First-order logic syntax and semantics
... The distinction between free and bound variables resembles the distinction between local and global variables in a procedure ...
... The distinction between free and bound variables resembles the distinction between local and global variables in a procedure ...
Full version - Villanova Computer Science
... A sequent is any finite sequence of formulas (in the sense of Slide 4.7). Here are the rules of one of several equivalent sequent calculus systems for classical propositional logic. Let us call it G1. In the following rules, E,F stand for any formulas, and G,H stand for any sequents. Above the horiz ...
... A sequent is any finite sequence of formulas (in the sense of Slide 4.7). Here are the rules of one of several equivalent sequent calculus systems for classical propositional logic. Let us call it G1. In the following rules, E,F stand for any formulas, and G,H stand for any sequents. Above the horiz ...
Proof Theory in Type Theory
... The starting motivation of this work was a comparison between the work of Tait/Schütte and the work of Lorenzen/Novikov on cut-elimination. From a constructive perspective, the analysis of sequent calculus is most elegantly expressed in Lorenzen/Novikov’s way, as the admissibility of the cut rule [ ...
... The starting motivation of this work was a comparison between the work of Tait/Schütte and the work of Lorenzen/Novikov on cut-elimination. From a constructive perspective, the analysis of sequent calculus is most elegantly expressed in Lorenzen/Novikov’s way, as the admissibility of the cut rule [ ...
Chapter 9: Quantified Formulas
... an exponential growth in the formula size (see Sect. 1.16), it is preferable to have a projection that works directly on the CNF, or better yet, on a general Boolean formula. We consider two techniques: binary resolution (see Definition 2.11), which works directly on CNF formulas, and expansion. Pro ...
... an exponential growth in the formula size (see Sect. 1.16), it is preferable to have a projection that works directly on the CNF, or better yet, on a general Boolean formula. We consider two techniques: binary resolution (see Definition 2.11), which works directly on CNF formulas, and expansion. Pro ...
Precalc Unit 10 Review Name
... can see that the first term, , is –7 and the common difference, d, is 1.4. Substitute these values into the formula: . Simplify the formula to ...
... can see that the first term, , is –7 and the common difference, d, is 1.4. Substitute these values into the formula: . Simplify the formula to ...
Comments on predicative logic
... The syntax of intuitionistic propositional second-order logic PSOLi consists of a set of propositional constants P , Q, R, etc, a denumerable set of propositional variables F , G, H, etc, together with the primitive logical signs of the conditional and the universal propositional quantifier, and pon ...
... The syntax of intuitionistic propositional second-order logic PSOLi consists of a set of propositional constants P , Q, R, etc, a denumerable set of propositional variables F , G, H, etc, together with the primitive logical signs of the conditional and the universal propositional quantifier, and pon ...
Chapter 1
... may be named by enclosing it in single quotes. (The latter convention may be more convenient for students should they wish to distinguish between formulas and formulanames in their own writing.) We have said that the alphabet of our object language is comprised of "symbols" and that its formulas are ...
... may be named by enclosing it in single quotes. (The latter convention may be more convenient for students should they wish to distinguish between formulas and formulanames in their own writing.) We have said that the alphabet of our object language is comprised of "symbols" and that its formulas are ...
Yakir-Vizel-Lecture1-Intro_to_SMT
... Use Partial Assignment • Consider a formula that has (10 ≤ x) and (x < 0) (represented by v and u respectively) – v and u are both assigned to TRUE – Every call to the T-Solver results in UnSAT • May also be due to other reasons ...
... Use Partial Assignment • Consider a formula that has (10 ≤ x) and (x < 0) (represented by v and u respectively) – v and u are both assigned to TRUE – Every call to the T-Solver results in UnSAT • May also be due to other reasons ...
(P Q). - Snistnote
... Ex: either P Q or Q P is included but not both. For two variables P and Q , there are 22 such formulas given by P Q, P ~ Q , ~ P Q and ~ P ~ Q these formulas are called minterms. ...
... Ex: either P Q or Q P is included but not both. For two variables P and Q , there are 22 such formulas given by P Q, P ~ Q , ~ P Q and ~ P ~ Q these formulas are called minterms. ...
Sub-Birkhoff
... Lemma 8 For a subequational logic L = hS,Ii with (axiom),(congruence) ∈ I we have ` s L t iff s →IS t. Here →IS is the closure of the rewrite relation →S induced by S, under all elements of I. The numbers of the remarks below correspond to the numbers of the summarizing table above. i. Completeness ...
... Lemma 8 For a subequational logic L = hS,Ii with (axiom),(congruence) ∈ I we have ` s L t iff s →IS t. Here →IS is the closure of the rewrite relation →S induced by S, under all elements of I. The numbers of the remarks below correspond to the numbers of the summarizing table above. i. Completeness ...
2 Lab 2 – October 10th, 2016
... U is the set of real numbers, P is interpreted as the property “to be a square root of −1”. Since no real number has the property I(P ), our sentence is true in hU, [[−]]i. On the other hand, consider the interpretation: U 0 is the set of natural numbers, P is interpreted as the property “to be even ...
... U is the set of real numbers, P is interpreted as the property “to be a square root of −1”. Since no real number has the property I(P ), our sentence is true in hU, [[−]]i. On the other hand, consider the interpretation: U 0 is the set of natural numbers, P is interpreted as the property “to be even ...
handout - Homepages of UvA/FNWI staff
... allows us to restrict our attention to cut free proofs and this in turn allows us to prove some non-obvious properties of logics, such as the disjunction property for intuitionistic logic. In this chapter we will try to do something similar for natural deduction. The analogue of a cut free proof wil ...
