Complexity of Existential Positive First-Order Logic
... clear that T has polynomial running time, and that Φ is true in Γ if and only if there exists a computation of T on Φ that computes a sentence that is true in Γ . We now show that E X P OS(Γ ) is hard for CSP(Γ )NP under ≤m -reductions. Let L be a problem with a non-deterministic polynomial-time man ...
... clear that T has polynomial running time, and that Φ is true in Γ if and only if there exists a computation of T on Φ that computes a sentence that is true in Γ . We now show that E X P OS(Γ ) is hard for CSP(Γ )NP under ≤m -reductions. Let L be a problem with a non-deterministic polynomial-time man ...
A Simple Exposition of Gödel`s Theorem
... Instead of simply going for this negative conclusion, Gödel massaged truth, to represent it in formal logic so far as possible. Truth itself cannot be represented, but provability-according-tothe-rules-of-formal-logic can. What is a proof in formal logic? It is a sequence of well-formed formulae, st ...
... Instead of simply going for this negative conclusion, Gödel massaged truth, to represent it in formal logic so far as possible. Truth itself cannot be represented, but provability-according-tothe-rules-of-formal-logic can. What is a proof in formal logic? It is a sequence of well-formed formulae, st ...
1 Formal Languages
... Note: the term ‘symbol’ may be replaced by the term ‘character’, as used in many computer languages, such as Basic, Pascal, and C++. Notice in this connection that the term ‘string’ is also used in many computer languages, usually to mean string of characters. Let us not worry too much about exactly ...
... Note: the term ‘symbol’ may be replaced by the term ‘character’, as used in many computer languages, such as Basic, Pascal, and C++. Notice in this connection that the term ‘string’ is also used in many computer languages, usually to mean string of characters. Let us not worry too much about exactly ...
Local Normal Forms for First-Order Logic with Applications to
... d that hold in S) does not work. A counterexample is given by the set of clique graphs. For every d, the spheres of a graph consisting of one 2d-clique fulfil exactly the same formulas of quantifier depth at most d as those of a graph which consists of two disjoint d-cliques. Nevertheless, it turns ...
... d that hold in S) does not work. A counterexample is given by the set of clique graphs. For every d, the spheres of a graph consisting of one 2d-clique fulfil exactly the same formulas of quantifier depth at most d as those of a graph which consists of two disjoint d-cliques. Nevertheless, it turns ...
Paper - Christian Muise
... ♦i ψ1 , . . . , ♦i ψm , i χ1 , . . . , i χn ), we have that ϕ |= ⊥ iff at least one of the following: (a) γ |= ⊥; (b) ψj ∧ χ1 ∧ . . . ∧ χn |= ⊥ (for some j); or (c) for all ∆i -combinations φ of ϕ, for some j, φ ∧ ψj |= ⊥ The final point means all ∆i -combinations conflict with at least one ψj . P ...
... ♦i ψ1 , . . . , ♦i ψm , i χ1 , . . . , i χn ), we have that ϕ |= ⊥ iff at least one of the following: (a) γ |= ⊥; (b) ψj ∧ χ1 ∧ . . . ∧ χn |= ⊥ (for some j); or (c) for all ∆i -combinations φ of ϕ, for some j, φ ∧ ψj |= ⊥ The final point means all ∆i -combinations conflict with at least one ψj . P ...
Computing Default Extensions by Reductions on OR
... propriate n; for computing default logic extensions this is what we refer to as “the second step” above. The existence of a logical equivalence of this sort is guaranteed by the Modal Reduction Theorem, and algorithms for determining the equivalence can be extracted from proofs of that theorem. The ...
... propriate n; for computing default logic extensions this is what we refer to as “the second step” above. The existence of a logical equivalence of this sort is guaranteed by the Modal Reduction Theorem, and algorithms for determining the equivalence can be extracted from proofs of that theorem. The ...
PROVING UNPROVABILITY IN SOME NORMAL MODAL LOGIC
... 2) T is complete with respect to all finite reflexive intransitive trees. 3) K4.3 is complete with respect to all finite irreflexive (strict) linear orderings. (It is also the logic of < N, >>.) 4) S4Grz is complete with respect to all finite linear orderings. (It is also the logic of < N, ≥>.) 5) K ...
... 2) T is complete with respect to all finite reflexive intransitive trees. 3) K4.3 is complete with respect to all finite irreflexive (strict) linear orderings. (It is also the logic of < N, >>.) 4) S4Grz is complete with respect to all finite linear orderings. (It is also the logic of < N, ≥>.) 5) K ...
