propositional logic, I
... If KB has the value True in the real world, then any sentence derived from KB by a sound inference procedure has also the value True in the real world. » Completeness of the inference mechanism is also ...
... If KB has the value True in the real world, then any sentence derived from KB by a sound inference procedure has also the value True in the real world. » Completeness of the inference mechanism is also ...
Lesson 12
... There is a subtle difference between entailment and inference. Version 2 CSE IIT, Kharagpur ...
... There is a subtle difference between entailment and inference. Version 2 CSE IIT, Kharagpur ...
Slides - AI-MAS
... Models are abstract mathematical structures that provide possible interpretations for each of the non-logical objects in a formal language. Given a model for a language - define what it is for a sentence in that language to be true (according to that model) or not. In any model in which the pr ...
... Models are abstract mathematical structures that provide possible interpretations for each of the non-logical objects in a formal language. Given a model for a language - define what it is for a sentence in that language to be true (according to that model) or not. In any model in which the pr ...
Autoepistemic Logic and Introspective Circumscription
... Thus, technically, the two systems appear to be quite different, and introspective circumscription, the younger and less known of the two, may have important advantages. The ease with which it handles quantification and equality is, in particular, of interest to logic programming. Since autoepistemi ...
... Thus, technically, the two systems appear to be quite different, and introspective circumscription, the younger and less known of the two, may have important advantages. The ease with which it handles quantification and equality is, in particular, of interest to logic programming. Since autoepistemi ...
Propositional logic, I (Lógica Proposicional, I)
... Validity: A wff is said to be valid if it has the value True under all possible interpretations. Ex. T,T∨P,¬P∨P,P⇒P,P⇒(Q⇒P),((P⇒ Q)⇒P)⇒P » A valid wff is a tautology (it is devoid of meaning about the world). Metatheorem 1: if ¬w is unsatisfiable, then the wff w is valid, and viceversa. ...
... Validity: A wff is said to be valid if it has the value True under all possible interpretations. Ex. T,T∨P,¬P∨P,P⇒P,P⇒(Q⇒P),((P⇒ Q)⇒P)⇒P » A valid wff is a tautology (it is devoid of meaning about the world). Metatheorem 1: if ¬w is unsatisfiable, then the wff w is valid, and viceversa. ...
Certamen 1 de Representación del Conocimiento
... 12 de Octubre, 2012 (a) [1/2 pto] Define a FOL signature S = {Ω, Π} for which formulas in Σ are well-formed. Solution: Ω = {A/0, B/0} and Π = {R/2, P/2} (b) [1/2 pto] Show that Σ is valid (provide an interpretation for S). Solution: Consider the interpretation I = (U, AI , B I , RI , P I ) where U = ...
... 12 de Octubre, 2012 (a) [1/2 pto] Define a FOL signature S = {Ω, Π} for which formulas in Σ are well-formed. Solution: Ω = {A/0, B/0} and Π = {R/2, P/2} (b) [1/2 pto] Show that Σ is valid (provide an interpretation for S). Solution: Consider the interpretation I = (U, AI , B I , RI , P I ) where U = ...
on fuzzy intuitionistic logic
... they m a y be t r u e 'in different ways'. By accepting different t r u t h values, we also break t h e true-false-dualism of classical logic. If we know t h e degree of t r u t h of a sentence we do not necessarily know t h e degree of falsehood of the sentence. In Fuzzy Intuitionistic Logic a half ...
... they m a y be t r u e 'in different ways'. By accepting different t r u t h values, we also break t h e true-false-dualism of classical logic. If we know t h e degree of t r u t h of a sentence we do not necessarily know t h e degree of falsehood of the sentence. In Fuzzy Intuitionistic Logic a half ...
Model Checking - Teaching-WIKI
... • A proof of a sequent is a proof tree whose nodes are sequents • The root is the sequent to be proven (the theorem) • For each sequent in the tree, all of its children are premises of some inference rule in which that sequent is a conclusion • A proof is complete when each sequent in the proof tree ...
