DOCSLIB.ORG

Sequent Calculus Cut Elimination

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sequent Calculus Cut Elimination Sequent Calculus Cut Elimination Akim Demaille [email protected] EPITA — École Pour l’Informatique et les Techniques Avancées June 10, 2016 Sequent Calculus Cut Elimination 1 Problems of Natural Deduction 2 Sequent Calculus 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 2 / 40 Preamble The following slides are implicitly dedicated to classical logic. A. Demaille Sequent Calculus, Cut Elimination 3 / 40 Problems of Natural Deduction 1 Problems of Natural Deduction 2 Sequent Calculus 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 4 / 40 Normalization A. Demaille Sequent Calculus, Cut Elimination 5 / 40 Proofs hard to find Some elimination rules used formulas coming out of the blue. [A] [B] · · · · · · · · · A _ B C C _E C A. Demaille Sequent Calculus, Cut Elimination 6 / 40 Negation is awkward A. Demaille Sequent Calculus, Cut Elimination 7 / 40 Sequent Calculus 1 Problems of Natural Deduction 2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 8 / 40 Gerhard Karl Erich Gentzen (1909–1945) German logician and mathematician. SA (1933) An assistant of David Hilbert in Göttingen (1935–1939). Joined the Nazi Party (1937). Worked on the V2. Died of malnutrition in a prison camp (August 4, 1945). A. Demaille Sequent Calculus, Cut Elimination 9 / 40 Syntax 1 Problems of Natural Deduction 2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 10 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Reading a Sequent If all the formulas of Γ are true, Γ ` ∆ then one of the formulas of ∆ is true. Commas on the left hand side stand for “and” Turnstile, `, stands for “implies” Commas on the right hand side stand for “or” A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent If all the formulas of Γ are true, Γ ` ∆ then one of the formulas of ∆ is true. Commas on the left hand side stand for “and” Turnstile, `, stands for “implies” Commas on the right hand side stand for “or” A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent If all the formulas of Γ are true, Γ ` ∆ then one of the formulas of ∆ is true. Commas on the left hand side stand for “and” Turnstile, `, stands for “implies” Commas on the right hand side stand for “or” A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent If all the formulas of Γ are true, Γ ` ∆ then one of the formulas of ∆ is true. Commas on the left hand side stand for “and” Turnstile, `, stands for “implies” Commas on the right hand side stand for “or” A. Demaille Sequent Calculus, Cut Elimination 12 / 40 ` AA is true Γ ` Γ is in contradiction A ` :A ` contradiction Some Special Sequents Γ ` AA is true under the hypotheses Γ A. Demaille Sequent Calculus, Cut Elimination 13 / 40 Γ ` Γ is in contradiction A ` :A ` contradiction Some Special Sequents Γ ` AA is true under the hypotheses Γ ` AA is true A. Demaille Sequent Calculus, Cut Elimination 13 / 40 A ` :A ` contradiction Some Special Sequents Γ ` AA is true under the hypotheses Γ ` AA is true Γ ` Γ is in contradiction A. Demaille Sequent Calculus, Cut Elimination 13 / 40 ` contradiction Some Special Sequents Γ ` AA is true under the hypotheses Γ ` AA is true Γ ` Γ is in contradiction A ` :A A. Demaille Sequent Calculus, Cut Elimination 13 / 40 Some Special Sequents Γ ` AA is true under the hypotheses Γ ` AA is true Γ ` Γ is in contradiction A ` :A ` contradiction A. Demaille Sequent Calculus, Cut Elimination 13 / 40 LK — Classical Sequent Calculus 1 Problems of Natural Deduction 2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 14 / 40 LK — Classical Sequent Calculus LK — Gentzen 1934 logistischerklassischer Kalkül. There are several possible exposures Different sets of inference rules We follow [Girard, 2011] A. Demaille Sequent Calculus, Cut Elimination 15 / 40 Identity Group Γ ` A; ∆ Γ0; A ` ∆0 Id Cut A ` A Γ; Γ0 ` ∆; ∆0 Gentzen’s Hauptsatz The cut rule is redundant, i.e., any sequent provable with