Logic
... Parentheses and Ambiguity • An ambiguous statements is a statement whose meaning is not clear due to its syntax. Example : ”P or Q and R” • In formal systems, an expression like P Q R is simply not allowed and considered unsyntactical. • Claims in our formal language are therefore never ambiguo ...
... Parentheses and Ambiguity • An ambiguous statements is a statement whose meaning is not clear due to its syntax. Example : ”P or Q and R” • In formal systems, an expression like P Q R is simply not allowed and considered unsyntactical. • Claims in our formal language are therefore never ambiguo ...
Parametric Polymorphism and Abstract Models of Storage
... Functional languages, e.g., PCF, satisfy Milner’s context lemma, which can be thought of as an extensionality principle. But imperative programming languages do not satisfy it. In Games semantics, we can see that functional languages can be interpreted by history-free strategies, but imperative lang ...
... Functional languages, e.g., PCF, satisfy Milner’s context lemma, which can be thought of as an extensionality principle. But imperative programming languages do not satisfy it. In Games semantics, we can see that functional languages can be interpreted by history-free strategies, but imperative lang ...
Constructive Set Theory and Brouwerian Principles1
... purpose of enhancing our account of Brouwerian intuitionism by contrast. The concept of algorithm or recursive function is fundamental to the Russian schools of Markov and Shanin. Contrary to Brouwer, this school takes the viewpoint that mathematical objects must be concrete, or at least have a cons ...
... purpose of enhancing our account of Brouwerian intuitionism by contrast. The concept of algorithm or recursive function is fundamental to the Russian schools of Markov and Shanin. Contrary to Brouwer, this school takes the viewpoint that mathematical objects must be concrete, or at least have a cons ...
An Axiomatization of G'3
... Hilbert Style Proof Systems. There are many different approaches that have been used to specify the meaning of logic formulas or, in other words, to define logics. In Hilbert style proof systems, also known as axiomatic systems, a logic is specified by giving a set of axioms (which is usually assume ...
... Hilbert Style Proof Systems. There are many different approaches that have been used to specify the meaning of logic formulas or, in other words, to define logics. In Hilbert style proof systems, also known as axiomatic systems, a logic is specified by giving a set of axioms (which is usually assume ...
deductive system
... A deductive system is a formal (mathematical) setup of reasoning. In order to describe a deductive system, a (formal) language system must first be in place, consisting of (well-formed) formulas, strings of symbols constructed according to some prescribed syntax. With the language in place, reasonin ...
... A deductive system is a formal (mathematical) setup of reasoning. In order to describe a deductive system, a (formal) language system must first be in place, consisting of (well-formed) formulas, strings of symbols constructed according to some prescribed syntax. With the language in place, reasonin ...
A SHORT PROOF FOR THE COMPLETENESS OF
... The resolution calculus is a classical tool for deciding if a given set of first order clauses are satisfiable or not. We refer to [2] as a standard monograph of the subject. Although the resolution calculus is comprehensive, elegant and complete, in its original form does not handle the equation sy ...
... The resolution calculus is a classical tool for deciding if a given set of first order clauses are satisfiable or not. We refer to [2] as a standard monograph of the subject. Although the resolution calculus is comprehensive, elegant and complete, in its original form does not handle the equation sy ...
P Q
... If this fails to lead to a proof, all possible detachments and replacements are applied to the goal and each of these results is tested for success using substitution; If it fails to match any of these with the goal, they are added to a subproblem list ...
... If this fails to lead to a proof, all possible detachments and replacements are applied to the goal and each of these results is tested for success using substitution; If it fails to match any of these with the goal, they are added to a subproblem list ...
Classicality as a Property of Predicate Symbols
... weaker than the law of excluded middle to intuitionistic logic [CZ]. For example, the Kreisel-Putnam logic is defined by adding the axiom schema: (¬A⊃B∨C)⊃(¬A ⊃B)∨(¬A ⊃C). Yet another approach to looking into the space between intuitionistic and classical logics is to view classicality as a property ...
