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Simplifying proofs in Fitchstyle natural deduction systems
, 2004
"... We present an algorithm for simplifying Fitchstyle natural deduction proofs in classical firstorder logic. We formalize Fitchstyle natural deduction as a denotational proof language, N DL, with a rigorous syntax and semantics. Based on that formalization, we define an array of simplifying transfo ..."
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We present an algorithm for simplifying Fitchstyle natural deduction proofs in classical firstorder logic. We formalize Fitchstyle natural deduction as a denotational proof language, N DL, with a rigorous syntax and semantics. Based on that formalization, we define an array of simplifying transformations and show them to be terminating and to respect the formal semantics of the language. We also show that the transformations never increase the size or complexity of a deduction—in the worst case, they produce deductions of the same size and complexity as the original. We present several examples of proofs containing various types of superfluous “detours, ” and explain how our procedure eliminates them, resulting in smaller and cleaner deductions. All of the transformations are fully implemented in SMLNJ, and the complete code listing is available. 1.1
Cut Formulae and Logic Programming
"... . In this paper we present a mechanism to define names for proofwitnesses of formulae and thus to use Gentzen's cutrule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proofsearch ..."
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. In this paper we present a mechanism to define names for proofwitnesses of formulae and thus to use Gentzen's cutrule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proofsearch by indicating when an instance of the cutrule should be attempted. By using the cutrule there are proofs that can be made dramatically shorter. We explain how this idea of using the cutrule can be applied to the logic of hereditary Harrop formulae. 1 Introduction The computation mechanisms both for logic and for functional programming are searches for cutfree proofs. First, in pure logic programming the achievement of a goal G w.r.t. a program P can be seen 1 as the search for a proof in Gentzen's intuitionistic sequent calculus LJ [Gen69], of the sequent P ) G, that by Gentzen's cutelimination theorem can be cutfree [Bee89], [Mil90]; a term found as a witness to a proof contains among...
A Dynamic Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded ..."
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded systems and cyberphysical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multiagent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic dynamics with hybrid systems. We survey dynamic logics for specifying and verifying properties for each of those classes of dynamical systems. A dynamic logic is a firstorder modal logic with a pair of parametrized modal operators for each dynamical system to express necessary or possible properties of their transition behavior. Due to their full basis of firstorder modal logic operators, dynamic logics can express a rich variety of system properties, including safety, controllability, reactivity, liveness, and quantified parametrized properties, even about
Proof Nets and the Identity of Proofs
, 2006
"... These are the notes for a 5lecturecourse given at ESSLLI 2006 in Malaga, Spain. The URL ..."
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These are the notes for a 5lecturecourse given at ESSLLI 2006 in Malaga, Spain. The URL
Notes Towards a Semantics for Proofsearch
"... Algorithmic proofsearch is an essential enabling technology throughout informatics. Proofsearch is the prooftheoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a welldeveloped semantics of proo ..."
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Algorithmic proofsearch is an essential enabling technology throughout informatics. Proofsearch is the prooftheoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a welldeveloped semantics of proofs, reductive logics are typically wellunderstood only operationally. Each deductive system can, typically, be read as a corresponding reductive system. We discuss some of the problems which must be addressed in order to provide a semantics of proofsearches of comparable value to the corresponding semantics of proofs. Just as the semantics of proofs is intimately related to the model theory of the underlying logic, so too should be the semantics of proofsearches. We discuss how to solve the problem of providing a semantics for proofsearches which adequately models both operational and logical aspects of the reductive system. 1
ANNALS OF PURE AND APPLIED LOGIC
, 1997
"... In [ 131 Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This was proved by noting that very large numbers can be actually constructed through very short proofs, ..."
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In [ 131 Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This was proved by noting that very large numbers can be actually constructed through very short proofs, A more refined analysis of these short proofs reveals the presence of cyclic paths in their logical graphs, Indeed, in [6] it is shown that cycles need to exist for the proofs to be short. Here, we present a new sequent calculus for classical logic which is close to linear logic in spirit, enjoys cutelimination, is acyclic and its proofs are just &~errtar~ ~ larger than proofs in LK. The proofs in the new calculus can bc obtained by a srn~ll perturhntim of proofs in LK and they represent a geometrical alternative for studying structural properties of LKproofs. They satisfy the constructive disjunction property and most important. simpler geometrical properties of their logical graphs. The geometrical counterpart to a cycle in LK is represented in the new setting by a spiwl which is passing through sets of formulas logically grouped together by the
AMeasureofInference in Classical and Intuitionistic Logics ∗
, 2011
"... This paper presents a measure of inference in classical and intuitionistic logics in the Gentzenstyle sequent calculi. The measure for a proof of a sequent is the width of the proof tree, that is, the number of leaves of the proof tree. Then the measure for a sequent is the minimum value of the wid ..."
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This paper presents a measure of inference in classical and intuitionistic logics in the Gentzenstyle sequent calculi. The measure for a proof of a sequent is the width of the proof tree, that is, the number of leaves of the proof tree. Then the measure for a sequent is the minimum value of the widths of possible proofs of the sequent; if it is unprovable, the assigned value is +∞ � It counts the indispensable cases for possible proofs of a sequent. By this measure, we can separate between sequents easy to be proved and ones difficult; we can go further than provability and/or unprovability � It is motivated by some economics/game theory problem (bounded rationality). However, it would be not straightforward to obtain the exact value of this measure for a given sequent. In this paper, we will develop a method of calculating the value of the measure. We will apply our measure to various classes of problems, for example, to evaluate the difficulty of proving contradictory sequents. We also exemplify our measure with a problem of game theoretical decision making. 1.