In proof theory, ludics is an analysis of the principles governing inference rules of mathematical logic. Key features of ludics include notion of compound connectives, using a technique known as focusing or focalisation (invented by the computer scientist Jean-Marc Andreoli), and its use of locations or loci over a base instead of propositions.
More precisely, ludics tries to retrieve known logical connectives and proof behaviours by following the paradigm of interactive computation, similarly to what is done in game semantics to which it is closely related. By abstracting the notion of formulae and focusing on their concrete uses—that is distinct occurrences—it provides an abstract syntax for computer science, as loci can be seen as pointers on memory.
The primary achievement of ludics is the discovery of a relationship between two natural, but distinct notions of type, or proposition.
The first view, which might be termed the proof-theoretic or Gentzen-style interpretation of propositions, says that the meaning of a proposition arises from its introduction and elimination rules. Focalization refines this viewpoint by distinguishing between positive propositions, whose meaning arises from their introduction rules, and negative propositions, whose meaning arises from their elimination rules. In focused calculi, it is possible to define positive connectives by giving only their introduction rules, with the shape of the elimination rules being forced by this choice. (Symmetrically, negative connectives can be defined in focused calculi by giving only the elimination rules, with the introduction rules forced by this choice.)
The second view, which might be termed the computational or Brouwer–Heyting–Kolmogorov interpretation of propositions, takes the view that we fix a computational system up front, and then give a realizability interpretation of propositions to give them constructive content. For example, a realizer for the proposition "A implies B" is a computable function that takes a realizer for A, and uses it to compute a realizer for B. Realizability models characterize realizers for propositions in terms of their visible behavior, and not in terms of their internal structure.
Girard shows that for second-orderaffinelinear logic, given a computational system with nontermination and error stops as effects, realizability and focalization give the same meaning to types.
Ludics was proposed by the logicianJean-Yves Girard. His paper introducing ludics, Locus solum: from the rules of logic to the logic of rules, has some features that may be seen as eccentric for a publication in mathematical logic (such as illustrations of skunks). It has to be noted that the intent of these features is to enforce the point of view of Jean-Yves Girard at the time of its writing. And, thus, it offers to readers the possibility to understand ludics independently of their backgrounds.[dubious–discuss]
Computation is any type of calculation that includes both arithmetical and non-arithmetical steps and which follows a well-defined model.
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
In mathematics, logic, and computer science, a type system is a formal system in which every term has a "type" which defines its meaning and the operations that may be performed on it. Type theory is the academic study of type systems.
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.
Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.
In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages. It does so by evaluating the meaning of syntactically valid strings defined by a specific programming language, showing the computation involved. In such a case that the evaluation would be of syntactically invalid strings, the result would be non-computation. Semantics describes the processes a computer follows when executing a program in that specific language. This can be shown by describing the relationship between the input and output of a program, or an explanation of how the program will be executed on a certain platform, hence creating a model of computation.
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics, as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction.
Game semantics is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes.
Noncommutative logic is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties, and the correctness criterion for its proof nets is given in terms of partial permutations. It also has a denotational semantics in which formulas are interpreted by modules over some specific Hopf algebras.
In proof theory, proof nets are a geometrical method of representing proofs that eliminates two forms of bureaucracy that differentiates proofs: (A) irrelevant syntactical features of regular proof calculi such as the natural deduction calculus and the sequent calculus, and (B) the order of rules applied in a derivation. In this way, the formal properties of proof identity correspond more closely to the intuitively desirable properties. Proof nets were introduced by Jean-Yves Girard.
In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof theory can be used to demonstrate such things as consistency, provide decision procedures, and allow mathematical or computational witnesses to be extracted as counterparts to theorems, the kind of task that is more often given to model theory.
Logics for computability are formulations of logic which capture some aspect of computability as a basic notion. This usually involves a mix of special logical connectives as well as semantics which explains how the logic is to be interpreted in a computational way.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another.
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and argument patterns symbolically, using formal systems such as first order logic. Within formal logic, mathematical logic studies the mathematical characteristics of logical systems, while philosophical logic applies them to philosophical problems such as the nature of meaning, knowledge, and existence. Systems of formal logic are also applied in other fields including linguistics, cognitive science, and computer science.
In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules. They were both proved inconsistent by Jean-Yves Girard in 1972. This result led to the realization that Martin-Löf's original 1971 type theory was inconsistent as it allowed the same "Type in Type" behaviour that Girard's paradox exploits.
In mathematical logic, the hypersequent framework is an extension of the proof-theoretical framework of sequent calculi used in structural proof theory to provide analytic calculi for logics which are not captured in the sequent framework. A hypersequent is usually taken to be a finite multiset of ordinary sequents, written
In mathematical logic, focused proofs are a family of analytic proofs that arise through goal-directed proof-search, and are a topic of study in structural proof theory and reductive logic. They form the most general definition of goal-directed proof-search—in which someone chooses a formula and performs hereditary reductions until the result meets some condition. The extremal case where reduction only terminates when axioms are reached forms the sub-family of uniform proofs.
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