Logic - Decision Procedures
... (3) I have not filed any of them that I can read; (4) None of them, that are written on one sheet, are undated; (5) All of them, that are not crossed, are in black ink; (6) All of them, written by Brown, begin with "Dear Sir"; (7) All of them, written on blue paper, are filed; (8) None of them, writ ...
... (3) I have not filed any of them that I can read; (4) None of them, that are written on one sheet, are undated; (5) All of them, that are not crossed, are in black ink; (6) All of them, written by Brown, begin with "Dear Sir"; (7) All of them, written on blue paper, are filed; (8) None of them, writ ...
predicate
... • Some people have more than one brother • x y1 y2 ( B(y1,x) B(y2,x) (y1 = y2) ...
... • Some people have more than one brother • x y1 y2 ( B(y1,x) B(y2,x) (y1 = y2) ...
Lecture 10. Model theory. Consistency, independence
... Soundness (of a logic): If ∆ has a model, then ∆ is consistent. Completeness (of a logic): If ∆ is consistent, then ∆ has a model. Because first-order logic is sound and complete, we can freely choose whether to give a semantic or syntactic argument of consistency or inconsistency. Suppose you are a ...
... Soundness (of a logic): If ∆ has a model, then ∆ is consistent. Completeness (of a logic): If ∆ is consistent, then ∆ has a model. Because first-order logic is sound and complete, we can freely choose whether to give a semantic or syntactic argument of consistency or inconsistency. Suppose you are a ...
Artificial Intelligence
... • To express the idea that everyone likes cheese, we might say: (∀x)(P(x)→L(x, C)) • The symbol ∀ is read “for all,” so the statement above could be read as “for every x it is true that if property P holds for x, then the relationship L holds between x and C,” or in plainer English: “every x that is ...
... • To express the idea that everyone likes cheese, we might say: (∀x)(P(x)→L(x, C)) • The symbol ∀ is read “for all,” so the statement above could be read as “for every x it is true that if property P holds for x, then the relationship L holds between x and C,” or in plainer English: “every x that is ...
Document
... 3. Accidentally Bound Variables and Miscellaneous Cases When a quantifier is dropped by UI or EI, all the variables thus freed must be uniformly replaced by free variables (or, in the case of UI, by free variables or constants). The rule UI and the third restriction on rule EI take care of this. ...
... 3. Accidentally Bound Variables and Miscellaneous Cases When a quantifier is dropped by UI or EI, all the variables thus freed must be uniformly replaced by free variables (or, in the case of UI, by free variables or constants). The rule UI and the third restriction on rule EI take care of this. ...
an interpretation of aristotle`s syllogistic and a certain fragment of set
... is aj ). This is not a syllogistic thesis. If, now, in some interpretation would be allowed using only at most k objects (i.e. k natural numbers in Leibnitz’s interpretation and k sets in Slupecki’s interpretation), then formula A would not be rejected. This can be seen especially well in Slupecki’s ...
... is aj ). This is not a syllogistic thesis. If, now, in some interpretation would be allowed using only at most k objects (i.e. k natural numbers in Leibnitz’s interpretation and k sets in Slupecki’s interpretation), then formula A would not be rejected. This can be seen especially well in Slupecki’s ...
A Revised Concept of Safety for General Answer Set Programs
... following relation of QEL to (ordinary) answer sets for logic programs with variables was established. If Π is a logic program, a total UNA-QHT model hU, T, T i of Π is an equilibrium model of Π iff it is an open answer set of Π. [7] provides a new definition of stable model for arbitrary first-orde ...
... following relation of QEL to (ordinary) answer sets for logic programs with variables was established. If Π is a logic program, a total UNA-QHT model hU, T, T i of Π is an equilibrium model of Π iff it is an open answer set of Π. [7] provides a new definition of stable model for arbitrary first-orde ...
Lindenbaum lemma for infinitary logics
... If ∆ and Π are singletons, we obtain the well known proof by cases property which in finitary logics coincides with its strong form but not in general it differs, see [2]. ...
