Higher Order Logic - Theory and Logic Group
... are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foun ...
... are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foun ...
Prime Implicates and Prime Implicants: From Propositional to Modal
... though this technique could prove highly relevant for modal and description logics, which generally suffer from an even higher computational complexity than propositional logic. As prime implicates are one of the better-known mechanisms for compiling formulae in propositional logic, it certainly mak ...
... though this technique could prove highly relevant for modal and description logics, which generally suffer from an even higher computational complexity than propositional logic. As prime implicates are one of the better-known mechanisms for compiling formulae in propositional logic, it certainly mak ...
Elements of Finite Model Theory
... although in recent years connections with other areas, such as formal methods and verification, and artificial intelligence, have been discovered. The birth of finite model theory is often identified with Trakhtenbrot’s result from 1950 stating that validity over finite models is not recursively enumerab ...
... although in recent years connections with other areas, such as formal methods and verification, and artificial intelligence, have been discovered. The birth of finite model theory is often identified with Trakhtenbrot’s result from 1950 stating that validity over finite models is not recursively enumerab ...
The Premiss-Based Approach to Logical Aggregation Franz Dietrich & Philippe Mongin
... and results make it possible to address unexplored sets or premisses, i .e, those which consist neither of propositional variables, nor of the whole agenda, and these are the truly salient cases for premiss-based aggregation. ...
... and results make it possible to address unexplored sets or premisses, i .e, those which consist neither of propositional variables, nor of the whole agenda, and these are the truly salient cases for premiss-based aggregation. ...
Divide and congruence: From decomposition of modal formulas to preservation of branching and eta-bisimilarity
... captures such observations. A modal characterisation of an equivalence on processes consists of a class C of modal formulas such that two processes are equivalent if and only if they satisfy the same formulas in C. For instance, HennessyMilner logic [23] is a modal characterisation of (strong) bisim ...
... captures such observations. A modal characterisation of an equivalence on processes consists of a class C of modal formulas such that two processes are equivalent if and only if they satisfy the same formulas in C. For instance, HennessyMilner logic [23] is a modal characterisation of (strong) bisim ...
Conditional XPath
... child and parent are just the XPath axis with the same name. right and left denote immediate sibling to the right and to the left, respectively. Given a tree, these expressions define paths or equivalently a set of pairs of nodes. For instance, the meaning of child is the set of all edges in the tre ...
... child and parent are just the XPath axis with the same name. right and left denote immediate sibling to the right and to the left, respectively. Given a tree, these expressions define paths or equivalently a set of pairs of nodes. For instance, the meaning of child is the set of all edges in the tre ...
Deductive Databases with Universally Quantified Conditions
... Deductive databases were introduced over 30 years ago ([Gallaire and Minker 1978],[Gallaire, Minker, and Nicolas 1981],[Gallaire, Minker, and Nicolas 1984]). A deductive database (or sometimes referred to as DATALOG programs) consists of a set of facts that correspond to a relational database and a ...
... Deductive databases were introduced over 30 years ago ([Gallaire and Minker 1978],[Gallaire, Minker, and Nicolas 1981],[Gallaire, Minker, and Nicolas 1984]). A deductive database (or sometimes referred to as DATALOG programs) consists of a set of facts that correspond to a relational database and a ...
Optimal acceptors and optimal proof systems
... Levin’s algorithm enumerates search algorithms, and for an acceptor we need decision algorithms, which may be faster for some inputs; the search-to-decision reduction adds a lot to the running time by running the decision algorithm for shorter formulas as well, which may be surprisingly much larger ...
... Levin’s algorithm enumerates search algorithms, and for an acceptor we need decision algorithms, which may be faster for some inputs; the search-to-decision reduction adds a lot to the running time by running the decision algorithm for shorter formulas as well, which may be surprisingly much larger ...
Mathematical Logic
... Example. (P → Q) → R ∧ ∀xR0 (x) has as s.p.p.’s the whole formula, R ∧ ∀xR0 (x), R, ∀xR0 (x), R0 (t). The positive subformulas are the s.p.p.’s and in addition P ; the negative subformulas are P → Q, Q. 2. Natural Deduction We introduce Gentzen’s system of natural deduction. To allow a direct corres ...