cuts is provable without. A. Demaille Sequent Calculus, Cut Elimination 16 / 40 Identity Group Γ ` A; ∆ Γ0; A ` ∆0 Id Cut A ` A Γ; Γ0 ` ∆; ∆0 Gentzen’s Hauptsatz The cut rule is redundant, i.e., any sequent provable with cuts is provable without. A. Demaille Sequent Calculus, Cut Elimination 16 / 40 Γ ` ∆ Γ ` ∆ `W W` Γ ` A; ∆ Γ; A ` ∆ Γ ` A; A; ∆ Γ; A; A ` ∆ `C C` Γ ` A; ∆ Γ; A ` ∆ Structural Group Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆ A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Γ ` A; A; ∆ Γ; A; A ` ∆ `C C` Γ ` A; ∆ Γ; A ` ∆ Structural Group Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆ Γ ` ∆ Γ ` ∆ `W W` Γ ` A; ∆ Γ; A ` ∆ A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Structural Group Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆ Γ ` ∆ Γ ` ∆ `W W` Γ ` A; ∆ Γ; A ` ∆ Γ ` A; A; ∆ Γ; A; A ` ∆ `C C` Γ ` A; ∆ Γ; A ` ∆ A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Logical Group: Negation Γ; A ` ∆ Γ ` A; ∆ `: :` Γ ` :A; ∆ Γ; :A ` ∆ A. Demaille Sequent Calculus, Cut Elimination 18 / 40 Additive ˘ Multiplicative ⊗ Γ; A ` ∆ l^` Γ; A ^ B ` ∆ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ Logical Group: Conjunction Γ ` A; ∆ Γ ` B; ∆ `^ Γ ` A ^ B; ∆ A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Additive ˘ Multiplicative ⊗ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ Logical Group: Conjunction Γ; A ` ∆ l^` Γ ` A; ∆ Γ ` B; ∆ Γ; A ^ B ` ∆ `^ Γ ` A ^ B; ∆ A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Multiplicative ⊗ Additive ˘ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ Logical Group: Conjunction Γ; A ` ∆ l^` Γ ` A; ∆ Γ ` B; ∆ Γ; A ^ B ` ∆ `^ Γ ` A ^ B; ∆ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Additive ˘ Multiplicative ⊗ Logical Group: Conjunction Γ; A ` ∆ l^` Γ ` A; ∆ Γ ` B; ∆ Γ; A ^ B ` ∆ `^ Γ ` A ^ B; ∆ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ A. Demaille Sequent Calculus, Cut Elimination 19 / 40 ˘ ⊗ Logical Group: Conjunction Γ; A ` ∆ l^` Γ ` A; ∆ Γ ` B; ∆ Γ; A ^ B ` ∆ Additive `^ Γ ` A ^ B; ∆ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ Multiplicative `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ A.
Load more

Recommended publications
  • Nested Sequents, a Natural Generalisation of Hypersequents, Allow Us to Develop a Systematic Proof Theory for Modal Logics
    Nested Sequents Habilitationsschrift Kai Br¨unnler Institut f¨ur Informatik und angewandte Mathematik Universit¨at Bern May 28, 2018 arXiv:1004.1845v1 [cs.LO] 11 Apr 2010 Abstract We see how nested sequents, a natural generalisation of hypersequents, allow us to develop a systematic proof theory for modal logics. As opposed to other prominent formalisms, such as the display calculus and labelled sequents, nested sequents stay inside the modal language and allow for proof systems which enjoy the subformula property in the literal sense. In the first part we study a systematic set of nested sequent systems for all normal modal logics formed by some combination of the axioms for seriality, reflexivity, symmetry, transitivity and euclideanness. We establish soundness and completeness and some of their good properties, such as invertibility of all rules, admissibility of the structural rules, termination of proof-search, as well as syntactic cut-elimination. In the second part we study the logic of common knowledge, a modal logic with a fixpoint modality. We look at two infinitary proof systems for this logic: an existing one based on ordinary sequents, for which no syntactic cut-elimination procedure is known, and a new one based on nested sequents. We see how nested sequents, in contrast to ordinary sequents, allow for syntactic cut-elimination and thus allow us to obtain an ordinal upper bound on the length of proofs. iii Contents 1 Introduction 1 2 Systems for Basic Normal Modal Logics 5 2.1 ModalAxiomsasLogicalRules . 6 2.1.1 TheSequentSystems .................... 6 2.1.2 Soundness........................... 12 2.1.3 Completeness........................