... If ∆ and Π are singletons, we obtain the well known proof by cases property which in finitary logics coincides with its strong form but not in general it differs, see [2]. ...
full text (.pdf)
... 1. THE FIRST-ORDERTHEORYOF RANDOM STRUCTURES In this section, we review for future reference several known theorems, due to Gaifman (1964), Glebskii et al. (1969), Fagin (1976), and Grandjean (1982), concerning the first-order theory of random structures with infinite or large finite universes. We p ...
... 1. THE FIRST-ORDERTHEORYOF RANDOM STRUCTURES In this section, we review for future reference several known theorems, due to Gaifman (1964), Glebskii et al. (1969), Fagin (1976), and Grandjean (1982), concerning the first-order theory of random structures with infinite or large finite universes. We p ...
Syntax and Semantics of Propositional and Predicate Logic
... The set of symbols of the first three kinds—constants, functions and relations—comprise the vocabulary. The vocabulary in use may change from time to time. One may also specify non-alphabetic symbols for the vocabulary, e.g., 0, +, <, etc. Note, however, that they do not carry any pre-defined meanin ...
... The set of symbols of the first three kinds—constants, functions and relations—comprise the vocabulary. The vocabulary in use may change from time to time. One may also specify non-alphabetic symbols for the vocabulary, e.g., 0, +, <, etc. Note, however, that they do not carry any pre-defined meanin ...
The semantics of predicate logic
... n-place (or n-ary) relation. Further, a set of singletons is a one-place (or unary) relation, or predicate. A zero-place (or nullary) relation corresponds to a nullary predicate, which we use to model atomic propositions, and is therefore either the constant T or the constant F . ...
... n-place (or n-ary) relation. Further, a set of singletons is a one-place (or unary) relation, or predicate. A zero-place (or nullary) relation corresponds to a nullary predicate, which we use to model atomic propositions, and is therefore either the constant T or the constant F . ...
22.1 Representability of Functions in a Formal Theory
... ∼Rp (x,y). Fortunately, we are not required to give a formal proof for the validity of these formulas.2 Instead, we can use the Peano axioms within a semantical argument that the formulas are valid in every model of Peano Arithmetic. We proceed by induction over x. (1) For x=0 we know p(x)=0. Consid ...
... ∼Rp (x,y). Fortunately, we are not required to give a formal proof for the validity of these formulas.2 Instead, we can use the Peano axioms within a semantical argument that the formulas are valid in every model of Peano Arithmetic. We proceed by induction over x. (1) For x=0 we know p(x)=0. Consid ...
PDF
... Q: What kind of functions can be represented in Peano Arithmetic? Let us consider a few examples: • Obviously addition, successor, and multiplication can be represented in Peano arithmetic. – Addition + is represented by the predicate R+ with R+ (x,y,z) ≡ x+y=z. – The successor function s is represe ...
... Q: What kind of functions can be represented in Peano Arithmetic? Let us consider a few examples: • Obviously addition, successor, and multiplication can be represented in Peano arithmetic. – Addition + is represented by the predicate R+ with R+ (x,y,z) ≡ x+y=z. – The successor function s is represe ...
Predicate Logic
... • Representing knowledge using logic is appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new state ...
... • Representing knowledge using logic is appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new state ...
Chapter 2
... + fire + retire). One of the nicest features of regular languages is that they have a dual characterization using fsa and regular expressions. Indeed, Kleene’s theorem says that a language L is regular iff it can be specified by a regular expression. There are several important variations of fsa tha ...
... + fire + retire). One of the nicest features of regular languages is that they have a dual characterization using fsa and regular expressions. Indeed, Kleene’s theorem says that a language L is regular iff it can be specified by a regular expression. There are several important variations of fsa tha ...
PDF
... where ∆ and Γ are finite sequences of formulas in L. The empty sequence is allowed, and is usually denoted by ∅, λ, or blank space. In the sequent above, ∆ is called the antecedent, and Γ the succedent. A formula in a sequent is a formula that occurs either in the antecedent or the succedent of the ...
