pdf
... It is known that the algebra of regular sets Reg Σ ∗ is the free Kleene algebra generated by Σ [18]. This is equivalent to the completeness of the axioms of KA for the standard language interpretation R of regular expressions. That is, for any two regular expressions e1 , e2 over Σ, if R(e1 ) = R(e2 ...
... It is known that the algebra of regular sets Reg Σ ∗ is the free Kleene algebra generated by Σ [18]. This is equivalent to the completeness of the axioms of KA for the standard language interpretation R of regular expressions. That is, for any two regular expressions e1 , e2 over Σ, if R(e1 ) = R(e2 ...
block_24 - Math GR. 9-12
... nonparallel sides are called legs • If the legs are congruent then the trapezoid is called isosceles trapezoid. ...
... nonparallel sides are called legs • If the legs are congruent then the trapezoid is called isosceles trapezoid. ...
Witness and Counterexample Automata for ACTL
... of reactive systems whose behaviour is characterized by the actions they perform and whose semantics is defined by means of LTS’s. ACTL is adequate with respect to strong bisimulation equivalence, this means that if p ∼ q, then p and q satisfy the same set of ACTL formulae. To define ACTL an auxilia ...
... of reactive systems whose behaviour is characterized by the actions they perform and whose semantics is defined by means of LTS’s. ACTL is adequate with respect to strong bisimulation equivalence, this means that if p ∼ q, then p and q satisfy the same set of ACTL formulae. To define ACTL an auxilia ...
On Herbrand`s Theorem for Intuitionistic Logic
... An indexed variable k v can be a free variable or parameter depending on v being weak or strong, respectively in a sequent S. For technical reasons only, if v is a weak (strong) variable, then k v (k v) denotes its free variable (parameter) ‘copy’. A formula φ is intuitionistically valid if, and onl ...
... An indexed variable k v can be a free variable or parameter depending on v being weak or strong, respectively in a sequent S. For technical reasons only, if v is a weak (strong) variable, then k v (k v) denotes its free variable (parameter) ‘copy’. A formula φ is intuitionistically valid if, and onl ...
Sequent calculus for predicate logic
... with ∈ {∧, ∨, →} and Q ∈ {∀, ∃}. Lemma 3.2. A cut free derivation π of a sequent σ in either the classical or intuitionistic sequent calculus only contains Gentzen subformulas of formulas occurring in σ. But note that the definition of Gentzen subformula is such that ∀x P x has infinitely many Gen ...
... with ∈ {∧, ∨, →} and Q ∈ {∀, ∃}. Lemma 3.2. A cut free derivation π of a sequent σ in either the classical or intuitionistic sequent calculus only contains Gentzen subformulas of formulas occurring in σ. But note that the definition of Gentzen subformula is such that ∀x P x has infinitely many Gen ...
S2 - CALCULEMUS.ORG
... In particular, the most important open mathematical problem „does P=NP?” is equivalent to some questions concerning finite model theory. Assuming that the way of a human brain’s work is based on the same rules as the way the machines made by a man work, this problem is also equivalent to a question ...
... In particular, the most important open mathematical problem „does P=NP?” is equivalent to some questions concerning finite model theory. Assuming that the way of a human brain’s work is based on the same rules as the way the machines made by a man work, this problem is also equivalent to a question ...
A Paedagogic Example of Cut-Elimination
... Definition 1 The set of symbols of LI consist of the following: 1. variables: x0 , x1 , x2 , . . . , 2. relation symbols: ∩, ∪, ≤, and 3. auxiliary symbols: ( and ). The set of terms of LI is the least set closed under the following formation rules: 1. Every variable is a term. 2. If X and Y are ter ...
... Definition 1 The set of symbols of LI consist of the following: 1. variables: x0 , x1 , x2 , . . . , 2. relation symbols: ∩, ∪, ≤, and 3. auxiliary symbols: ( and ). The set of terms of LI is the least set closed under the following formation rules: 1. Every variable is a term. 2. If X and Y are ter ...
A Proof of Cut-Elimination Theorem for U Logic.
... Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He wanted to interpret implication as formal provability. To protect his system against the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural deduction[3, p. 8]. The first sequent calculus for BPL ...
... Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He wanted to interpret implication as formal provability. To protect his system against the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural deduction[3, p. 8]. The first sequent calculus for BPL ...
On Equivalent Transformations of Infinitary Formulas under the
... One of the reasons why stable models of infinitary formulas are important is that they are closely related to aggregates in answer set programming (ASP). The semantics of aggregates proposed by Ferraris [1, Section 4.1] treats a ground aggregate as shorthand for a propositional formula. An aggregate ...
... One of the reasons why stable models of infinitary formulas are important is that they are closely related to aggregates in answer set programming (ASP). The semantics of aggregates proposed by Ferraris [1, Section 4.1] treats a ground aggregate as shorthand for a propositional formula. An aggregate ...
Ambient Logic II.fm
... often omitted in the contexts n[0] and M.0, yielding n[] and M. Composition has the weakest binding power, so that the expression (νn)P | Q is read ((νn)P) | Q, the expression !P | Q is read (!P) | Q, the expression M.P | Q is read (M.P) | Q, and the expression (n).P | Q is read ((n).P) | Q. Structu ...
... often omitted in the contexts n[0] and M.0, yielding n[] and M. Composition has the weakest binding power, so that the expression (νn)P | Q is read ((νn)P) | Q, the expression !P | Q is read (!P) | Q, the expression M.P | Q is read (M.P) | Q, and the expression (n).P | Q is read ((n).P) | Q. Structu ...