    [Show full text]
  • Sequent Calculus and the Specification of Computation
    Sequent Calculus and the Specification of Computation Lecture Notes Draft: 5 September 2001 These notes are being used with lectures for CSE 597 at Penn State in Fall 2001. These notes had been prepared to accompany lectures given at the Marktoberdorf Summer School, 29 July – 10 August 1997 for a course titled “Proof Theoretic Specification of Computation”. An earlier draft was used in courses given between 27 January and 11 February 1997 at the University of Siena and between 6 and 21 March 1997 at the University of Genoa. A version of these notes was also used at Penn State for CSE 522 in Fall 2000. For the most recent version of these notes, contact the author. °c Dale Miller 321 Pond Lab Computer Science and Engineering Department The Pennsylvania State University University Park, PA 16802-6106 USA Office: +(814)865-9505 [email protected] http://www.cse.psu.edu/»dale Contents 1 Overview and Motivation 5 1.1 Roles for logic in the specification of computations . 5 1.2 Desiderata for declarative programming . 6 1.3 Relating algorithm and logic . 6 2 First-Order Logic 8 2.1 Types and signatures . 8 2.2 First-order terms and formulas . 8 2.3 Sequent calculus . 10 2.4 Classical, intuitionistic, and minimal logics . 10 2.5 Permutations of inference rules . 12 2.6 Cut-elimination . 12 2.7 Consequences of cut-elimination . 13 2.8 Additional readings . 13 2.9 Exercises . 13 3 Logic Programming Considered Abstractly 15 3.1 Problems with proof search . 15 3.2 Interpretation as goal-directed search .
    [Show full text]
  • Chapter 1 Preliminaries
    Chapter 1 Preliminaries 1.1 First-order theories In this section we recall some basic definitions and facts about first-order theories. Most proofs are postponed to the end of the chapter, as exercises for the reader. 1.1.1 Definitions Definition 1.1 (First-order theory) —A first-order theory T is defined by: A first-order language L , whose terms (notation: t, u, etc.) and formulas (notation: φ, • ψ, etc.) are constructed from a fixed set of function symbols (notation: f , g, h, etc.) and of predicate symbols (notation: P, Q, R, etc.) using the following grammar: Terms t ::= x f (t1,..., tk) | Formulas φ, ψ ::= t1 = t2 P(t1,..., tk) φ | | > | ⊥ | ¬ φ ψ φ ψ φ ψ x φ x φ | ⇒ | ∧ | ∨ | ∀ | ∃ (assuming that each use of a function symbol f or of a predicate symbol P matches its arity). As usual, we call a constant symbol any function symbol of arity 0. A set of closed formulas of the language L , written Ax(T ), whose elements are called • the axioms of the theory T . Given a closed formula φ of the language of T , we say that φ is derivable in T —or that φ is a theorem of T —and write T φ when there are finitely many axioms φ , . , φ Ax(T ) such ` 1 n ∈ that the sequent φ , . , φ φ (or the formula φ φ φ) is derivable in a deduction 1 n ` 1 ∧ · · · ∧ n ⇒ system for classical logic. The set of all theorems of T is written Th(T ). Conventions 1.2 (1) In this course, we only work with first-order theories with equality.
    [Show full text]
  • Sequent-Type Calculi for Systems of Nonmonotonic Paraconsistent Logics
    Sequent-Type Calculi for Systems of Nonmonotonic Paraconsistent Logics Tobias Geibinger Hans Tompits Databases and Artificial Intelligence Group, Knowledge-Based Systems Group, Institute of Logic and Computation, Institute of Logic and Computation, Technische Universit¨at Wien, Technische Universit¨at Wien, Favoritenstraße 9-11, A-1040 Vienna, Austria Favoritenstraße 9-11, A-1040 Vienna, Austria [email protected] [email protected] Paraconsistent logics constitute an important class of formalisms dealing with non-trivial reasoning from inconsistent premisses. In this paper, we introduce uniform axiomatisations for a family of nonmonotonic paraconsistent logics based on minimal inconsistency in terms of sequent-type proof systems. The latter are prominent and widely-used forms of calculi well-suited for analysing proof search. In particular, we provide sequent-type calculi for Priest’s three-valued minimally inconsistent logic of paradox, and for four-valued paraconsistent inference relations due to Arieli and Avron. Our calculi follow the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti and Olivetti, whose distinguishing feature is the use of a so-called rejection calculus for axiomatising invalid formulas. In fact, we present a general method to obtain sequent systems for any many-valued logic based on minimal inconsistency, yielding the calculi for the logics of Priest and of Arieli and Avron as special instances. 1 Introduction Paraconsistent logics reject the principle of explosion, also known as ex falso sequitur quodlibet, which holds in classical logic and allows the derivation of any assertion from a contradiction. The motivation behind paraconsistent logics is simple, as contradictory theories may still contain useful information, hence we would like to be able to draw non-trivial conclusions from said theories.
