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Journal articles on the topic 'Higher-order logic'

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Published: 4 June 2021
Last updated: 31 January 2023

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1

Bruce, Kim, Johan van Benthem, and Kees Doets. "Higher-order Logic." Journal of Symbolic Logic 54, no. 3 (September 1989): 1090. http://dx.doi.org/10.2307/2274769.

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2

Forster, Thomas. "A Consistent Higher-Order Theory Without a (Higher-Order) Model." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 35, no. 5 (1989): 385–86. http://dx.doi.org/10.1002/malq.19890350502.

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3

Andrews, James H. "An untyped higher order logic with Y combinator." Journal of Symbolic Logic 72, no. 4 (December 2007): 1385–404. http://dx.doi.org/10.2178/jsl/1203350794.

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AbstractWe define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.
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4

Audenaert, Pieter. "The Higher-Order-Logic Formath." Bulletin of the Belgian Mathematical Society - Simon Stevin 15, no. 2 (May 2008): 335–67. http://dx.doi.org/10.36045/bbms/1210254829.

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5

Czajka, Łukasz. "Higher-Order Illative Combinatory Logic." Journal of Symbolic Logic 78, no. 3 (September 2013): 837–72. http://dx.doi.org/10.2178/jsl.7803080.

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AbstractWe show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.
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6

Charalambidis, Angelos, Konstantinos Handjopoulos, Panagiotis Rondogiannis, and William W. Wadge. "Extensional Higher-Order Logic Programming." ACM Transactions on Computational Logic 14, no. 3 (August 2013): 1–40. http://dx.doi.org/10.1145/2499937.2499942.

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7

Cropper, Andrew, Rolf Morel, and Stephen Muggleton. "Learning higher-order logic programs." Machine Learning 109, no. 7 (December 3, 2019): 1289–322. http://dx.doi.org/10.1007/s10994-019-05862-7.

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AbstractA key feature of inductive logic programming is its ability to learn first-order programs, which are intrinsically more expressive than propositional programs. In this paper, we introduce techniques to learn higher-order programs. Specifically, we extend meta-interpretive learning (MIL) to support learning higher-order programs by allowing for higher-order definitions to be used as background knowledge. Our theoretical results show that learning higher-order programs, rather than first-order programs, can reduce the textual complexity required to express programs, which in turn reduces the size of the hypothesis space and sample complexity. We implement our idea in two new MIL systems: the Prolog system $$\text {Metagol}_{ho}$$ Metagol ho and the ASP system $$\text {HEXMIL}_{ho}$$ HEXMIL ho . Both systems support learning higher-order programs and higher-order predicate invention, such as inventing functions for and conditions for . We conduct experiments on four domains (robot strategies, chess playing, list transformations, and string decryption) that compare learning first-order and higher-order programs. Our experimental results support our theoretical claims and show that, compared to learning first-order programs, learning higher-order programs can significantly improve predictive accuracies and reduce learning times.
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8

Hetzl, Stefan, Alexander Leitsch, and Daniel Weller. "CERES in higher-order logic." Annals of Pure and Applied Logic 162, no. 12 (December 2011): 1001–34. http://dx.doi.org/10.1016/j.apal.2011.06.005.

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9

Awodey, S., and C. Butz. "Topological completeness for higher-order logic." Journal of Symbolic Logic 65, no. 3 (September 2000): 1168–82. http://dx.doi.org/10.2307/2586693.

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AbstractUsing recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces—so-called “topological semantics”. The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.
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10

Simons, Peter. "Who's Afraid of Higher-Order Logic?" Grazer Philosophische Studien 44 (1993): 253–64. http://dx.doi.org/10.5840/gps19934443.

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11

Bell, J. L., J. Lambek, and P. J. Scott. "Introduction to Higher Order Categorical Logic." Journal of Symbolic Logic 54, no. 3 (September 1989): 1113. http://dx.doi.org/10.2307/2274784.