... Example. (P → Q) → R ∧ ∀xR0 (x) has as s.p.p.’s the whole formula, R ∧ ∀xR0 (x), R, ∀xR0 (x), R0 (t). The positive subformulas are the s.p.p.’s and in addition P ; the negative subformulas are P → Q, Q. 2. Natural Deduction We introduce Gentzen’s system of natural deduction. To allow a direct corres ...
Introduction to Modal and Temporal Logic
... Rule Instances: Uniformly replace formula and set variables with formulae and formula sets Derivation of ϕ0 from assumptions Γ0: is a finite tree of judgments with: 1. a root node Γ0 ⊢ ϕ0 2. only (Ax) judgment instances and (Id) instances as leaves ...
... Rule Instances: Uniformly replace formula and set variables with formulae and formula sets Derivation of ϕ0 from assumptions Γ0: is a finite tree of judgments with: 1. a root node Γ0 ⊢ ϕ0 2. only (Ax) judgment instances and (Id) instances as leaves ...
On modal logics of group belief
... (called AL for Acceptance Logic) which enables to reason about acceptances of agents and groups of agents. We call the former individual acceptances, and the latter collective acceptances. In Section 4, we will study the logical properties of the notion of acceptance and its interactions with classi ...
... (called AL for Acceptance Logic) which enables to reason about acceptances of agents and groups of agents. We call the former individual acceptances, and the latter collective acceptances. In Section 4, we will study the logical properties of the notion of acceptance and its interactions with classi ...
Preferences and Unrestricted Rebut
... determine whether one argument attacks another. Suppose one considers arguments constructed using reasons, represented as rules. Many rules will be defeasible (for instance, when applying the argument scheme of expert opinion), but some rules will be strict (for instance, Modus ponens).The idea in f ...
... determine whether one argument attacks another. Suppose one considers arguments constructed using reasons, represented as rules. Many rules will be defeasible (for instance, when applying the argument scheme of expert opinion), but some rules will be strict (for instance, Modus ponens).The idea in f ...
Logic and Proof
... Aristotle observed that the correctness of this inference has nothing to do with the truth or falsity of the individual statements, but, rather, the general pattern: Every A is B. Every B is C. Therefore every A is C. We can substitute various properties for A, B, and C; try substituting the propert ...
... Aristotle observed that the correctness of this inference has nothing to do with the truth or falsity of the individual statements, but, rather, the general pattern: Every A is B. Every B is C. Therefore every A is C. We can substitute various properties for A, B, and C; try substituting the propert ...
The Computer Modelling of Mathematical Reasoning Alan Bundy
... While there are now several textbooks on Artificial Intelligence techniques and, more particularly, on Problem Solving and Theorem Proving, I felt the need for a book concentrating on applications of these techniques to Mathematics. There was certainly enough material, but it was scattered in resear ...
... While there are now several textbooks on Artificial Intelligence techniques and, more particularly, on Problem Solving and Theorem Proving, I felt the need for a book concentrating on applications of these techniques to Mathematics. There was certainly enough material, but it was scattered in resear ...
Classical Propositional Logic
... logic, soundness was an intuitive notion, and asked rule-by-rule; the assumption seems to have been that a logical system is sound if and only if all its rules are sound. ...
... logic, soundness was an intuitive notion, and asked rule-by-rule; the assumption seems to have been that a logical system is sound if and only if all its rules are sound. ...
Modal fixpoint logic: some model theoretic questions
... of recursive principle. At the end of the 1970s, Amir Pnueli [Pnu77] argued that linear temporal logic (LTL), which is obtained by restricting to models based on the natural numbers and by adding the “until” operator to modal logic, could be a useful formalism in that respect. Since then, other temp ...
... of recursive principle. At the end of the 1970s, Amir Pnueli [Pnu77] argued that linear temporal logic (LTL), which is obtained by restricting to models based on the natural numbers and by adding the “until” operator to modal logic, could be a useful formalism in that respect. Since then, other temp ...
