ACM Transactions on Computational Logic最新文献

{"title":"A Category Theoretic View of Contextual Types: From Simple Types to Dependent Types","authors":"Jason Z. S. Hu, Brigitte Pientka, Ulrich Schöpp","doi":"https://dl.acm.org/doi/10.1145/3545115","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3545115","url":null,"abstract":"<p>We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of <span>Cocon</span>, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes <span>Cocon</span> different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann’s work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model <span>Cocon</span>. In particular, we can capture the contextual objects of <span>Cocon</span> using a comonad ♭ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies <span>Cocon</span> as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of <span>Cocon</span> without recursor in the Fitch-style dependent modal type theory presented by Birkedal et al.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"39 8","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Syntactic Completeness of Proper Display Calculi","authors":"Jinsheng Chen, Giuseppe Greco, Alessandra Palmigiano, Apostolos Tzimoulis","doi":"https://dl.acm.org/doi/10.1145/3529255","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3529255","url":null,"abstract":"<p>A recent strand of research in structural proof theory aims at exploring the notion of <i>analytic calculi</i> (i.e., those calculi that support general and modular proof-strategies for cut elimination) and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of <i>proper display calculi</i> as one possible design framework for proof calculi in which the analyticity desiderata are realized in a particularly transparent way. Recently, the theory of <i>properly displayable</i> logics (i.e., those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (a.k.a. unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by <i>analytic inductive axioms</i>, which can be equivalently and algorithmically transformed into analytic structural rules so the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination, and the subformula property. In this context, the proof that the given calculus is <i>complete</i> w.r.t. the original logic is usually carried out <i>syntactically</i>, i.e., by showing that a (cut-free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far, this proof strategy for <i>syntactic completeness</i> has been implemented on a case-by-case base and not in general. In this article, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut-free derivation can be effectively generated that has a specific shape, referred to as <i>pre-normal form</i>.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"38 12","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unifying Operational Weak Memory Verification: An Axiomatic Approach","authors":"Simon Doherty, Sadegh Dalvandi, Brijesh Dongol, Heike Wehrheim","doi":"https://dl.acm.org/doi/10.1145/3545117","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3545117","url":null,"abstract":"<p>In this article, we propose an approach to program verification using an abstract characterisation of weak memory models. Our approach is based on a hierarchical axiom scheme that captures the <i>observational properties</i> of a memory model. In particular, we show that it is possible to prove correctness of a program with respect to a particular axiom scheme, and we show this proof to suffice for <i>any</i> memory model that satisfies the axioms. Our axiom scheme is developed using a characterisation of <i>weakest liberal preconditions</i> for weak memory. This characterisation naturally extends to Hoare logic and Owicki-Gries reasoning by lifting weakest liberal preconditions (defined over read/write events) to the level of programs. We study three memory models (SC, TSO, and RC11-RAR) as example instantiations of the axioms, then we demonstrate the applicability of our reasoning technique on a number of litmus tests. The majority of the proofs in this article are supported by mechanisation within Isabelle/HOL.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"85 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Witnesses for Answer Sets of Logic Programs","authors":"Yisong Wang, Thomas Eiter, Yuanlin Zhang, Fangzhen Lin","doi":"10.1145/3568955","DOIUrl":"https://doi.org/10.1145/3568955","url":null,"abstract":"In this article, we consider Answer Set Programming (ASP). It is a declarative problem solving paradigm that can be used to encode a problem as a logic program whose answer sets correspond to the solutions of the problem. It has been widely applied in various domains in AI and beyond. Given that answer sets are supposed to yield solutions to the original problem, the question of “why a set of atoms is an answer set” becomes important for both semantics understanding and program debugging. It has been well investigated for normal logic programs. However, for the class of disjunctive logic programs, which is a substantial extension of that of normal logic programs, this question has not been addressed much. In this article, we propose a notion of reduct for disjunctive logic programs and show how it can provide answers to the aforementioned question. First, we show that for each answer set, its reduct provides a resolution proof for each atom in it. We then further consider minimal sets of rules that will be sufficient to provide resolution proofs for sets of atoms. Such sets of rules will be called witnesses and are the focus of this article. We study complexity issues of computing various witnesses and provide algorithms for computing them. In particular, we show that the problem is tractable for normal and headcycle-free disjunctive logic programs, but intractable for general disjunctive logic programs. We also conducted some experiments and found that for many well-known ASP and SAT benchmarks, computing a minimal witness for an atom of an answer set is often feasible.