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12

Simons, Peter. "WHO’S AFRAID OF HIGHER-ORDER LOGIC?" Grazer Philosophische studien 44, no. 1 (August 13, 1993): 253–64. http://dx.doi.org/10.1163/18756735-90000529.

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13

Rota, Gian-Carlo. "Introduction to higher order categorical logic." Advances in Mathematics 67, no. 2 (February 1988): 239. http://dx.doi.org/10.1016/0001-8708(88)90045-x.

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14

Andrews, Peter B. "On connections and higher-order logic." Journal of Automated Reasoning 5, no. 3 (September 1989): 257–91. http://dx.doi.org/10.1007/bf00248320.

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15

Ballard, David. "Independence in higher-order subclassical logic." Notre Dame Journal of Formal Logic 26, no. 4 (October 1985): 444–54. http://dx.doi.org/10.1305/ndjfl/1093870936.

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16

Kumar, Ramana, Rob Arthan, Magnus O. Myreen, and Scott Owens. "Self-Formalisation of Higher-Order Logic." Journal of Automated Reasoning 56, no. 3 (February 15, 2016): 221–59. http://dx.doi.org/10.1007/s10817-015-9357-x.

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17

Sági, Gábor. "Ultraproducts and Higher Order Formulas." MLQ 48, no. 2 (February 2002): 261–75. http://dx.doi.org/10.1002/1521-3870(200202)48:2<261::aid-malq261>3.0.co;2-f.

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18

Marshall R., M. Victoria. "Higher order reflection principles." Journal of Symbolic Logic 54, no. 2 (June 1989): 474–89. http://dx.doi.org/10.2307/2274862.

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In [1] and [2] there is a development of a class theory, whose axioms were formulated by Bernays and based on a reflection principle. See [3]. These axioms are formulated in first order logic with ∈:(A1)Extensionality.(A2)Class specification. Ifϕis a formula andAis not free inϕ, thenNote that “xis a set“ can be written as “∃u(x∈u)”.(A3)Subsets.Note also that “B⊆A” can be written as “∀x(x∈B→x∈A)”.(A4)Reflection principle. Ifϕ(x)is a formula, thenwhere “uis a transitive set” is the formula “∃v(u∈v) ∧ ∀x∀y(x∈y∧y∈u→x∈u)” andϕPuis the formulaϕrelativized to subsets ofu.(A5)Foundation.(A6)Choice for sets.We denote byB1the theory with axioms (A1) to (A6).The existence of weakly compact and-indescribable cardinals for everynis established inB1by the method of defining all metamathematical concepts forB1in a weaker theory of classes where the natural numbers can be defined and using the reflection principle to reflect the satisfaction relation; see [1]. There is a proof of the consistency ofB1assuming the existence of a measurable cardinal; see [4] and [5]. In [6] several set and class theories with reflection principles are developed. In them, the existence of inaccessible cardinals and some kinds of indescribable cardinals can be proved; and also there is a generalization of indescribability for higher-order languages using only class parameters.The purpose of this work is to develop higher order reflection principles, including higher order parameters, in order to obtain other large cardinals.
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19

Benzmüller, Christoph, Dov Gabbay, Valerio Genovese, and Daniele Rispoli. "Embedding and automating conditional logics in classical higher-order logic." Annals of Mathematics and Artificial Intelligence 66, no. 1-4 (September 25, 2012): 257–71. http://dx.doi.org/10.1007/s10472-012-9320-z.

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20

HANUS, MICHAEL, and CHRISTIAN PREHOFER. "Higher-order narrowing with definitional trees." Journal of Functional Programming 9, no. 1 (January 1999): 33–75. http://dx.doi.org/10.1017/s0956796899003330.