The greatest common divisor: a case study for program extraction
... Yiannis Moschovakis suggested the following example of a classical existence proof with a quantifier–free kernel which does not obviously contain an algorithm: the gcd of two natural numbers a1 and a2 is a linear combination of the two. Here we treat that example as a case study for program extracti ...
... Yiannis Moschovakis suggested the following example of a classical existence proof with a quantifier–free kernel which does not obviously contain an algorithm: the gcd of two natural numbers a1 and a2 is a linear combination of the two. Here we treat that example as a case study for program extracti ...
Modal Languages and Bounded Fragments of Predicate Logic
... with restricted choices of objects in each move – which has a natural generalization to the case with whole families of n-ary accessibility relations.) In the above theorem, the first-order formula may contain any other relation symbols, or equality, too. A formula φ with one free variable is invari ...
... with restricted choices of objects in each move – which has a natural generalization to the case with whole families of n-ary accessibility relations.) In the above theorem, the first-order formula may contain any other relation symbols, or equality, too. A formula φ with one free variable is invari ...
Relevant Logic A Philosophical Examination of Inference Stephen Read February 21, 2012
... approach, however, directly replaces the Classical Account with the Relevant Account, and extracts the notion of relevance from the new criterion for validity. For if the conclusion really does follow from the premises, then they must be (logically) relevant to it. Recognising the Relevant Account o ...
... approach, however, directly replaces the Classical Account with the Relevant Account, and extracts the notion of relevance from the new criterion for validity. For if the conclusion really does follow from the premises, then they must be (logically) relevant to it. Recognising the Relevant Account o ...
Predicate logic definitions
... Sentences P and Q of PLE are quantificationally equivalent if there is no interpretation on which P and Q have different truth values. A set Γ of sentences of PLE is quantificationally consistent if there is an interpretation on which every member of Γ is true. A set Γ of sentences of PLE quantific ...
... Sentences P and Q of PLE are quantificationally equivalent if there is no interpretation on which P and Q have different truth values. A set Γ of sentences of PLE is quantificationally consistent if there is an interpretation on which every member of Γ is true. A set Γ of sentences of PLE quantific ...
Chiron: A Set Theory with Types, Undefinedness, Quotation, and
... The usefulness of a logic is often measured by its expressivity: the more that can be expressed in the logic, the more useful the logic is. By a logic, we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may al ...
... The usefulness of a logic is often measured by its expressivity: the more that can be expressed in the logic, the more useful the logic is. By a logic, we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may al ...
Modal Consequence Relations
... a set Σ to a formula δ is valid in ` iff hΣ, δi ∈ `, for which we write Σ ` δ. δ is a tautology of ` if ∅ ` δ, for which we also write ` δ. In the early years, research into modal logic was concerned with the question of finding the correct inference rules. This research line is still there but has ...
... a set Σ to a formula δ is valid in ` iff hΣ, δi ∈ `, for which we write Σ ` δ. δ is a tautology of ` if ∅ ` δ, for which we also write ` δ. In the early years, research into modal logic was concerned with the question of finding the correct inference rules. This research line is still there but has ...
datalog - FORTH-ICS
... One way to think about this operator is that it applies rules on existing facts to get new facts according to the head of those rules. In general, for a recursive Datalog program, the same operator can be repeatedly applied on facts produced by previous applications of it. It is easy to see that the ...
... One way to think about this operator is that it applies rules on existing facts to get new facts according to the head of those rules. In general, for a recursive Datalog program, the same operator can be repeatedly applied on facts produced by previous applications of it. It is easy to see that the ...
Henkin`s Method and the Completeness Theorem
... sentence ϕ in the alphabet A, we will use the standard notation “` ϕ” for ϕ is provable in L (that is, ϕ is derivable from the axioms of L by the use of the inference rules of L); and “|= ϕ” for ϕ is valid (that is, ϕ is satisfied in every interpretation of L). The soundness theorem for L states tha ...
... sentence ϕ in the alphabet A, we will use the standard notation “` ϕ” for ϕ is provable in L (that is, ϕ is derivable from the axioms of L by the use of the inference rules of L); and “|= ϕ” for ϕ is valid (that is, ϕ is satisfied in every interpretation of L). The soundness theorem for L states tha ...