    [Show full text]
  • Theorem Proving in Classical Logic
    MEng Individual Project Imperial College London Department of Computing Theorem Proving in Classical Logic Supervisor: Dr. Steffen van Bakel Author: David Davies Second Marker: Dr. Nicolas Wu June 16, 2021 Abstract It is well known that functional programming and logic are deeply intertwined. This has led to many systems capable of expressing both propositional and first order logic, that also operate as well-typed programs. What currently ties popular theorem provers together is their basis in intuitionistic logic, where one cannot prove the law of the excluded middle, ‘A A’ – that any proposition is either true or false. In classical logic this notion is provable, and the_: corresponding programs turn out to be those with control operators. In this report, we explore and expand upon the research about calculi that correspond with classical logic; and the problems that occur for those relating to first order logic. To see how these calculi behave in practice, we develop and implement functional languages for propositional and first order logic, expressing classical calculi in the setting of a theorem prover, much like Agda and Coq. In the first order language, users are able to define inductive data and record types; importantly, they are able to write computable programs that have a correspondence with classical propositions. Acknowledgements I would like to thank Steffen van Bakel, my supervisor, for his support throughout this project and helping find a topic of study based on my interests, for which I am incredibly grateful. His insight and advice have been invaluable. I would also like to thank my second marker, Nicolas Wu, for introducing me to the world of dependent types, and suggesting useful resources that have aided me greatly during this report.
    [Show full text]
  • Systematic Construction of Natural Deduction Systems for Many-Valued Logics
    23rd International Symposium on Multiple Valued Logic. Sacramento, CA, May 1993 Proceedings. (IEEE Press, Los Alamitos, 1993) pp. 208{213 Systematic Construction of Natural Deduction Systems for Many-valued Logics Matthias Baaz∗ Christian G. Ferm¨ullery Richard Zachy Technische Universit¨atWien, Austria Abstract sion.) Each position i corresponds to one of the truth values, vm is the distinguished truth value. The in- A construction principle for natural deduction sys- tended meaning is as follows: Derive that at least tems for arbitrary finitely-many-valued first order log- one formula of Γm takes the value vm under the as- ics is exhibited. These systems are systematically ob- sumption that no formula in Γi takes the value vi tained from sequent calculi, which in turn can be au- (1 i m 1). ≤ ≤ − tomatically extracted from the truth tables of the log- Our starting point for the construction of natural ics under consideration. Soundness and cut-free com- deduction systems are sequent calculi. (A sequent is pleteness of these sequent calculi translate into sound- a tuple Γ1 ::: Γm, defined to be satisfied by an ness, completeness and normal form theorems for the interpretationj iffj for some i 1; : : : ; m at least one 2 f g natural deduction systems. formula in Γi takes the truth value vi.) For each pair of an operator 2 or quantifier Q and a truth value vi 1 Introduction we construct a rule introducing a formula of the form 2(A ;:::;A ) or (Qx)A(x), respectively, at position i The study of natural deduction systems for many- 1 n of a sequent.