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 46"},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46529749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Precise Subtyping for Asynchronous Multiparty Sessions","authors":"Silvia Ghilezan, Jovanka Pantović, Ivan Prokić, Alceste Scalas, Nobuko Yoshida","doi":"https://dl.acm.org/doi/10.1145/3568422","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3568422","url":null,"abstract":"<p>Session subtyping is a cornerstone of refinement of communicating processes: a process implementing a session type (i.e., a communication protocol) <i>T</i> can be safely used whenever a process implementing one of its supertypes <i>T</i>′ is expected, in any context, without introducing deadlocks nor other communication errors. As a consequence, whenever <i>T</i> ≤ <i>T</i>′ holds, it is safe to replace an implementation of <i>T</i>′ with an implementation of the subtype <i>T</i>, which may allow for more optimised communication patterns. We present the first formalisation of the <i>precise</i> subtyping relation for <i>asynchronous multiparty</i> sessions. We show that our subtyping relation is <i>sound</i>\u0000(i.e., guarantees safe process replacement, as outlined above) and also <i>complete</i>: any extension of the relation is unsound. To achieve our results, we develop a novel <i>session decomposition</i> technique, from <i>full</i>\u0000session types (including internal/external choices) into <i>single input/output session trees</i> (without choices). We cover <i>multiparty</i> sessions with <i>asynchronous</i>\u0000interaction, where messages are transmitted via FIFO queues (as in the TCP/IP protocol), and prove that our subtyping is both operationally and denotationally precise. Our session decomposition technique expresses the subtyping relation as a composition of refinement relations between single input/output trees, and providing a simple reasoning principle for asynchronous message optimisations.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"40 7","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Generalized Realizability and Intuitionistic Logic","authors":"Aleksandr Yu. Konovalov","doi":"https://dl.acm.org/doi/10.1145/3565367","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3565367","url":null,"abstract":"<p>Let <i>V</i> be a set of number-theoretical functions. We define a notion of <i>V</i>-realizability for predicate formulas in such a way that the indices of functions in <i>V</i> are used for interpreting the implication and the universal quantifier. In this paper we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of <i>V</i>-realizability if and only if some natural conditions for <i>V</i> hold.</p>","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"6 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reducible Theories and Amalgamations of Models","authors":"Bahar Aameri, M. Grüninger","doi":"10.1145/3565364","DOIUrl":"https://doi.org/10.1145/3565364","url":null,"abstract":"Within knowledge representation in artificial intelligence, a first-order ontology is a theory in first-order logic that axiomatizes the concepts in some domain. Ontology verification is concerned with the relationship between the intended models of an ontology and the models of the axiomatization of the ontology. In particular, we want to characterize the models of an ontology up to isomorphism and determine whether or not these models are equivalent to the intended models of the ontology. Unfortunately, it can be quite difficult to characterize the models of an ontology up to isomorphism. In the first half of this article, we review the different metalogical relationships between first-order theories and identify which relationship is needed for ontology verification. In particular, we will demonstrate that the notion of logical synonymy is needed to specify a representation theorem for the class of models of one first-order ontology with respect to another. In the second half of the article, we discuss the notion of reducible theories and show we can specify representation theorems by which models are constructed by amalgamating models of the constituent ontologies.