... weaker than the law of excluded middle to intuitionistic logic [CZ]. For example, the Kreisel-Putnam logic is defined by adding the axiom schema: (¬A⊃B∨C)⊃(¬A ⊃B)∨(¬A ⊃C). Yet another approach to looking into the space between intuitionistic and classical logics is to view classicality as a property ...
Lecture 9. Model theory. Consistency, independence, completeness
... ∆ is satisfiable, equivalently ∆ has a model, iff there is a model M such that for all δ ∈ ∆: M ╞ δ. ∆ is closed under logical consequence (under entailment) iff: if ∆ ╞ ϕ then ϕ ∈ ∆. If ϕ is a sentence (i.e. has no free variables) and M ╞ ϕ , then we say that M is a model for ϕ, or ϕ holds in (or o ...
... ∆ is satisfiable, equivalently ∆ has a model, iff there is a model M such that for all δ ∈ ∆: M ╞ δ. ∆ is closed under logical consequence (under entailment) iff: if ∆ ╞ ϕ then ϕ ∈ ∆. If ϕ is a sentence (i.e. has no free variables) and M ╞ ϕ , then we say that M is a model for ϕ, or ϕ holds in (or o ...
Introduction to Functional Programming (1)
... The library file Prelude.hs provides a large number of standard functions. In addition to the familiar numeric functions such as + and ∗, the library also provides many useful functions on lists. Prelude> head [1,2,3,4,5] ...
... The library file Prelude.hs provides a large number of standard functions. In addition to the familiar numeric functions such as + and ∗, the library also provides many useful functions on lists. Prelude> head [1,2,3,4,5] ...
Functional Programming Big Picture
... ? a set of functional forms to construct complex functions, ? a function application operation, ? some structure to represent data In functional programming, functions are viewed as values themselves, which can be computed by other functions and can be parameters to other functions ? Functions are f ...
... ? a set of functional forms to construct complex functions, ? a function application operation, ? some structure to represent data In functional programming, functions are viewed as values themselves, which can be computed by other functions and can be parameters to other functions ? Functions are f ...
X - Al Akhawayn University
... Queen can move any number of spaces vertically, horizontally or diagonally Two queens are in check if they are in the same row, column or diagonal, so that one could move to the other’s square ...
... Queen can move any number of spaces vertically, horizontally or diagonally Two queens are in check if they are in the same row, column or diagonal, so that one could move to the other’s square ...
4. Propositional Logic Using truth tables
... Problems: Use the truth table method to solve the following problems: 1. Decide whether p0→p1 is equivalent to ¬(p1→p0) or not. 2. Decide whether ¬p0 ∨p1 is equivalent to ¬(p0 ∧p1) or not. ...
... Problems: Use the truth table method to solve the following problems: 1. Decide whether p0→p1 is equivalent to ¬(p1→p0) or not. 2. Decide whether ¬p0 ∨p1 is equivalent to ¬(p0 ∧p1) or not. ...
A Revised Concept of Safety for General Answer Set Programs
... implying that the rule is not safe. [2] also uses a similar concept of restricted variable but only imposes restrictions on variables occurring in the head of the rule or within the scope of negation; so according to this account, (1) is safe. In this paper we propose a new concept of safety that, ...
... implying that the rule is not safe. [2] also uses a similar concept of restricted variable but only imposes restrictions on variables occurring in the head of the rule or within the scope of negation; so according to this account, (1) is safe. In this paper we propose a new concept of safety that, ...
Notes and exercises on First Order Logic
... Example 0.4 Identify the scope of each quantifier and all bound and free variables. (i) B(x, y) → ∀z∀x[B(x, y) → B(z, y)] Answer: The first x is free, the second and third are bound. All occurrences of y are free, and all occurrences of z are bound. The scope of ∀z is ∀x[B(x, y) → B(z, y)], and the ...
... Example 0.4 Identify the scope of each quantifier and all bound and free variables. (i) B(x, y) → ∀z∀x[B(x, y) → B(z, y)] Answer: The first x is free, the second and third are bound. All occurrences of y are free, and all occurrences of z are bound. The scope of ∀z is ∀x[B(x, y) → B(z, y)], and the ...