    [Show full text]
  • Linear Logic Programming Dale Miller INRIA/Futurs & Laboratoire D’Informatique (LIX) Ecole´ Polytechnique, Rue De Saclay 91128 PALAISEAU Cedex FRANCE
    1 An Overview of Linear Logic Programming Dale Miller INRIA/Futurs & Laboratoire d’Informatique (LIX) Ecole´ polytechnique, Rue de Saclay 91128 PALAISEAU Cedex FRANCE Abstract Logic programming can be given a foundation in sequent calculus by viewing computation as the process of building a cut-free sequent proof bottom-up. The first accounts of logic programming as proof search were given in classical and intuitionistic logic. Given that linear logic allows richer sequents and richer dynamics in the rewriting of sequents during proof search, it was inevitable that linear logic would be used to design new and more expressive logic programming languages. We overview how linear logic has been used to design such new languages and describe briefly some applications and implementation issues for them. 1.1 Introduction It is now commonplace to recognize the important role of logic in the foundations of computer science. When a major new advance is made in our understanding of logic, we can thus expect to see that advance ripple into many areas of computer science. Such rippling has been observed during the years since the introduction of linear logic by Girard in 1987 [Gir87]. Since linear logic embraces computational themes directly in its design, it often allows direct and declarative approaches to compu- tational and resource sensitive specifications. Linear logic also provides new insights into the many computational systems based on classical and intuitionistic logics since it refines and extends these logics. There are two broad approaches by which logic, via the theory of proofs, is used to describe computation [Mil93]. One approach is the proof reduction paradigm, which can be seen as a foundation for func- 1 2 Dale Miller tional programming.
    [Show full text]
  • Permutability of Proofs in Intuitionistic Sequent Calculi
    Permutabilityofproofsinintuitionistic sequent calculi Roy Dyckho Scho ol of Mathematical Computational Sciences St Andrews University St Andrews Scotland y Lus Pinto Departamento de Matematica Universidade do Minho Braga Portugal Abstract Weprove a folklore theorem that two derivations in a cutfree se quent calculus for intuitioni sti c prop ositional logic based on Kleenes G are interp ermutable using a set of basic p ermutation reduction rules derived from Kleenes work in i they determine the same natu ral deduction The basic rules form a conuentandweakly normalisin g rewriting system We refer to Schwichtenb ergs pro of elsewhere that a mo dication of this system is strongly normalising Key words intuitionistic logic pro of theory natural deduction sequent calcu lus Intro duction There is a folklore theorem that twointuitionistic sequent calculus derivations are really the same i they are interp ermutable using p ermutations as de scrib ed by Kleene in Our purp ose here is to make precise and provesuch a p ermutability theorem Prawitz showed howintuitionistic sequent calculus derivations determine LJ to NJ here we consider only natural deductions via a mapping from the cutfree derivations and normal natural deductions resp ectively and in eect that this mapping is surjectiveby constructing a rightinverse of from NJ to LJ Zucker showed that in the negative fragment of the calculus c LJ ie LJ including cut two derivations have the same image under i they are interconvertible using a sequence of p ermutativeconversions eg p
    [Show full text]
  • Propositional Calculus
    CSC 438F/2404F Notes Winter, 2014 S. Cook REFERENCES The first two references have especially influenced these notes and are cited from time to time: [Buss] Samuel Buss: Chapter I: An introduction to proof theory, in Handbook of Proof Theory, Samuel Buss Ed., Elsevier, 1998, pp1-78. [B&M] John Bell and Moshe Machover: A Course in Mathematical Logic. North- Holland, 1977. Other logic texts: The first is more elementary and readable. [Enderton] Herbert Enderton: A Mathematical Introduction to Logic. Academic Press, 1972. [Mendelson] E. Mendelson: Introduction to Mathematical Logic. Wadsworth & Brooks/Cole, 1987. Computability text: [Sipser] Michael Sipser: Introduction to the Theory of Computation. PWS, 1997. [DSW] M. Davis, R. Sigal, and E. Weyuker: Computability, Complexity and Lan- guages: Fundamentals of Theoretical Computer Science. Academic Press, 1994. Propositional Calculus Throughout our treatment of formal logic it is important to distinguish between syntax and semantics. Syntax is concerned with the structure of strings of symbols (e.g. formulas and formal proofs), and rules for manipulating them, without regard to their meaning. Semantics is concerned with their meaning. 1 Syntax Formulas are certain strings of symbols as specified below. In this chapter we use formula to mean propositional formula. Later the meaning of formula will be extended to first-order formula. (Propositional) formulas are built from atoms P1;P2;P3;:::, the unary connective :, the binary connectives ^; _; and parentheses (,). (The symbols :; ^ and _ are read \not", \and" and \or", respectively.) We use P; Q; R; ::: to stand for atoms. Formulas are defined recursively as follows: Definition of Propositional Formula 1) Any atom P is a formula.
