First-Order Theorem Proving and Vampire

... For representing these problems it uses the TPTP syntax, which is understood by all modern theorem provers, including Vampire. In the TPTP syntax this group theory problem can be written down as follows: %---- 1 * x = 1 fof(left identity,axiom, ! [X] : mult(e,X) = X). %---- i(x) * x = 1 fof(left inv ...

... For representing these problems it uses the TPTP syntax, which is understood by all modern theorem provers, including Vampire. In the TPTP syntax this group theory problem can be written down as follows: %---- 1 * x = 1 fof(left identity,axiom, ! [X] : mult(e,X) = X). %---- i(x) * x = 1 fof(left inv ...

a-logic - Digital [email protected] State University

... he standard logic today is the logic of Frege’s Begriffschrift (1879) and subsequently of Russell and Whitehead’s great book, Principia Mathematica (1913) of Quine’s Mathematical Logic (1940) and Methods of Logic (4th ed.,1982) and of hundreds of other textbooks and treatises which have the same set ...

... he standard logic today is the logic of Frege’s Begriffschrift (1879) and subsequently of Russell and Whitehead’s great book, Principia Mathematica (1913) of Quine’s Mathematical Logic (1940) and Methods of Logic (4th ed.,1982) and of hundreds of other textbooks and treatises which have the same set ...

Principia Logico-Metaphysica (Draft/Excerpt)

... In this and subsequent chapters, our metalanguage makes use of informal notions and principles about numbers and sets so as to more precisely articulate and render certain definitions. These are not primitive notions or principles of our metaphysical system; ultimately, they are to be understood by ...

... In this and subsequent chapters, our metalanguage makes use of informal notions and principles about numbers and sets so as to more precisely articulate and render certain definitions. These are not primitive notions or principles of our metaphysical system; ultimately, they are to be understood by ...

Announcement as effort on topological spaces

... in multi-agent subset space logics, but that are also relative to each state. This amounts to, when shifting the viewpoint from x to y ∈ Ui , in (x, Ui , Uj ), we simultaneously have to shift the neighbourhood (and not merely the point in the actual neighbourhood) for the other agent. So we then go ...

... in multi-agent subset space logics, but that are also relative to each state. This amounts to, when shifting the viewpoint from x to y ∈ Ui , in (x, Ui , Uj ), we simultaneously have to shift the neighbourhood (and not merely the point in the actual neighbourhood) for the other agent. So we then go ...

Color - Alex Kocurek

... Very often, these inexpressibility claims are justified in the literature only by example: all of the most straightforward attempts at formalizing these English sentences into first-order modal logic fail. While this style of argument may be convincing, it does not constitute a proof. One can someti ...

... Very often, these inexpressibility claims are justified in the literature only by example: all of the most straightforward attempts at formalizing these English sentences into first-order modal logic fail. While this style of argument may be convincing, it does not constitute a proof. One can someti ...

JUXTAPOSITION - Brown University

... combining logical systems, called “juxtaposition”. I prove general metalogical results concerning the combination of logics by juxtaposition. Second, I examine the particular case of combining classical and intuitionist logics. I show how the general results can be applied to shed light on the pheno ...

... combining logical systems, called “juxtaposition”. I prove general metalogical results concerning the combination of logics by juxtaposition. Second, I examine the particular case of combining classical and intuitionist logics. I show how the general results can be applied to shed light on the pheno ...

Functional Programming in Haskell

... Everything we will talk about here can be done in C or even assembly language. The question is not whether it can be done but how easily it can be done. It’s all about expressiveness of the language. ...

... Everything we will talk about here can be done in C or even assembly language. The question is not whether it can be done but how easily it can be done. It’s all about expressiveness of the language. ...

Graphical Representation of Canonical Proof: Two case studies

... it seems as if non-confluence may be an even more strongly intrinsic property of classical proof normalisation than previously thought. Secondly, canonical proof representations, such as proof nets for linear logic, hold the promise of unlocking the computational content of logics. The reasoning to ...

... it seems as if non-confluence may be an even more strongly intrinsic property of classical proof normalisation than previously thought. Secondly, canonical proof representations, such as proof nets for linear logic, hold the promise of unlocking the computational content of logics. The reasoning to ...

A Representation Theorem for Second-Order Functionals

... Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a datatypegeneric representation theorem. More precisely, we prove a representation theorem for a wide class of second-o ...

... Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a datatypegeneric representation theorem. More precisely, we prove a representation theorem for a wide class of second-o ...

Independence logic and tuple existence atoms

... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...

... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...

a semantic perspective - Institute for Logic, Language and

... basic tools needed in modal model theory (such as the standard translation, generated submodels, bounded morphisms, and so on). Basic results about these concepts are stated and some simple proofs are given. But we have a second, more ambitious, goal: to help the reader think semantically. We want t ...

... basic tools needed in modal model theory (such as the standard translation, generated submodels, bounded morphisms, and so on). Basic results about these concepts are stated and some simple proofs are given. But we have a second, more ambitious, goal: to help the reader think semantically. We want t ...

COS220lec52_FP

... The rest list elements are the operands. Two advantages of using such a notation: • It permits to describe procedures with an arbitrary number of arguments (operands). ...

... The rest list elements are the operands. Two advantages of using such a notation: • It permits to describe procedures with an arbitrary number of arguments (operands). ...

Scala - Dave Reed

... ● Scala is a pure object-oriented language as everything is an object o Java is not a pure OO language - has primitive types (like boolean and int) ● Even numbers and functions are objects o Since numbers are objects they have methods ...

... ● Scala is a pure object-oriented language as everything is an object o Java is not a pure OO language - has primitive types (like boolean and int) ● Even numbers and functions are objects o Since numbers are objects they have methods ...

A counterexample-guided abstraction

... by an equivalence relation (of finite index) on the states of the concrete system. The states of the abstract model will be the equivalence classes of this relation, and each abstract state will have transitions corresponding to the transitions of each of the concrete states in the equivalence class ...

... by an equivalence relation (of finite index) on the states of the concrete system. The states of the abstract model will be the equivalence classes of this relation, and each abstract state will have transitions corresponding to the transitions of each of the concrete states in the equivalence class ...

5 model theory of modal logic

... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related t ...

... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related t ...

Structural Proof Theory

... of proof. Sequent calculus, instead, has been developed in various directions. One line leads from Gentzen through Ketonen, Kleene, Dragalin, and Troelstra to what are known as contraction-free systems of sequent calculus. Each of these logicians added some essential discovery, until a gem emerged. ...

... of proof. Sequent calculus, instead, has been developed in various directions. One line leads from Gentzen through Ketonen, Kleene, Dragalin, and Troelstra to what are known as contraction-free systems of sequent calculus. Each of these logicians added some essential discovery, until a gem emerged. ...

CS 345 - Programming Languages

... Function (method) is the basic primitive in all languages we have seen so far • F(x)=y – function F takes x and return y ...

... Function (method) is the basic primitive in all languages we have seen so far • F(x)=y – function F takes x and return y ...

Model Theory of Modal Logic, Chapter in: Handbook of Modal Logic

... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related to ...

... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related to ...

Q-Midi - Q - Equational Programming Language

... two data structures are very similar. Q provides a variety of built-in operations on lists and tuples, such as determining the length, indexing and concatenation. The most important construct in the Q language is the function application which is simply denoted by juxtaposition. Thus, e.g., sin X de ...

... two data structures are very similar. Q provides a variety of built-in operations on lists and tuples, such as determining the length, indexing and concatenation. The most important construct in the Q language is the function application which is simply denoted by juxtaposition. Thus, e.g., sin X de ...

A Logical Foundation for Session

... Over the years, computation systems have evolved from monolithic single-threaded machines to concurrent and distributed environments with multiple communicating threads of execution, for which writing correct programs becomes substantially harder than in the more traditional sequential setting. Thes ...

... Over the years, computation systems have evolved from monolithic single-threaded machines to concurrent and distributed environments with multiple communicating threads of execution, for which writing correct programs becomes substantially harder than in the more traditional sequential setting. Thes ...

How to Go Nonmonotonic Contents David Makinson

... different directions. The few available textbooks tend to perpetuate this impression. Our main purpose is to take some of the mystery out of the subject and show that it is not as unfamiliar as may at first sight seem. In fact, it is easily accessible to anybody with a minimal background in classica ...

... different directions. The few available textbooks tend to perpetuate this impression. Our main purpose is to take some of the mystery out of the subject and show that it is not as unfamiliar as may at first sight seem. In fact, it is easily accessible to anybody with a minimal background in classica ...

Foundations of Databases - Free University of Bozen

... Datalog Semantics via Least Fixpoint The semantics of P on database instance I of edb(P ) is a special fixpoint: Theorem. Let P be a datalog program and I be a database instance. Then ...

... Datalog Semantics via Least Fixpoint The semantics of P on database instance I of edb(P ) is a special fixpoint: Theorem. Let P be a datalog program and I be a database instance. Then ...