... where ∆ and Γ are finite sequences of formulas in L. The empty sequence is allowed, and is usually denoted by ∅, λ, or blank space. In the sequent above, ∆ is called the antecedent, and Γ the succedent. A formula in a sequent is a formula that occurs either in the antecedent or the succedent of the ...
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 ...
Lecture 10 Notes
... philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important factors in the technology of knowledge creation. For PC we have a clear computational semantics for understanding the logical ...
... philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important factors in the technology of knowledge creation. For PC we have a clear computational semantics for understanding the logical ...
Propositional Logic First Order Logic
... (3) I have not filed any of them that I can read; (4) None of them, that are written on one sheet, are undated; (5) All of them, that are not crossed, are in black ink; (6) All of them, written by Brown, begin with "Dear Sir"; (7) All of them, written on blue paper, are filed; (8) None of them, writ ...
... (3) I have not filed any of them that I can read; (4) None of them, that are written on one sheet, are undated; (5) All of them, that are not crossed, are in black ink; (6) All of them, written by Brown, begin with "Dear Sir"; (7) All of them, written on blue paper, are filed; (8) None of them, writ ...
First-Order Predicate Logic (2) - Department of Computer Science
... We can rewrite Qterm into a program Aterm that terminates for input P ] if Qterm outputs “no” for input P ] . Otherwise it does not terminate. Then • Aterm terminates for input A]term if, and only if (by def. of Aterm ) • Qterm outputs “no” for input A]term if, and only if (by def. of Qterm ) • Ater ...
... We can rewrite Qterm into a program Aterm that terminates for input P ] if Qterm outputs “no” for input P ] . Otherwise it does not terminate. Then • Aterm terminates for input A]term if, and only if (by def. of Aterm ) • Qterm outputs “no” for input A]term if, and only if (by def. of Qterm ) • Ater ...
Overview of proposition and predicate logic Introduction
... Below ϕ, ψ, χ are formulas in the language of proposition logic, i.e., they are constructed from elementary propositions p, q, r, . . ., and from the logical connectives ¬, ∧, ∨, →. For every connective there are one or two introduction rules (on the left) and elimination rules (on the right). We wi ...
... Below ϕ, ψ, χ are formulas in the language of proposition logic, i.e., they are constructed from elementary propositions p, q, r, . . ., and from the logical connectives ¬, ∧, ∨, →. For every connective there are one or two introduction rules (on the left) and elimination rules (on the right). We wi ...
First-Order Logic
... N = {P(a), ¬P(x) ∨ P(f (x)), Q(y , z), ¬P(f (f (x)))} By convention, ∀-quantifiers are not written. An explicitly quantified formula can be restored by first connecting the clauses by ∧ and then ∀-quantifying over all variables, or the other way round. ∀x. ∀y . ∀z. (P(a) ∧ (¬P(x) ∨ P(f (x))) ∧ Q(y , ...
... N = {P(a), ¬P(x) ∨ P(f (x)), Q(y , z), ¬P(f (f (x)))} By convention, ∀-quantifiers are not written. An explicitly quantified formula can be restored by first connecting the clauses by ∧ and then ∀-quantifying over all variables, or the other way round. ∀x. ∀y . ∀z. (P(a) ∧ (¬P(x) ∨ P(f (x))) ∧ Q(y , ...
Cocktail
... verification conditions for a given program extended with some annotations (directly or indirectly) Proving the verification condition(s) yields higher reliability of the program Using tools like ESC/Java helps to find common bugs in software otherwise unnoticed ...
... verification conditions for a given program extended with some annotations (directly or indirectly) Proving the verification condition(s) yields higher reliability of the program Using tools like ESC/Java helps to find common bugs in software otherwise unnoticed ...
Knowledge Representation
... • These are joined by logical connectives (and, or, implication) e.g., P Q; Q R • Given some statements in the logic we can deduce new facts (e.g., from above deduce R) ...
... • These are joined by logical connectives (and, or, implication) e.g., P Q; Q R • Given some statements in the logic we can deduce new facts (e.g., from above deduce R) ...