First-Order Logic
... Note: In F3 , ∀y is in the scope of ∀x, therefore the order of quantifiers must be · · · ∀x · · · ∀y · · · F3 ⇔ F and F30 ⇔ F Note: However G < F ...
... Note: In F3 , ∀y is in the scope of ∀x, therefore the order of quantifiers must be · · · ∀x · · · ∀y · · · F3 ⇔ F and F30 ⇔ F Note: However G < F ...
Arithmetic Series
... • So to find a50 I need to take d, which is 3, and add it to my a1, which is 2, 49 times. That’s a lot of adding. • But if we think back to elementary school, repetitive adding is just multiplication. ...
... • So to find a50 I need to take d, which is 3, and add it to my a1, which is 2, 49 times. That’s a lot of adding. • But if we think back to elementary school, repetitive adding is just multiplication. ...
A Resolution Method for Modal Logic S5
... an agent knows A then she knows that she knows A (the positive introspection axiom). Given an S5 formula A, the set of subformulæ of A is defined inductively as follows: • A is a subformula of A; • if B C is a subformula of A, then so are B and C, for = ∧, ∨, →; • if B is a subformula of A, the ...
... an agent knows A then she knows that she knows A (the positive introspection axiom). Given an S5 formula A, the set of subformulæ of A is defined inductively as follows: • A is a subformula of A; • if B C is a subformula of A, then so are B and C, for = ∧, ∨, →; • if B is a subformula of A, the ...
Guarded fragments with constants - Institute for Logic, Language
... returns a formula α0 using variables in {z1 , . . . , zw }, where z1 , . . . , zw do not appear in α. By renaming translation we mean a translation that only renames the variables of the input formula. We do the proof by induction on the quantifier depth qd(α) in the input formula α. If qd(α) = 0, t ...
... returns a formula α0 using variables in {z1 , . . . , zw }, where z1 , . . . , zw do not appear in α. By renaming translation we mean a translation that only renames the variables of the input formula. We do the proof by induction on the quantifier depth qd(α) in the input formula α. If qd(α) = 0, t ...
вдгжеиз © ¢ on every class of ordered finite struc
... on every class of ordered finite structures, that is to say, if is a class of ordered finite structures, then the class of polynomial-time computable queries on coincides with the class of queries definable in least fixedpoint logic on . Least fixed-point logic LFP is the extension of first- ...
... on every class of ordered finite structures, that is to say, if is a class of ordered finite structures, then the class of polynomial-time computable queries on coincides with the class of queries definable in least fixedpoint logic on . Least fixed-point logic LFP is the extension of first- ...
Completed Notes
... f (n) = value of the term you are looking for (nth term) f (n 1) = value of the previous term d = common difference Explicit Formula: An explicit formula is used if you don’t know the previous term’s value and/or you want to find a term far out in the sequence (ex: you want to find the value of th ...
... f (n) = value of the term you are looking for (nth term) f (n 1) = value of the previous term d = common difference Explicit Formula: An explicit formula is used if you don’t know the previous term’s value and/or you want to find a term far out in the sequence (ex: you want to find the value of th ...
Assumption/guarantee specifications in linear-time
... which will be abbreviated as g + cp. A formula cp is oalid, denoted k cp (or simply cp when it is clear that validity is intended), if cp is satisfied by every sequence. Quantification deserves special attention. Each variable is either rigid (having the same interpretation in all states of a sequen ...
... which will be abbreviated as g + cp. A formula cp is oalid, denoted k cp (or simply cp when it is clear that validity is intended), if cp is satisfied by every sequence. Quantification deserves special attention. Each variable is either rigid (having the same interpretation in all states of a sequen ...
Blank Notes
... f (n) = value of the term you are looking for (nth term) f (n 1) = value of the previous term d = common difference Explicit Formula: An explicit formula is used if you don’t know the previous term’s value and/or you want to find a term far out in the sequence (ex: you want to find the value of th ...
... f (n) = value of the term you are looking for (nth term) f (n 1) = value of the previous term d = common difference Explicit Formula: An explicit formula is used if you don’t know the previous term’s value and/or you want to find a term far out in the sequence (ex: you want to find the value of th ...
Propositional Logic .
... Databases – query languages Programming languages (e.g. prolog) Design Validation and verification AI (e.g. inference systems) ...
... Databases – query languages Programming languages (e.g. prolog) Design Validation and verification AI (e.g. inference systems) ...
Eliminating past operators in Metric Temporal Logic
... the above modalities are of the form pUI q, pSI q, or pSq, where p and q are propositions. We call this the “flat” or “non-recursive” version of MTL. The idea we use is quite simple: to flatten UI formulas for example, we introduce new propositions p0 and p1 for each subformula of the form ϕUI ψ, re ...