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 24"},"PeriodicalIF":0.5,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48732481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hardness Characterisations and Size-width Lower Bounds for QBF Resolution","authors":"Olaf Beyersdorff, Joshua Blinkhorn, M. Mahajan, Tomás Peitl","doi":"10.1145/3565286","DOIUrl":"https://doi.org/10.1145/3565286","url":null,"abstract":"We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) via circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems [16], but leaving open the most important case of QBF resolution. Different from the Frege case, our characterisation uses a new version of decision lists as its circuit model, which is stronger than the CNFs the system works with. Our decision list model is well suited to compute countermodels for QBFs. Our characterisation works for both Q-Resolution and QU-Resolution. Using our characterisation, we obtain a size-width relation for QBF resolution in the spirit of the celebrated result for propositional resolution [4]. However, our result is not just a replication of the propositional relation—intriguingly ruled out for QBF in previous research [12]—but shows a different dependence between size, width, and quantifier complexity. An essential ingredient is an improved relation between the size and width of term decision lists; this may be of independent interest. We demonstrate that our new technique elegantly reproves known QBF hardness results and unifies previous lower-bound techniques in the QBF domain.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"24 1","pages":"1 - 30"},"PeriodicalIF":0.5,"publicationDate":"2022-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41890179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extensible Proof Systems for Infinite-State Systems","authors":"J. Keiren, R. Cleaveland","doi":"10.48550/arXiv.2207.12953","DOIUrl":"https://doi.org/10.48550/arXiv.2207.12953","url":null,"abstract":"This paper revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46662838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Proof Complexity of Resolution over Polynomial Calculus","authors":"Erfan Khaniki","doi":"https://dl.acm.org/doi/10.1145/3506702","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3506702","url":null,"abstract":"&lt;p&gt;The proof system &lt;sans-serif&gt;Res (PC&lt;/sans-serif&gt;&lt;sub&gt;&lt;i&gt;d,R&lt;/i&gt;&lt;/sub&gt;) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree &lt;i&gt;d&lt;/i&gt; multivariate polynomials over a ring &lt;i&gt;R&lt;/i&gt; with Boolean variables. Proving super-polynomial lower bounds for the size of &lt;sans-serif&gt;Res&lt;/sans-serif&gt;(&lt;sans-serif&gt;PC&lt;/sans-serif&gt;&lt;sub&gt;1,&lt;i&gt;R&lt;/i&gt;&lt;/sub&gt;)-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for &lt;sans-serif&gt;Res&lt;/sans-serif&gt;(&lt;sans-serif&gt;PC&lt;/sans-serif&gt;&lt;sub&gt;1,𝔽&lt;/sub&gt;) when 𝔽 is a finite field, such as 𝔽&lt;sub&gt;2&lt;/sub&gt;. In this article, we investigate &lt;sans-serif&gt;Res&lt;/sans-serif&gt;(&lt;sans-serif&gt;PC&lt;/sans-serif&gt;&lt;sub&gt;&lt;i&gt;d,R&lt;/i&gt;&lt;/sub&gt;) and tree-like &lt;sans-serif&gt;Res&lt;/sans-serif&gt;(&lt;sans-serif&gt;PC&lt;/sans-serif&gt;&lt;sub&gt;&lt;i&gt;d,R&lt;/i&gt;&lt;/sub&gt;) and prove size-width relations for them when &lt;i&gt;R&lt;/i&gt; is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows:\u0000&lt;p&gt;&lt;table border=\"0\" list-type=\"ordered\" width=\"95%\"&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(1)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;We prove almost quadratic lower bounds for &lt;sans-serif&gt;Res&lt;/sans-serif&gt;(&lt;sans-serif&gt;PC&lt;/sans-serif&gt;&lt;sub&gt;&lt;i&gt;d&lt;/i&gt;&lt;/sub&gt;,𝔽)-refutations for every fixed &lt;i&gt;d&lt;/i&gt;. The new lower bounds are for the following CNFs:&lt;/p&gt;&lt;p&gt;&lt;table border=\"0\" list-type=\"ordered\" width=\"95%\"&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(a)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;Mod &lt;i&gt;q&lt;/i&gt; Tseitin formulas (&lt;i&gt;char&lt;/i&gt;(𝔽)≠ &lt;i&gt;q&lt;/i&gt;) and Flow formulas,&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(b)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;Random &lt;i&gt;k&lt;/i&gt;-CNFs with linearly many clauses.&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(2)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;We also prove super-polynomial (more than &lt;i&gt;n&lt;/i&gt;&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt; for any fixed &lt;i&gt;k&lt;/i&gt;) and also exponential (2&lt;i&gt;&lt;sup&gt;nϵ&lt;/sup&gt;&lt;/i&gt; for an ϵ &gt; 0) lower bounds for tree-like &lt;sans-serif&gt;Res&lt;/sans-serif&gt;(&lt;sans-serif&gt;PC&lt;/sans-serif&gt;&lt;sub&gt;&lt;i&gt;d&lt;/i&gt;,𝔽&lt;/sub&gt;)-refutations based on how big &lt;i&gt;d&lt;/i&gt; is with respect to &lt;i&gt;n&lt;/i&gt; for the following CNFs:&lt;/p&gt;&lt;p&gt;&lt;table border=\"0\" list-type=\"ordered\" width=\"95%\"&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(a)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;Mod &lt;i&gt;q&lt;/i&gt; Tseitin formulas (&lt;i&gt;char&lt;/i&gt;(𝔽)≠ &lt;i&gt;q&lt;/i&gt;) and Flow formulas,&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(b)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;Random &lt;i&gt;k&lt;/i&gt;-CNFs of suitable densities,&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td valign=\"top\"&gt;&lt;p&gt;(c)&lt;/p&gt;&lt;/td&gt;&lt;td colspan=\"5\" valign=\"top\"&gt;&lt;p&gt;Pigeonhole principle and Counting mod &lt;i&gt;q&lt;/i&gt; principle.&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;/p&gt; The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case &lt;i&gt;d&lt;/i&gt;=1. The lower bounds for the tree-like systems were known for the case &lt;i&gt;d&lt;/i&gt;=1 (except for the Counting mod &lt;i&gt;q&lt;/i&gt; principle, in which","PeriodicalId":50916,"journal":{"name":"ACM Transactions on Computational Logic","volume":"60 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138542040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}