    [Show full text]
  • Focusing in Linear Meta-Logic: Extended Report Vivek Nigam, Dale Miller
    Focusing in linear meta-logic: extended report Vivek Nigam, Dale Miller To cite this version: Vivek Nigam, Dale Miller. Focusing in linear meta-logic: extended report. [Research Report] Maybe this file need a better style. Specially the proofs., 2008. inria-00281631 HAL Id: inria-00281631 https://hal.inria.fr/inria-00281631 Submitted on 24 May 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Focusing in linear meta-logic: extended report Vivek Nigam and Dale Miller INRIA & LIX/Ecole´ Polytechnique, Palaiseau, France nigam at lix.polytechnique.fr dale.miller at inria.fr May 24, 2008 Abstract. It is well known how to use an intuitionistic meta-logic to specify natural deduction systems. It is also possible to use linear logic as a meta-logic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusing annotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and non-normal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations.
    [Show full text]
  • On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem
    On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem Yannick Forster Dominik Kirst Gert Smolka Saarland University Saarland University Saarland University Saarbrücken, Germany Saarbrücken, Germany Saarbrücken, Germany [email protected] [email protected] [email protected] Abstract like decidability, enumerability, and reductions are avail- We formalise the computational undecidability of validity, able without reference to a concrete model of computation satisfiability, and provability of first-order formulas follow- such as Turing machines, general recursive functions, or ing a synthetic approach based on the computation native the λ-calculus. For instance, representing a given decision to Coq’s constructive type theory. Concretely, we consider problem by a predicate p on a type X, a function f : X ! B Tarski and Kripke semantics as well as classical and intu- with 8x: p x $ f x = tt is a decision procedure, a function itionistic natural deduction systems and provide compact д : N ! X with 8x: p x $ ¹9n: д n = xº is an enumer- many-one reductions from the Post correspondence prob- ation, and a function h : X ! Y with 8x: p x $ q ¹h xº lem (PCP). Moreover, developing a basic framework for syn- for a predicate q on a type Y is a many-one reduction from thetic computability theory in Coq, we formalise standard p to q. Working formally with concrete models instead is results concerning decidability, enumerability, and reducibil- cumbersome, given that every defined procedure needs to ity without reference to a concrete model of computation. be shown representable by a concrete entity of the model.
    [Show full text]
  • Gentzen Sequent Calculus GL
    Chapter 11 (Part 2) Gentzen Sequent Calculus GL The proof system GL for the classical propo- sitional logic is a version of the original Gentzen (1934) systems LK. A constructive proof of the completeness the- orem for the system GL is very similar to the proof of the completeness theorem for the system RS. Expressions of the system like in the original Gentzen system LK are Gentzen sequents. Hence we use also a name Gentzen sequent calculus. 1 Language of GL: L = Lf[;\;);:;g. We add a new symbol to the alphabet: ¡!. It is called a Gentzen arrow. The sequents are built out of ¯nite sequences (empty included) of formulas, i.e. elements of F¤, and the additional sign ¡!. We denote, as in the RS system, the ¯nite sequences of formulas by Greek capital let- ters ¡; ¢; §, with indices if necessary. Sequent de¯nition: a sequent is the expres- sion ¡ ¡! ¢; where ¡; ¢ 2 F¤. Meaning of sequents Intuitively, we interpret a sequent A1; :::; An ¡! B1; :::; Bm; where n; m ¸ 1 as a formula (A1 \ ::: \ An) ) (B1 [ ::: [ Bm): The sequent: A1; :::; An ¡! (where n ¸ 1) means that A1 \ ::: \ An yields a contra- diction. The sequent ¡! B1; :::; Bm (where m ¸ 1) means j= (B1 [ ::: [ Bm). The empty sequent: ¡! means a contra- diction. 2 Given non empty sequences: ¡, ¢, we de- note by σ¡ any conjunction of all formulas of ¡, and by ±¢ any disjunction of all formulas of ¢. The intuitive semantics (meaning, interpre- tation) of a sequent ¡ ¡! ¢ (where ¡; ¢ are nonempty) is ¡ ¡! ¢ ´ (σ¡ ) ±¢): 3 Formal semantics for sequents (expressions of GL) Let v : V AR ¡! fT;F g be a truth assign- ment, v¤ its (classical semantics) extension to the set of formulas F.
    [Show full text]