slides
... a list with the parameters as elements - Predicate Functions: (#T and () are true and false) 1. EQ? takes two symbolic parameters; it returns #T if both parameters are atoms and the two are the same e.g., (EQ? 'A 'A) yields #T (EQ? 'A '(A B)) yields () Note that if EQ? is called with list parameters ...
... a list with the parameters as elements - Predicate Functions: (#T and () are true and false) 1. EQ? takes two symbolic parameters; it returns #T if both parameters are atoms and the two are the same e.g., (EQ? 'A 'A) yields #T (EQ? 'A '(A B)) yields () Note that if EQ? is called with list parameters ...
Chapter 14 Functional Programming Languages
... 2. The values of the parameters are substituted into the function body 3. The function body is evaluated 4. The value of the last expression in the body is the value of the function (Special forms use a different evaluation process) ...
... 2. The values of the parameters are substituted into the function body 3. The function body is evaluated 4. The value of the last expression in the body is the value of the function (Special forms use a different evaluation process) ...
A Paedagogic Example of Cut-Elimination
... left and right rank. We prove the theorem by induction on the rank n of the cut. 1. If n = 1, then Y ≡ x and we have an inference from two axioms: x≤x x≤x x≤x which can be replaced by the axiom x≤x and is thus provable without cut. 2. If n > 1 we distinguish cases according to the form of Y : (a) Y ...
... left and right rank. We prove the theorem by induction on the rank n of the cut. 1. If n = 1, then Y ≡ x and we have an inference from two axioms: x≤x x≤x x≤x which can be replaced by the axiom x≤x and is thus provable without cut. 2. If n > 1 we distinguish cases according to the form of Y : (a) Y ...
Functional Programming Languages (LISP/ Scheme)
... 1) A set of primitive functions (car, cdr, cons, etc.) 2) A set of functional forms to form complex functions from primitive ones. (define, defun, etc.) 3) A function application operation. (eval) 4) Some structure or structures for storing data. (Define, setq, etc.) ...
... 1) A set of primitive functions (car, cdr, cons, etc.) 2) A set of functional forms to form complex functions from primitive ones. (define, defun, etc.) 3) A function application operation. (eval) 4) Some structure or structures for storing data. (Define, setq, etc.) ...
Introduction to Artificial Intelligence
... In propositional logic there are two truth values: t for “true” and f for “false”. Is a formula, such as A ∧ B true? The answer depends on whether the variables A and B are true. Example: If A stands for “It is raining today” and B for “It is cold today” and these are both true, then A ∧ B is true. ...
... In propositional logic there are two truth values: t for “true” and f for “false”. Is a formula, such as A ∧ B true? The answer depends on whether the variables A and B are true. Example: If A stands for “It is raining today” and B for “It is cold today” and these are both true, then A ∧ B is true. ...
byd.1 Second-Order logic
... generality the axiom schema of comprehension is impredicative: it allows one to assert the existence of a relation R(x1 , . . . , xk ) that is “defined” by a formula with second-order quantifiers; and these quantifiers range over the set of all such relations—a set which includes R itself! Around t ...
... generality the axiom schema of comprehension is impredicative: it allows one to assert the existence of a relation R(x1 , . . . , xk ) that is “defined” by a formula with second-order quantifiers; and these quantifiers range over the set of all such relations—a set which includes R itself! Around t ...
Compound term
... The ability to tamper with control flow in a Prolog program is a deficiency because it is detrimental to one of the advantages of logic programming: programs do not specify how to find solutions Using the ability for flow control, clutters the simplicity of logic programs with details of order con ...
... The ability to tamper with control flow in a Prolog program is a deficiency because it is detrimental to one of the advantages of logic programming: programs do not specify how to find solutions Using the ability for flow control, clutters the simplicity of logic programs with details of order con ...
Haskell review
... [3,5..] creates [3,5,7,9,11,13,15,17 and so forth] • Haskell uses lazy evaluation. It will not create a list until it needs to do so take 4 [3,5..] returns [3,5,7,9] • Haskell only had to create the first four elements of the list ...
... [3,5..] creates [3,5,7,9,11,13,15,17 and so forth] • Haskell uses lazy evaluation. It will not create a list until it needs to do so take 4 [3,5..] returns [3,5,7,9] • Haskell only had to create the first four elements of the list ...