... allows us to restrict our attention to cut free proofs and this in turn allows us to prove some non-obvious properties of logics, such as the disjunction property for intuitionistic logic. In this chapter we will try to do something similar for natural deduction. The analogue of a cut free proof wil ...
Syntax and Semantics of Propositional and Predicate Logic
... When doing formal logic, we use ordinary mathematical concepts and tools—functions, variables, deductions, etc. This creates a potential confusion: when we mention a mathematical notion, do we mean that we are using the notion, or that we are talking about it? For example, consider the variable ’x’. ...
... When doing formal logic, we use ordinary mathematical concepts and tools—functions, variables, deductions, etc. This creates a potential confusion: when we mention a mathematical notion, do we mean that we are using the notion, or that we are talking about it? For example, consider the variable ’x’. ...
predicate
... pairs of binary strings. Is there a sequence of indices i1,i2,…,in such that si1 sin = ti1 tin • Example • s1 = 1, s2 = 10, s3 = 011 • t1 = 101, t2 = 00, t3 = 11 ...
... pairs of binary strings. Is there a sequence of indices i1,i2,…,in such that si1 sin = ti1 tin • Example • s1 = 1, s2 = 10, s3 = 011 • t1 = 101, t2 = 00, t3 = 11 ...
(p q r) (p q r) (p q r) (p q r) ( p q r)
... Ex.: p q Complete elementary disjunction (CED) of a given set S of elementary propositional symbols is an elementary disjunction in which each symbol (element of S) occurs just once: Ex.: p q Disjunctive normal form (DNF) of a formula F is a formula F’ such that F’ is equivalent to F and F’ ha ...
... Ex.: p q Complete elementary disjunction (CED) of a given set S of elementary propositional symbols is an elementary disjunction in which each symbol (element of S) occurs just once: Ex.: p q Disjunctive normal form (DNF) of a formula F is a formula F’ such that F’ is equivalent to F and F’ ha ...
Section 8
... Example: Write out the first three terms and the last term, then find the sum (using the formula from this section): ...
... Example: Write out the first three terms and the last term, then find the sum (using the formula from this section): ...
Lesson 3
... Complete elementary disjunction (CED) of a given set S of elementary propositional symbols is an elementary disjunction in which each symbol (element of S) occurs just once: Ex.: p q Disjunctive normal form (DNF) of a formula F is a formula F’ such that F’ is equivalent to F and F’ has the form o ...
... Complete elementary disjunction (CED) of a given set S of elementary propositional symbols is an elementary disjunction in which each symbol (element of S) occurs just once: Ex.: p q Disjunctive normal form (DNF) of a formula F is a formula F’ such that F’ is equivalent to F and F’ has the form o ...
Automata for the modal µ-calculus and related results
... It was proved that this fixpoint language has the same expressive power as the monadic second order logic of n successors. Hence adding explicit conjunction to this language will not increase its expressive power. These considerations lead to the notion of disjunctive formulas which are formulas whe ...
... It was proved that this fixpoint language has the same expressive power as the monadic second order logic of n successors. Hence adding explicit conjunction to this language will not increase its expressive power. These considerations lead to the notion of disjunctive formulas which are formulas whe ...
A group of 3?
... A. Describe the pattern that allows you to find the next term in the sequence 2, 6, 18, 54, 162, … Write a recursive formula for the sequence. Pattern: Multiply a term by 3 to find the next term. Recursive Formula: an = an – 1 • 3, where a1 = 2. ...
... A. Describe the pattern that allows you to find the next term in the sequence 2, 6, 18, 54, 162, … Write a recursive formula for the sequence. Pattern: Multiply a term by 3 to find the next term. Recursive Formula: an = an – 1 • 3, where a1 = 2. ...
EQ: What is the formula to find the sum of an arithmetic sequence?
... n terms of an infinite sequence Ex 5. Find the 150th partial sum of the arithmetic sequence 5, 16, 27, 38, 49, … ...
... n terms of an infinite sequence Ex 5. Find the 150th partial sum of the arithmetic sequence 5, 16, 27, 38, 49, … ...
Warm-up Finding Terms of a Sequence
... What is a recursive sequence? Definition: A recursive sequence is the process in which each step of a pattern is dependent on the step or steps before it. ...
... What is a recursive sequence? Definition: A recursive sequence is the process in which each step of a pattern is dependent on the step or steps before it. ...
Chapter 9
... Typically: we first do the proof for Case 1 where = and function symbols are absent, then Case 2 where identity is present, and finally Case 3 where both identity and function symbols are present. In this last case, there is often a subsidiary induction on complexity to prove that all terms have som ...
... Typically: we first do the proof for Case 1 where = and function symbols are absent, then Case 2 where identity is present, and finally Case 3 where both identity and function symbols are present. In this last case, there is often a subsidiary induction on complexity to prove that all terms have som ...
Part 1: Propositional Logic
... Obviously, A(F ) depends only on the values of those finitely many variables in F under A. If F contains n distinct propositional variables, then it is sufficient to check 2n valuations to see whether F is satisfiable or not. ⇒ truth table. So the satisfiability problem is clearly decidable (but, by ...
... Obviously, A(F ) depends only on the values of those finitely many variables in F under A. If F contains n distinct propositional variables, then it is sufficient to check 2n valuations to see whether F is satisfiable or not. ⇒ truth table. So the satisfiability problem is clearly decidable (but, by ...