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Functional logic languages with a sound and complete operational semantics are mainly based on an inference rule called narrowing. Narrowing extends functional evaluation by goal solving capabilities, as in logic programming. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for first-order functional logic programs due to its optimality properties wrt the length of derivations and the number of computed solutions. In this paper, we extend the needed narrowing strategy to higher-order functions and λ-terms as data structures. By the use of definitional trees, our strategy computes only independent solutions. Thus, it is the first calculus for higher-order functional logic programming which provides for such an optimality result. Since we allow higher-order logical variables denoting λ-terms, applications go beyond current functional and logic programming languages. We show soundness and completeness of our strategy with respect to LNT reductions, a particular form of higher-order reductions defined via definitional trees. A general completeness result is only provided for terminating rewrite systems due to the lack of an overall theory of higher-order reduction which is outside the scope of this paper.
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21

Paulson, Lawrence C. "Natural deduction as higher-order resolution." Journal of Logic Programming 3, no. 3 (October 1986): 237–58. http://dx.doi.org/10.1016/0743-1066(86)90015-4.

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22

Afshari, Bahareh, Stefan Hetzl, and Graham E. Leigh. "Herbrand's theorem as higher order recursion." Annals of Pure and Applied Logic 171, no. 6 (June 2020): 102792. http://dx.doi.org/10.1016/j.apal.2020.102792.

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23

Kartzow, Alexander. "First-Order Logic on Higher-Order Nested Pushdown Trees." ACM Transactions on Computational Logic 14, no. 2 (June 2013): 1–45. http://dx.doi.org/10.1145/2480759.2480760.

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24

Ognjanovic, Zoran, and Nebojsa Ikodinovic. "A logic with higher order conditional probabilities." Publications de l'Institut Math?matique (Belgrade), no. 96 (2007): 141–54. http://dx.doi.org/10.2298/pim0796141o.

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We investigate probability logic with the conditional probability operators This logic, denoted LCP, allows making statements such as: P?s?, CP?s(? | ?) CP?0(? | ?) with the intended meaning "the probability of ? is at least s" "the conditional probability of ? given ? is at least s", "the conditional probability of ? given ? at most 0". A possible-world approach is proposed to give semantics to such formulas. Every world of a given set of worlds is equipped with a probability space and conditional probability is derived in the usual way: P(? | ?) = P(?^?)/P(?), P(?) > 0, by the (unconditional) probability measure that is defined on an algebra of subsets of possible worlds. Infinitary axiomatic system for our logic which is sound and complete with respect to the mentioned class of models is given. Decidability of the presented logic is proved.
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25

Benzmüller, Christoph, Chad E. Brown, and Michael Kohlhase. "Higher-order semantics and extensionality." Journal of Symbolic Logic 69, no. 4 (December 2004): 1027–88. http://dx.doi.org/10.2178/jsl/1102022211.

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Abstract.In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes.
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26

Antos, Carolin. "Foundations of Higher-Order Forcing." Bulletin of Symbolic Logic 24, no. 4 (December 2018): 457. http://dx.doi.org/10.1017/bsl.2018.38.

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27

Sochor, A. "Constructibility in higher order arithmetics." Archive for Mathematical Logic 32, no. 6 (November 1993): 381–89. http://dx.doi.org/10.1007/bf01270463.

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28

DOWEK, GILLES, THERESE HARDIN, and CLAUDE KIRCHNER. "HOL-λσ: an intentional first-order expression of higher-order logic." Mathematical Structures in Computer Science 11, no. 1 (February 2001): 21–45. http://dx.doi.org/10.1017/s0960129500003236.

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We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on λ-calculus, that is, a proposition can be proved without the extensionality axioms in one theory if and only if it can be in the other. We show that the Extended Narrowing and Resolution first-order proof-search method can be applied to this theory. In this way we get a step-by-step simulation of higher-order resolution. Hence, expressing higher-order logic as a first-order theory and applying a first-order proof search method is a relevant alternative to a direct implementation. In particular, the well-studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover, as we stay in a first-order setting, extensions, such as equational higher-order resolution, may be easier to handle.
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29

Basu, Sankha S., and Stephen G. Simpson. "Mass problems and intuitionistic higher-order logic." Computability 5, no. 1 (February 11, 2016): 29–47. http://dx.doi.org/10.3233/com-150041.

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30

Periyasamy, K. "High-Level Text on Higher Order Logic." IEEE Software 13, no. 1 (January 1996): 117. http://dx.doi.org/10.1109/ms.1996.476302.

