Towards automated reasoning in Herbrand structures

Liron Cohen; Reuben N S Rowe; Yoni Zohar

doi:10.1093/logcom/exz011

Towards automated reasoning in Herbrand structures

Journal of Logic and Computation ◽

10.1093/logcom/exz011 ◽

2019 ◽

Vol 29 (5) ◽

pp. 693-721

Author(s):

Liron Cohen ◽

Reuben N S Rowe ◽

Yoni Zohar

Keyword(s):

Inductive Reasoning ◽

Salient Feature ◽

Proof System ◽

Finite Approximations ◽

Completeness Result ◽

Height System ◽

Infinite Height ◽

Cyclic Fragment ◽

Effective Proof ◽

Induction Scheme

Abstract Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally exhibit the induction scheme, thus providing a congenial, computationally oriented framework for formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof system for them incomplete. Furthermore, the fact that they are not compact poses yet another proof-theoretic challenge. This paper offers several layers for coping with the inherent incompleteness and non-compactness of these logics. First, two types of infinitary proof system are introduced—one of infinite width and one of infinite height—which manipulate infinite sequents and are sound and complete for the intended semantics. The restriction of these systems to finite sequents induces a completeness result for finite entailments. Then, in search of effectiveness, two finite approximations of these systems are presented and explored. Interestingly, the approximation of the infinite-width system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the infinite-height system.

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Reasoning Inside The Box: Deduction in Herbrand Logics

10.29007/kx2m ◽

2018 ◽

Author(s):

Liron Cohen ◽

Yoni Zohar

Keyword(s):

Automated Reasoning ◽

Proof System ◽

First Order ◽

Proof Systems ◽

The Rich ◽

Order Structures ◽

Effective Proof

Herbrand structures are a subclass of standard first-order structures commonly used in logic and automated reasoning due to their strong definitional character. This paper is devoted to the logics induced by them: Herbrand and semi-Herbrand logics, with and without equality. The rich expressiveness of these logics entails that there is no adequate effective proof system for them. We therefore introduce infinitary proof systems for Herbrand logics, and prove their completeness. Natural and sound finitary approximations of the infinitary systems are also presented.

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Non-well-founded Deduction for Induction and Coinduction

Automated Deduction – CADE 28 - Lecture Notes in Computer Science ◽

10.1007/978-3-030-79876-5_1 ◽

2021 ◽

pp. 3-24

Author(s):

Liron Cohen

Keyword(s):

Computer Science ◽

Proof Theory ◽

Inductive Reasoning ◽

Inference Rules ◽

Closure Operators ◽

Robust Approach ◽

Induction Scheme ◽

Automated Proof ◽

General Method ◽

The Way

AbstractInduction and coinduction are both used extensively within mathematics and computer science. Algebraic formulations of these principles make the duality between them apparent, but do not account well for the way they are commonly used in deduction. Generally, the formalization of these reasoning methods employs inference rules that express a general explicit (co)induction scheme. Non-well-founded proof theory provides an alternative, more robust approach for formalizing implicit (co)inductive reasoning. This approach has been extremely successful in recent years in supporting implicit inductive reasoning, but is not as well-developed in the context of coinductive reasoning. This paper reviews the general method of non-well-founded proofs, and puts forward a concrete natural framework for (co)inductive reasoning, based on (co)closure operators, that offers a concise framework in which inductive and coinductive reasoning are captured as we intuitively understand and use them. Through this framework we demonstrate the enormous potential of non-well-founded deduction, both in the foundational theoretical exploration of (co)inductive reasoning and in the provision of proof support for (co)inductive reasoning within (semi-)automated proof tools.

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Adenomatoid Tumor of the Testis, A Mesonephric Hamartoma

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100065638 ◽

1971 ◽

Vol 29 ◽

pp. 318-319

Author(s):

Raoul Fresco ◽

Mary Chang-Lo

Keyword(s):

Electron Microscopy ◽

Epithelial Cells ◽

Fibrous Tissue ◽

Salient Feature ◽

Luminal Surface ◽

Adenomatoid Tumor ◽

Ultrastructural Studies ◽

Epithelial Origin ◽

Brush Borders ◽

Cell Processes

Confusion surrounds the nature of the “adenomatoid tumor” of the testis, as evidenced by the large number of synonyms which have been ascribed to it. Various authors have considered the tumor to be of endothelial, mesothelial or epithelial origin. There appears to be no controversy as to the stromal elements of the tumor, which consists mainly of smooth muscle and fibrous tissue. It is the irregular gland-like spaces which have given rise to the numerous theories as to its histogenesis, and even recent ultrastructural studies fail to agree on the origin of these structures.Electron microscopy of a typical intrascrotal adenomatoid tumor showed the gland-like spaces to be lined by epithelial cells (Fig. 1), rich in cytoplasmic tonofibrils and united to each other by numerous desmosomes (Fig. 2). The most salient feature of these epithelial cells was the presence on their luminal surface of numerous long and repeatedly branching microvillous structures of the type known as stereocilia (Fig. 3). These are extremely long slender cell processes which are as much as three to four times the length of those in brush borders.

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Use of 2½ D imaging in analysis of NiTi martensite

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100110192 ◽

1978 ◽

Vol 36 (1) ◽

pp. 610-611

Author(s):

G. M. Michal

Keyword(s):

Memory Effect ◽

Monoclinic Phase ◽

Reaction Path ◽

Structural Characteristics ◽

Salient Feature ◽

Martensite Phase ◽

Orientation Relationships ◽

Low Symmetry ◽

Hydride Phases ◽

Unusual Shape

Several TEM investigations have attempted to correlate the structural characteristics to the unusual shape memory effect in NiTi, the consensus being the essence of the memory effect is ostensible manifest in the structure of NiTi transforming martensitic- ally from a B2 ordered lattice to a low temperature monoclinic phase. Commensurate with the low symmetry of the martensite phase, many variants may form from the B2 lattice explaining the very complex transformed microstructure. The microstructure may also be complicated by the enhanced formation of oxide or hydride phases and precipitation of intermetallic compounds by electron beam exposure. Variants are typically found in selfaccommodation groups with members of a group internally twinned and the twins themselves are often observed to be internally twinned. Often the most salient feature of a group of variants is their close clustering around a given orientation. Analysis of such orientation relationships may be a key to determining the nature of the reaction path that gives the transformation its apparently perfect reversibility.

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The Potentials of Scanning Microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100101505 ◽

1968 ◽

Vol 26 ◽

pp. 352-353

Author(s):

A. V. Crewe

Keyword(s):

Salient Feature ◽

Secondary Emission ◽

Resolving Power ◽

X Rays ◽

Scanning Microscope ◽

Forming Process ◽

Additional Tool ◽

Detection Systems ◽

Conventional Microscope ◽

Present Information

If the resolving power of a scanning electron microscope can be improved until it is comparable to that of a conventional microscope, it would serve as a valuable additional tool in many investigations.The salient feature of scanning microscopes is that the image-forming process takes place before the electrons strike the specimen. This means that several different detection systems can be employed in order to present information about the specimen. In our own particular work we have concentrated on the use of energy loss information in the beam which is transmitted through the specimen, but there are also numerous other possibilities (such as secondary emission, generation of X-rays, and cathode luminescence).Another difference between the pictures one would obtain from the scanning microscope and those obtained from a conventional microscope is that the diffraction phenomena are totally different. The only diffraction phenomena which would be seen in the scanning microscope are those which exist in the beam itself, and not those produced by the specimen.

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