Elements of Modal Logic - University of Victoria
... and R is a set of inference rules. An inference rule has the form, α1 . . . αn β A set Σ is said to be closed under an inference rule iff β ∈ Σ whenever all of the αi ’s are in Σ. Each system S determines a logic L(S), which is defined as the smallest set containing A that is closed under the rules ...
... and R is a set of inference rules. An inference rule has the form, α1 . . . αn β A set Σ is said to be closed under an inference rule iff β ∈ Σ whenever all of the αi ’s are in Σ. Each system S determines a logic L(S), which is defined as the smallest set containing A that is closed under the rules ...
Logics of Truth - Project Euclid
... truth are then the standard Tarski ones, where propositions are the objects of the truth predicate. In an important sense these axioms reflect the minimal conditions one would expect of any theory of truth. The other theories we shall consider all have the above theory as a consequence. 3.1 Models o ...
... truth are then the standard Tarski ones, where propositions are the objects of the truth predicate. In an important sense these axioms reflect the minimal conditions one would expect of any theory of truth. The other theories we shall consider all have the above theory as a consequence. 3.1 Models o ...
CITS3211 FUNCTIONAL PROGRAMMING 5. Higherorder functions
... Abstraction is the process of generalising a program by making its components into parameters of the program − e.g. by naming an explicit constant − this gives concise programs and reusable code − it’s better to write one program and use it ten times than it is to write ten slightly ...
... Abstraction is the process of generalising a program by making its components into parameters of the program − e.g. by naming an explicit constant − this gives concise programs and reusable code − it’s better to write one program and use it ten times than it is to write ten slightly ...
Classical First-Order Logic Introduction
... Monadic predicate logic (i.e. only unary predicates and no function symbols) is decidable. The Bernays-Schönfinkel class of formulas (i.e. formulas that can be written with all quantiers appearing at the beginning of the formula with existentials before universals and that do not contain any functi ...
... Monadic predicate logic (i.e. only unary predicates and no function symbols) is decidable. The Bernays-Schönfinkel class of formulas (i.e. formulas that can be written with all quantiers appearing at the beginning of the formula with existentials before universals and that do not contain any functi ...
Monadic Second Order Logic and Automata on Infinite Words
... MSOL0 [S] formulas φ, ψ := S(X, Y ) | X ⊆ Y | X ⊆ Qa | ¬ψ | ψ ∧ φ | ∃X.ψ I will show how to translate between MSOL0 [S] and MSOL[S]. We can then conclude that MSOL0 [S] defines the same languages as MSOL[S]. First we do the translation from MSOL[S] to MSOL0 [S]. The usual conditions on variable bind ...
... MSOL0 [S] formulas φ, ψ := S(X, Y ) | X ⊆ Y | X ⊆ Qa | ¬ψ | ψ ∧ φ | ∃X.ψ I will show how to translate between MSOL0 [S] and MSOL[S]. We can then conclude that MSOL0 [S] defines the same languages as MSOL[S]. First we do the translation from MSOL[S] to MSOL0 [S]. The usual conditions on variable bind ...
A Uniform Proof Procedure for Classical and Non
... In this paper we present a proof procedure which allows a uniform treatment of classical, intuitionistic, and modal logics. It is based on a unified representation of Wallen’s matrix characterizations and generalizes Bibel’s connection method [4, 5] for classical predicate logic accordingly. In orde ...
... In this paper we present a proof procedure which allows a uniform treatment of classical, intuitionistic, and modal logics. It is based on a unified representation of Wallen’s matrix characterizations and generalizes Bibel’s connection method [4, 5] for classical predicate logic accordingly. In orde ...
Extending modal logic
... 2. K is closed under bounded morphic images and disjoint unions 3. The Horn conditions can be written so that their left hand sides are tree-shaped. 4. K is definable by Sahlqvist formulas ...
... 2. K is closed under bounded morphic images and disjoint unions 3. The Horn conditions can be written so that their left hand sides are tree-shaped. 4. K is definable by Sahlqvist formulas ...