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31

Might, Matthew. "Logic-flow analysis of higher-order programs." ACM SIGPLAN Notices 42, no. 1 (January 17, 2007): 185–98. http://dx.doi.org/10.1145/1190215.1190247.

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32

Chen, Weidong, and David Scott Warren. "Predicate abstractions in higher-order logic programming." New Generation Computing 14, no. 2 (June 1996): 195–236. http://dx.doi.org/10.1007/bf03037499.

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33

Bizjak, Aleš, and Lars Birkedal. "On Models of Higher-Order Separation Logic." Electronic Notes in Theoretical Computer Science 336 (April 2018): 57–78. http://dx.doi.org/10.1016/j.entcs.2018.03.016.

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34

Back, R. J. R., and J. von Wright. "Refinement concepts formalised in higher order logic." Formal Aspects of Computing 2, no. 1 (March 1990): 247–72. http://dx.doi.org/10.1007/bf01888227.

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35

Owens, Scott, and Konrad Slind. "Adapting functional programs to higher order logic." Higher-Order and Symbolic Computation 21, no. 4 (October 30, 2008): 377–409. http://dx.doi.org/10.1007/s10990-008-9038-0.

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36

Mehta, Farhad, and Tobias Nipkow. "Proving pointer programs in higher-order logic." Information and Computation 199, no. 1-2 (May 2005): 200–227. http://dx.doi.org/10.1016/j.ic.2004.10.007.

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37

RONDOGIANNIS, P., and W. W. WADGE. "Higher-order functional languages and intensional logic." Journal of Functional Programming 9, no. 5 (September 1999): 527–64. http://dx.doi.org/10.1017/s0956796899003445.

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In this paper we demonstrate that a broad class of higher-order functional programs can be transformed into semantically equivalent multidimensional intensional programs that contain only nullary variable definitions. The proposed algorithm systematically eliminates user-defined functions from the source program, by appropriately introducing context manipulation (i.e. intensional) operators. The transformation takes place in M steps, where M is the order of the initial functional program. During each step the order of the program is reduced by one, and the final outcome of the algorithm is an M-dimensional intensional program of order zero. As the resulting intensional code can be executed in a purely tagged-dataflow way, the proposed approach offers a promising new technique for the implementation of higher-order functional languages.
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38

Aguirre, Alejandro, Gilles Barthe, Marco Gaboardi, Deepak Garg, and Pierre-Yves Strub. "A relational logic for higher-order programs." Proceedings of the ACM on Programming Languages 1, ICFP (August 29, 2017): 1–29. http://dx.doi.org/10.1145/3110265.

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39

von Wright, J. "Representing higher-order logic proofs in HOL." Computer Journal 38, no. 2 (February 1, 1995): 171–79. http://dx.doi.org/10.1093/comjnl/38.2.171.

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40

Varming, Carsten, and Lars Birkedal. "Higher-Order Separation Logic in Isabelle/HOLCF." Electronic Notes in Theoretical Computer Science 218 (October 2008): 371–89. http://dx.doi.org/10.1016/j.entcs.2008.10.022.

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41

Kobayashi, Naoki, Étienne Lozes, and Florian Bruse. "On the relationship between higher-order recursion schemes and higher-order fixpoint logic." ACM SIGPLAN Notices 52, no. 1 (May 11, 2017): 246–59. http://dx.doi.org/10.1145/3093333.3009854.

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42

Baliga, Ganesh, John Case, Sanjay Jain, and Mandayam Suraj. "Machine learning of higher-order programs." Journal of Symbolic Logic 59, no. 2 (June 1994): 486–500. http://dx.doi.org/10.2307/2275402.

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AbstractA generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To motivate these studies partially, it is shown that, in some cases, interesting global properties for computable functions can be proved from suitable generator programs which cannot be proved from any ordinary programs for them. The power (for variants of various learning criteria from the literature) of learning generator programs is compared with the power of learning ordinary programs. The learning power in these cases is also compared to that of learning limiting programs, i.e., programs allowed finitely many mind changes about their correct outputs.
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43

Zhao, Chunna, Murong Jiang, and Yaqun Huang. "Formal Verification of Fractional-Order PID Control Systems Using Higher-Order Logic." Fractal and Fractional 6, no. 9 (August 30, 2022): 485. http://dx.doi.org/10.3390/fractalfract6090485.