Rewriting in the partial algebra of typed terms modulo AC
... top down automaton (called the typing policy). In this work, we show that the reachability and property reachability remain decidable when lifted to the higher order. The proof techniques share many common points with the original study of PRS, however some simplifications are obtained by an extensi ...
... top down automaton (called the typing policy). In this work, we show that the reachability and property reachability remain decidable when lifted to the higher order. The proof techniques share many common points with the original study of PRS, however some simplifications are obtained by an extensi ...
Sec 13.1 Arithmethic and Geometric Sequences
... Often in applications we will want the sum of a certain number of terms in an arithmetic sequence. The story is told of a grade school teacher In the 1700's that wanted to keep her class busy while she graded papers so she asked them to add up all of the numbers from 1 to 100. These numbers are an ...
... Often in applications we will want the sum of a certain number of terms in an arithmetic sequence. The story is told of a grade school teacher In the 1700's that wanted to keep her class busy while she graded papers so she asked them to add up all of the numbers from 1 to 100. These numbers are an ...
Lesson 12
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
Sample pages 1 PDF
... denoted. To emphasize this distinction, for instance for A = (A, +, <, 0), it is better to write A = (A, +A ,
... denoted. To emphasize this distinction, for instance for A = (A, +, <, 0), it is better to write A = (A, +A ,
Object-Based Unawareness
... is already familiar with this literature, we still urge that he skims through this section, as we shall introduce some notation and terminology here. First, we note that DLR do not commit to a specific definition of unawareness. Rather, their impossibility theorem applies to all definitions that ar ...
... is already familiar with this literature, we still urge that he skims through this section, as we shall introduce some notation and terminology here. First, we note that DLR do not commit to a specific definition of unawareness. Rather, their impossibility theorem applies to all definitions that ar ...
First Order Predicate Logic
... A path in a tableaux is contradictory or closed if some atomic formulae α and ~ α appear on the same path. If all the paths of a tableau are closed, then it is called a contradictory tableaux. A tableau proof of a formula α is a contradictory tableau with root as ~ α . Let α be any formula. If table ...
... A path in a tableaux is contradictory or closed if some atomic formulae α and ~ α appear on the same path. If all the paths of a tableau are closed, then it is called a contradictory tableaux. A tableau proof of a formula α is a contradictory tableau with root as ~ α . Let α be any formula. If table ...
(A B) |– A
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
(A B) |– A
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
Standardization of Formulæ
... An existential quantifier can be removed by replacing the variable it bounds by a Skolem function of the form f (x1 , ..xn ), where: f is a fresh function symbol x1 , .., xn are the variables which are universally quantified before the quantifier to be removed ∀x∃y (p(x) → ¬q(y )) ∃x∀z(q(x, z) ∨ r ( ...
... An existential quantifier can be removed by replacing the variable it bounds by a Skolem function of the form f (x1 , ..xn ), where: f is a fresh function symbol x1 , .., xn are the variables which are universally quantified before the quantifier to be removed ∀x∃y (p(x) → ¬q(y )) ∃x∀z(q(x, z) ∨ r ( ...
“Sometimes” and “Not Never” Revisited
... temporal modalities are provided for describing events along a single time path (cf. [ 141).In contrast, in a logic of branching time, the modalities reflect the branching nature of time by allowing quantification over possible futures (cf. [ 1, 71). Some controversy has arisen in the computer scien ...
... temporal modalities are provided for describing events along a single time path (cf. [ 141).In contrast, in a logic of branching time, the modalities reflect the branching nature of time by allowing quantification over possible futures (cf. [ 1, 71). Some controversy has arisen in the computer scien ...
Expressive Completeness for Metric Temporal Logic
... arises by annotating the temporal modalities of LTL with intervals with rational endpoints, representing metric constraints. Since the MTL operators are definable in FO(<, +Q), it is immediate that one can translate MTL into FO(<, +Q). The main result of this paper shows the converse, that MTL is ex ...
... arises by annotating the temporal modalities of LTL with intervals with rational endpoints, representing metric constraints. Since the MTL operators are definable in FO(<, +Q), it is immediate that one can translate MTL into FO(<, +Q). The main result of this paper shows the converse, that MTL is ex ...
slides (modified) - go here for webmail
... Q: Can we satisfy these constraints? Q: Which graph problem does this remind you of ? ...
... Q: Can we satisfy these constraints? Q: Which graph problem does this remind you of ? ...
Boolean unification with predicates
... We will henceforth refer to the 2nd problem in this list as Boolean unification (BU). In this article, we extend the research on BU by analysing the following more general problem: Problem (Boolean unification with predicates (BUP)) For an input formula F[X ] in first-order logic with equality conta ...
... We will henceforth refer to the 2nd problem in this list as Boolean unification (BU). In this article, we extend the research on BU by analysing the following more general problem: Problem (Boolean unification with predicates (BUP)) For an input formula F[X ] in first-order logic with equality conta ...
pdf [local copy]
... F1. Suppose that the schematic paraphrase A[P, Q] contains more than one occurrence of P. Then the description elimination system is not terminating: one can find infinite description elimination sequences. (Theorem 5.7) F2. Suppose that the schematic paraphrase A[P, Q] contains only one occurrence ...
... F1. Suppose that the schematic paraphrase A[P, Q] contains more than one occurrence of P. Then the description elimination system is not terminating: one can find infinite description elimination sequences. (Theorem 5.7) F2. Suppose that the schematic paraphrase A[P, Q] contains only one occurrence ...
pdf
... However, in many seemingly analogous cases we do have termination nevertheless, e.g. for the rewrite system for the Hydra battle [Mos09, Fle07], since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s f ...
... However, in many seemingly analogous cases we do have termination nevertheless, e.g. for the rewrite system for the Hydra battle [Mos09, Fle07], since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s f ...