Short Introduction to the Lambda
... domain and codomain; this is a mathematical notation to specify a function without assigning to it a particular name (an anonymous function). In Lambda-calculus we describe the relation input-output by means of the lambda-abstraction operator: λx. x*x (anonymous function associating x to x*x). Other ...
... domain and codomain; this is a mathematical notation to specify a function without assigning to it a particular name (an anonymous function). In Lambda-calculus we describe the relation input-output by means of the lambda-abstraction operator: λx. x*x (anonymous function associating x to x*x). Other ...
CHAPTER 5 SOME EXTENSIONAL SEMANTICS
... If T is the only designated value, the third value ⊥ corresponds to some notion of incomplete information, like undefined or unknown and is often denoted by the symbol U or I. If, on the other hand, ⊥ corresponds to inconsistent information, i.e. its meaning is something like known to be both true a ...
... If T is the only designated value, the third value ⊥ corresponds to some notion of incomplete information, like undefined or unknown and is often denoted by the symbol U or I. If, on the other hand, ⊥ corresponds to inconsistent information, i.e. its meaning is something like known to be both true a ...
Discrete Mathematics - Lecture 4: Propositional Logic and Predicate
... Lecture 4: Propositional Logic and Predicate Logic ...
... Lecture 4: Propositional Logic and Predicate Logic ...
Completeness and Decidability of a Fragment of Duration Calculus
... Duration Calculus (DC) was introduced by Zhou, Hoare and Ravn in 1991 as a logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a re ...
... Duration Calculus (DC) was introduced by Zhou, Hoare and Ravn in 1991 as a logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a re ...
1 Introduction 2 Formal logic
... Formal logic as we understand it in these lectures is an approach to making informal mathematical reasoning precise. It has three main ingredients: • A formal language in which to express the mathematical statements we want to reason about. • A semantics that explains the meaning of statements in ou ...
... Formal logic as we understand it in these lectures is an approach to making informal mathematical reasoning precise. It has three main ingredients: • A formal language in which to express the mathematical statements we want to reason about. • A semantics that explains the meaning of statements in ou ...
Expressing C++ Template Metaprograms as Lambda expressions
... We examine how functional languages could be used to write template metaprograms in, letting developers use a better syntax for writing and maintaining metaprograms. Since lambda expressions are capable of expressing any functional program we show how lambda expressions can be used to express C++ te ...
... We examine how functional languages could be used to write template metaprograms in, letting developers use a better syntax for writing and maintaining metaprograms. Since lambda expressions are capable of expressing any functional program we show how lambda expressions can be used to express C++ te ...
A Simple and Practical Valuation Tree Calculus for First
... the development of correct programs (at least the mission critical ones), and so the search for them is a worthwhile research in applied logic. The reader will note that the cut elimination is not a central issue in IPAs because natural mathematical proofs without cuts (in the form of lemmas) are im ...
... the development of correct programs (at least the mission critical ones), and so the search for them is a worthwhile research in applied logic. The reader will note that the cut elimination is not a central issue in IPAs because natural mathematical proofs without cuts (in the form of lemmas) are im ...
4.6 Lisp - University of Hawaii
... pure functions to the greatest extent possible • Process of computation is fundamentally different – In an imperative language, operations are executed and the results are stored in variables for later use – Management of variables is a constant concern and source of complexity for imperative progra ...
... pure functions to the greatest extent possible • Process of computation is fundamentally different – In an imperative language, operations are executed and the results are stored in variables for later use – Management of variables is a constant concern and source of complexity for imperative progra ...
Document
... A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions Predicates can be quantified using the universal quantifier (“for all”) or the existential quantifier (“there exists”) Quantified predicates can be negated as follows x P(x) ...
... A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions Predicates can be quantified using the universal quantifier (“for all”) or the existential quantifier (“there exists”) Quantified predicates can be negated as follows x P(x) ...
19. Introduction to evaluation order
... As already mentioned above, the value v of an expression e is a particular simple expression which is semantically equivalent with e. The expression v is obtained from e by a number of rewriting steps. Normal forms represent our intuition of the value of an expression Here is the definition of a nor ...
... As already mentioned above, the value v of an expression e is a particular simple expression which is semantically equivalent with e. The expression v is obtained from e by a number of rewriting steps. Normal forms represent our intuition of the value of an expression Here is the definition of a nor ...