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Fractional-order PID control is a landmark in the development of fractional-order control theory. It can improve the control precision and accuracy of systems and achieve more robust control results. As a theorem-proving formal verification method, it can be applied to an arbitrary system represented by a mathematical model. It is the ideal verification method because it is not subject to limits on state numbers. This paper presents the higher-order logic (HOL) formal verification and modeling of fractional-order PID controller systems. Firstly, a fractional-order PID controller was designed. The accuracy of fractional-order PID control can be supported by simulation, comparing integral-order PID controls. Secondly, the superior property of fractional-order PID control is validated via higher-order logic theorem proofs. An important basic property, the relationship between fractional-order differential calculus and integral-order differential calculus, was analyzed via a higher-order logic theorem proof. Then, the relations between the fractional-order PID controller and integral-order PID controller were verified based on the fractional-order Grünwald–Letnikov definition for higher-order logic theorem proofs. Formalization models of the fractional-order PID controller and the fractional-order closed-loop control system were established. Finally, the stability of the fractional-order control systems was verified based on established formal models and theorems. The results show that the fractional-order PID controllers can be conducive to the control performance of control systems, and the higher-order logic formal verification method can ensure the reliability and security of fractional-order control systems.
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44

Steen, Alexander. "Higher-order theorem proving and its applications." it - Information Technology 61, no. 4 (August 27, 2019): 187–91. http://dx.doi.org/10.1515/itit-2019-0001.

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Abstract Automated theorem proving systems validate or refute whether a conjecture is a logical consequence of a given set of assumptions. Higher-order provers have been successfully applied in academic and industrial applications, such as planning, software and hardware verification, or knowledge-based systems. Recent studies moreover suggest that automation of higher-order logic, in particular, yields effective means for reasoning within expressive non-classical logics, enabling a whole new range of applications, including computer-assisted formal analysis of arguments in metaphysics. My work focuses on the theoretical foundations, effective implementation and practical application of higher-order theorem proving systems. This article briefly introduces higher-order reasoning in general and presents an overview of the design and implementation of the higher-order theorem prover Leo-III. In the second part, some example applications of Leo-III are discussed.
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45

Grellois, Charles, and Paul-André Melliès. "Indexed linear logic and higher-order model checking." Electronic Proceedings in Theoretical Computer Science 177 (March 17, 2015): 43–52. http://dx.doi.org/10.4204/eptcs.177.4.

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46

Biering, Bodil, Lars Birkedal, and Noah Torp-Smith. "BI-hyperdoctrines, higher-order separation logic, and abstraction." ACM Transactions on Programming Languages and Systems 29, no. 5 (August 2, 2007): 24. http://dx.doi.org/10.1145/1275497.1275499.

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47

Camilleri, A. J. "Mechanizing CSP trace theory in higher order logic." IEEE Transactions on Software Engineering 16, no. 9 (1990): 993–1004. http://dx.doi.org/10.1109/32.58786.

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48

Krebbers, Robbert, Amin Timany, and Lars Birkedal. "Interactive proofs in higher-order concurrent separation logic." ACM SIGPLAN Notices 52, no. 1 (May 11, 2017): 205–17. http://dx.doi.org/10.1145/3093333.3009855.

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49

Paulson, L. "Mechanizing coinduction and corecursion in higher-order logic." Journal of Logic and Computation 7, no. 2 (April 1, 1997): 175–204. http://dx.doi.org/10.1093/logcom/7.2.175.

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50

Chen, Weidong, Michael Kifer, and David S. Warren. "HiLog: A foundation for higher-order logic programming." Journal of Logic Programming 15, no. 3 (February 1993): 187–230. http://dx.doi.org/10.1016/0743-1066(93)90039-j.

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