... the above modalities are of the form pUI q, pSI q, or pSq, where p and q are propositions. We call this the “flat” or “non-recursive” version of MTL. The idea we use is quite simple: to flatten UI formulas for example, we introduce new propositions p0 and p1 for each subformula of the form ϕUI ψ, re ...
MUltseq: a Generic Prover for Sequents and Equations*
... logics. This means that it takes as input the rules of a many-valued sequent calculus as well as a many-sided sequent and searches – automatically or interactively – for a proof of the latter. For the sake of readability, the output of MUltseq is typeset as a LATEX document. Though the sequent rules ...
... logics. This means that it takes as input the rules of a many-valued sequent calculus as well as a many-sided sequent and searches – automatically or interactively – for a proof of the latter. For the sake of readability, the output of MUltseq is typeset as a LATEX document. Though the sequent rules ...
Classicality as a Property of Predicate Symbols
... that one of the two components contain predicate symbols from D only. A⊃B ~ ¬B⊃¬A - holds if all symbols from B belong to D ¬(A&B) ~ ¬Α∨¬Β - holds if all symbols from A or all symbols from B belong to D ¬(A⊃B) ~ A&¬B - holds if all symbols from A belong to D A⊃B ~ ¬A∨B - holds if all symbols from A ...
... that one of the two components contain predicate symbols from D only. A⊃B ~ ¬B⊃¬A - holds if all symbols from B belong to D ¬(A&B) ~ ¬Α∨¬Β - holds if all symbols from A or all symbols from B belong to D ¬(A⊃B) ~ A&¬B - holds if all symbols from A belong to D A⊃B ~ ¬A∨B - holds if all symbols from A ...
Disjunctive Normal Form
... A direct proof of a conditional statement p q is constructed when the first step is the assumption that p is true, subsequent steps using rules of inference, with the final step showing q must also be true. Indirect proof – if we prove the theorem without starting with the premises and end with th ...
... A direct proof of a conditional statement p q is constructed when the first step is the assumption that p is true, subsequent steps using rules of inference, with the final step showing q must also be true. Indirect proof – if we prove the theorem without starting with the premises and end with th ...
Logic - Decision Procedures
... Databases – query languages Programming languages (e.g. prolog) Design Validation and verification AI (e.g. inference systems) ...
... Databases – query languages Programming languages (e.g. prolog) Design Validation and verification AI (e.g. inference systems) ...
PHIL12A Section answers, 9 February 2011
... 2. How many different ternary sentential connectives are there? How did you arrive at this number? You should not try to list them all! We calculate the number of ternary connectives in the same way as we calculated the number of binary connectives in the last question. A truth table for a ternary ...
... 2. How many different ternary sentential connectives are there? How did you arrive at this number? You should not try to list them all! We calculate the number of ternary connectives in the same way as we calculated the number of binary connectives in the last question. A truth table for a ternary ...
Adding the Everywhere Operator to Propositional Logic (pdf file)
... metavariables have been replaced by formula variables. The inference rules of C include those of C (even to requiring that, in Textual Substitution, γ and β be concrete). Inference rule Uniform Substitution is used only for replacing formula variables: αβγ denotes a copy of the formula denoted by α ...
... metavariables have been replaced by formula variables. The inference rules of C include those of C (even to requiring that, in Textual Substitution, γ and β be concrete). Inference rule Uniform Substitution is used only for replacing formula variables: αβγ denotes a copy of the formula denoted by α ...
A A S x
... that |= x A(x) x A(x): negating the formula: x A(x) x A(x), Renaming variables: x A(x) y A(y), Moving quantifiers to the left: x [A(x) y A(y)] x y [A(x) A(y)], and Skolemize: x [A(x) A(f(x))]. But the terms x and f(x) cannot be unified! Thus we did not prove the above t ...
... that |= x A(x) x A(x): negating the formula: x A(x) x A(x), Renaming variables: x A(x) y A(y), Moving quantifiers to the left: x [A(x) y A(y)] x y [A(x) A(y)], and Skolemize: x [A(x) A(f(x))]. But the terms x and f(x) cannot be unified! Thus we did not prove the above t ...
.pdf
... which all occurrences (even those within the scope of 2 ) of the variables of v are replaced by the formulas denoted by the corresponding variables of . This method for eliminating axiom schemes does not work in the case of Schematic C of Table 2, because (17) does not preserve C-validity. For exa ...
... which all occurrences (even those within the scope of 2 ) of the variables of v are replaced by the formulas denoted by the corresponding variables of . This method for eliminating axiom schemes does not work in the case of Schematic C of Table 2, because (17) does not preserve C-validity. For exa ...