... • A proof of a sequent is a proof tree whose nodes are sequents • The root is the sequent to be proven (the theorem) • For each sequent in the tree, all of its children are premises of some inference rule in which that sequent is a conclusion • A proof is complete when each sequent in the proof tree ...
Algebraic Proof Systems
... A proof system f1 polynomially simulates a proof system f2 , if there exists a polynomial time computable function g such that for all ā ∈ {0, 1}∗ , f1 (g (ā)) = f2 (ā). Meaning: Given a proof ā of f2 (ā) in the second system, we can construct a proof g (ā) of the same tautology in the first s ...
... A proof system f1 polynomially simulates a proof system f2 , if there exists a polynomial time computable function g such that for all ā ∈ {0, 1}∗ , f1 (g (ā)) = f2 (ā). Meaning: Given a proof ā of f2 (ā) in the second system, we can construct a proof g (ā) of the same tautology in the first s ...
characterization of classes of frames in modal language
... If a logic consists of K, φ → φ, φ → φ, grz, then it is characterized by the class of reflexive, transitive and antisymmetric Kripke frames which do not contain any infinite ascending chains of distinct points. S4 is valid in frames defined by grz. S4 laws in K ∪ grz were proved around 1979 by W. J ...
... If a logic consists of K, φ → φ, φ → φ, grz, then it is characterized by the class of reflexive, transitive and antisymmetric Kripke frames which do not contain any infinite ascending chains of distinct points. S4 is valid in frames defined by grz. S4 laws in K ∪ grz were proved around 1979 by W. J ...
PDF
... from the language Lc of PLc under this system. In Li , the logical connectives consist of →, ¬, ∧, ∨, whereas in Lc , only → is used. The other connectives are introduced as abbreviational devices: ¬A is A →⊥, A ∨ B is ¬A → B, and A ∧ B is ¬(A → ¬B). So it doesn’t make much sense to say that PLi < P ...
... from the language Lc of PLc under this system. In Li , the logical connectives consist of →, ¬, ∧, ∨, whereas in Lc , only → is used. The other connectives are introduced as abbreviational devices: ¬A is A →⊥, A ∨ B is ¬A → B, and A ∧ B is ¬(A → ¬B). So it doesn’t make much sense to say that PLi < P ...
Automata theory
... In this chapter we present a logical formalism for the declarative description of regular languages. We use logical formulas to describe properties of words, and logical operators to construct complex properties out of simpler ones. We then show how to automatically translate a formula describing a ...
... In this chapter we present a logical formalism for the declarative description of regular languages. We use logical formulas to describe properties of words, and logical operators to construct complex properties out of simpler ones. We then show how to automatically translate a formula describing a ...
Local Normal Forms for First-Order Logic with Applications to
... Hanf [Han65] and Gaifman [Gai82]. Hanf showed that, for every first-order formula ψ, there is an r such that whether ψ holds in a structure A (“A |= ψ”) only depends on the multiset of isomorphism types of all r-spheres in A. Here an r-sphere is a substructure of A which is induced by all elements o ...
... Hanf [Han65] and Gaifman [Gai82]. Hanf showed that, for every first-order formula ψ, there is an r such that whether ψ holds in a structure A (“A |= ψ”) only depends on the multiset of isomorphism types of all r-spheres in A. Here an r-sphere is a substructure of A which is induced by all elements o ...
A SHORT PROOF FOR THE COMPLETENESS OF
... and Wos extended it by a new inference rule called paramodulation. In [4] they proved that this extended calculus is sound and refutation complete over the so called E-interpretations (E refers to the special treatment of the equality symbol; see Chapter 8.2 of [2] for the details). In the present p ...
... and Wos extended it by a new inference rule called paramodulation. In [4] they proved that this extended calculus is sound and refutation complete over the so called E-interpretations (E refers to the special treatment of the equality symbol; see Chapter 8.2 of [2] for the details). In the present p ...
Lectures 12 to 14: Soundness and completeness
... V(E) = T for all valuations of constituent propositions Each ‘valuation’ is called a ‘model’. ...
... V(E) = T for all valuations of constituent propositions Each ‘valuation’ is called a ‘model’. ...
