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A Framework for Defining Logics
 JOURNAL OF THE ASSOCIATION FOR COMPUTING MACHINERY
, 1993
"... The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed calculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof's system of arities. T ..."
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Cited by 695 (39 self)
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The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed calculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higherorder judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logicindependent tools such as proof editors and proof checkers can be constructed.
Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing b ..."
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Cited by 113 (22 self)
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
Lower Bounds to the Size of ConstantDepth Propositional Proofs
, 1994
"... 1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp ..."
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Cited by 53 (6 self)
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1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp(O(log 2 n)) but such that every depth d refutation must have the size at least exp(n\Omega\Gamma21 ). The sets T d n express a weaker form of the pigeonhole principle. It is a fundamental problem of mathematical logic and complexity theory whether there exists a proof system for propositional logic in which every tautology has a short proof, where the length (equivalently the size) of a proof is measured essentially by the total number of symbols in it and short means polynomial in the length of the tautology. Equivalently one can ask whether for every theory T there is another theory S (both first order and reasonably axiomatized, e.g. by schemes) having the property that if a statement...
What's so special about Kruskal's Theorem AND THE ORDINAL Γ0? A SURVEY OF SOME RESULTS IN PROOF THEORY
 ANNALS OF PURE AND APPLIED LOGIC, 53 (1991), 199260
, 1991
"... This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, an ..."
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Cited by 43 (3 self)
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This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of secondorder logic, slowgrowing and fastgrowing hierarchies including Girard’s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the “tree theorem”, as well as a “finite miniaturization ” of Kruskal’s theorem due to Harvey Friedman. These versions of Kruskal’s theorem are remarkable from a prooftheoretic point of view because they are not provable in relatively strong logical systems. They are examples of socalled “natural independence phenomena”, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal’s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of KnuthBendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ0 and Kruskal’s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence.
Feature Logics
 HANDBOOK OF LOGIC AND LANGUAGE, EDITED BY VAN BENTHEM & TER MEULEN
, 1994
"... Feature logics form a class of specialized logics which have proven especially useful in classifying and constraining the linguistic objects known as feature structures. Linguistically, these structures have their origin in the work of the Prague school of linguistics, followed by the work of Chom ..."
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Cited by 33 (0 self)
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Feature logics form a class of specialized logics which have proven especially useful in classifying and constraining the linguistic objects known as feature structures. Linguistically, these structures have their origin in the work of the Prague school of linguistics, followed by the work of Chomsky and Halle in The Sound Pattern of English [16]. Feature structures have been reinvented several times by computer scientists: in the theory of data structures, where they are known as record structures, in artificial intelligence, where they are known as frame or slotvalue structures, in the theory of data bases, where they are called "complex objects", and in computati
An Equivalence between Second Order Bounded Domain Bounded Arithmetic and First Order Bounded Arithmetic
, 1993
"... We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and pr ..."
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Cited by 28 (4 self)
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We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and preserve the syntactic structure of bounded formulae. As a corollary, for the bounded domain case we obtain Buss's result concerning 1 expressibility in V 2 as a direct consequence of his main result for rst order theories. Using only plain corollaries of the cut elimination theorem, we show that V 2 (BD) prove the same formulae where 8 stand for rst order quanti ers. Combined with the above mentioned result this gives an alternative proof of Buss's characterization of 2 functions. All this readily extends to the case V k (BD) vs. S k+1 (i; k 1).
On transforming intuitionistic matrix proofs into standardsequent proofs
 TABLEAUX–95, LNAI 918
, 1995
"... We present a procedure transforming intuitionistic matrix proofs into proofs within the intuitionistic standard sequent calculus. The transformation is based on L. Wallen’s proof justifying his matrix characterization for the validity of intuitionistic formulae. Since this proof makes use of Fitting ..."
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Cited by 26 (15 self)
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We present a procedure transforming intuitionistic matrix proofs into proofs within the intuitionistic standard sequent calculus. The transformation is based on L. Wallen’s proof justifying his matrix characterization for the validity of intuitionistic formulae. Since this proof makes use of Fitting‘s nonstandard sequent calculus our procedure consists of two steps. First a nonstandard sequent proof will be extracted from a given matrix proof. Secondly we transform each nonstandard proof into a standard proof in a structure preserving way. To simplify the latter step we introduce an extended standard calculus which is shown to be sound and complete.
Generic Automatic Proof Tools
, 1997
"... This article explores a synthesis between two distinct traditions in automated reasoning: resolution and interaction. In particular it discusses Isabelle, an interactive theorem prover based upon a form of resolution. It aims to demonstrate the value of proof tools that, compared with traditional re ..."
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Cited by 26 (9 self)
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This article explores a synthesis between two distinct traditions in automated reasoning: resolution and interaction. In particular it discusses Isabelle, an interactive theorem prover based upon a form of resolution. It aims to demonstrate the value of proof tools that, compared with traditional resolution systems, seem absurdly limited. Isabelle's classical reasoner searches for proofs using a tableau approach. The reasoner is generic: it accepts rules proved in applied theories, involving defined connectives. The reasoner works in a variety of domains without reducing them to firstorder logic. Resolution systems such as Otter [13], setheo [11] and pttp [34] represent automatic theorem proving at its highest point of refinement. They achieve extremely high inference rates and can run continuously for days without running out of storage. They can crack many of the toughest challenge problems that have been circulated. While they exploit many specialized algorithms, data structures and optimizations, they rely crucially on unification. Interactive systems let the user direct each step of the proof. They can implement complicated formalisms, chosen for maximum expressiveness, and typically based on the typed calculus. hol [7, 8] and pvs [23] are used for verification of hardware and realtime systems, while Coq [4] is used for formalizing mathematics. Large numbers of axioms  say, the description of a cpu design  do not overwhelm them, because finding the proof is the user's job. Partial automation is sometimes provided, but a resolution enthusiast would regret the lack of uniform search procedures based on unification. One procedure provided by most interactive provers is rewriting. Rewrite rules have many advantages. Unlike programmed inference rules, they are ...
The Undecidability of kProvability
 Annals of Pure and Applied Logic
, 1989
"... The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X... ..."
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Cited by 25 (0 self)
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The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X...
Dependent choices, ‘quote’ and the clock
 Th. Comp. Sc
, 2003
"... When using the CurryHoward correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem: to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an inst ..."
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Cited by 24 (9 self)
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When using the CurryHoward correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem: to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an instruction of our programming language. This problem