Bilattices In Logic Programming
... if we had A ← true. We will need something more general; we assume that we have constants in our language L(B), particular atomic formulas, that designate members of the bilattice B. To keep things simple we assume members of B themselves occur in the language L(B), to avoid details on how such memb ...
... if we had A ← true. We will need something more general; we assume that we have constants in our language L(B), particular atomic formulas, that designate members of the bilattice B. To keep things simple we assume members of B themselves occur in the language L(B), to avoid details on how such memb ...
Comp 411 Principles of Programming Languages Lecture 7 Meta-interpreters
... – The defect is that the output of an actual interpreter is restricted to values that can be characterized syntactically. (How do you output a function?) – On the other hand, interpreters naturally introduce a simple form of functional abstraction. A recursive interpreter accepts an extra input, an ...
... – The defect is that the output of an actual interpreter is restricted to values that can be characterized syntactically. (How do you output a function?) – On the other hand, interpreters naturally introduce a simple form of functional abstraction. A recursive interpreter accepts an extra input, an ...
Mathematical Logic Deciding logical consequence Complexity of
... syntax: a precisely defined symbolic language with procedures for transforming symbolic statements into other statements, based solely on their form. No intuition or interpretation is needed, merely applications of agreed upon rules to a set of agreed upon ...
... syntax: a precisely defined symbolic language with procedures for transforming symbolic statements into other statements, based solely on their form. No intuition or interpretation is needed, merely applications of agreed upon rules to a set of agreed upon ...
(A B) |– A
... if T, A |– B and |– A, then T |– B. It is not necessary to state theorems in the assumptions. if A |– B, then T, A |– B. (Monotonicity of proving) if T |– A and T, A |– B, then T |– B. if T |– A and A |– B, then T |– B. if T |– A; T |– B; A, B |– C then T |– C. if T |– A and T |– B, then T |– A B. ...
... if T, A |– B and |– A, then T |– B. It is not necessary to state theorems in the assumptions. if A |– B, then T, A |– B. (Monotonicity of proving) if T |– A and T, A |– B, then T |– B. if T |– A and A |– B, then T |– B. if T |– A; T |– B; A, B |– C then T |– C. if T |– A and T |– B, then T |– A B. ...
A simple proof of Parsons` theorem
... where we are identifying the terms with their interpretations in M. Note that all elements c, d1 , d2 , . . . are members of the above subset because the variables vj appear in the enumeration of terms. It is also clear that the above subset defines a substructure M∗ of M. Using the fact that U is a ...
... where we are identifying the terms with their interpretations in M. Note that all elements c, d1 , d2 , . . . are members of the above subset because the variables vj appear in the enumeration of terms. It is also clear that the above subset defines a substructure M∗ of M. Using the fact that U is a ...
mj cresswell
... everything w i l l b e 0 . A n d he thought th is was false because even i f everything now existing will always be 0 it does not follow that always it will be that everything then existing is 0 . But you don't have to interpret BF that way. (See Cresswell 1990, p.96) You can interpret v as ranging ...
... everything w i l l b e 0 . A n d he thought th is was false because even i f everything now existing will always be 0 it does not follow that always it will be that everything then existing is 0 . But you don't have to interpret BF that way. (See Cresswell 1990, p.96) You can interpret v as ranging ...
Logic - UNM Computer Science
... and b are both 1, i.e., high voltage, then both switches are turned on, essentially making c grounded, i.e., 0. If either a and b are 0, i.e., a low voltage, then there is no current flowing from the VCC to the ground. This keeps c at high voltage, i.e. 1. With the NAND implemented physically, it is ...
... and b are both 1, i.e., high voltage, then both switches are turned on, essentially making c grounded, i.e., 0. If either a and b are 0, i.e., a low voltage, then there is no current flowing from the VCC to the ground. This keeps c at high voltage, i.e. 1. With the NAND implemented physically, it is ...