We give an exact characterization of the computational complexity of the @math CFA hierarchy. For any @math , we prove that the control flow decision problem is complete for deterministic exponential time. This theorem validates empirical observations that such control flow analysis is intractable. It also provides more general insight into the complexity of abstract interpretation.
Although @math CFA and ML type inference are two static analyses complete for @cite_6 , the proofs of these respective theorems is fundamentally different. The ML proof relies on type inference simulating exact normalization (analogous to the -completeness proof for 0CFA), hence subverting the approximation of the analysis. In contrast, the @math CFA proof harnesses the approximation that results from nonlinearity.
@cite_6
2021217869 2059860151 2053667957 2071787143
A well known but incorrect piece of functional programming folklore is that ML expressions can be efficiently typed in polynomial time. In probing the truth of that folklore, various researchers, including Wand, Buneman, Kanellakis, and Mitchell, constructed simple counterexamples consisting of typable ML programs having length n , with principal types having O(2 cn ) distinct type variables and length O(2 2cn ). When the types associated with these ML constructions were represented as directed acyclic graphs, their sizes grew as O(2 cn ). The folklore was even more strongly contradicted by the recent result of Kanellakis and Mitchell that simply deciding whether or not an ML expression is typable is PSPACE-hard. We improve the latter result, showing that deciding ML typability is DEXPTIME-hard. As Kanellakis and Mitchell have shown containment in DEXPTIME, the problem is DEXPTIME-complete. The proof of DEXPTIME-hardness is carried out via a generic reduction: it consists of a very straightforward simulation of any deterministic one-tape Turing machine M with input k running in O ( c |k| ) time by a polynomial-sized ML formula P M,k , such that M accepts k iff P M,k is typable. The simulation of the transition function δ of the Turing Machine is realized uniquely through terms in the lambda calculus without the use of the polymorphic let construct. We use let for two purposes only: to generate an exponential amount of blank tape for the Turing Machine simulation to begin, and to compose an exponential number of applications of the ML formula simulating state transition. It is purely the expressive power of ML polymorphism to succinctly express function composition which results in a proof of DEXPTIME-hardness. We conjecture that lower bounds on deciding typability for extensions to the typed lambda calculus can be regarded precisely in terms of this expressive capacity for succinct function composition. To further understand this lower bound, we relate it to the problem of proving equality of type variables in a system of type equations generated from an ML expression with let-polymorphism. We show that given an oracle for solving this problem, deciding typability would be in PSPACE, as would be the actual computation of the principal type of the expression, were it indeed typable. Useful type inference must be faster than normalization. Otherwise, you could check safety conditions by running the program. We analyze the relationship between bounds on normalization and type inference. We show how the success of type inference is fundamentally related to the amnesia of the type system: the nonlinearity by which all instances of a variable are constrained to have the same type.Recent work on intersection types has advocated their usefulness for static analysis and modular compilation. We analyze System-I (and some instances of its descendant, System E), an intersection type system with a type inference algorithm. Because System-I lacks idempotency, each occurrence of a variable requires a distinct type. Consequently, type inference is equivalent to normalization in every single case, and time bounds on type inference and normalization are identical. Similar relationships hold for other intersection type systems without idempotency.The analysis is founded on an investigation of the relationship between linear logic and intersection types. We show a lockstep correspondence between normalization and type inference. The latter shows the promise of intersection types to facilitate static analyses of varied granularity, but also belies an immense challenge: to add amnesia to such analysis without losing all of its benefits. Multimodal information processing has received considerable attention in recent years. The focus of existing research in this area has been predominantly on the use of fusion technology. In this paper, we suggest that cross-modal association can provide a new set of powerful solutions in this area. We investigate different cross-modal association methods using the linear correlation model. We also introduce a novel method for cross-modal association called Cross-modal Factor Analysis (CFA). Our earlier work on Latent Semantic Indexing (LSI) is extended for applications that use off-line supervised training. As a promising research direction and practical application of cross-modal association, cross-modal information retrieval where queries from one modality are used to search for content in another modality using low-level features is then discussed in detail. Different association methods are tested and compared using the proposed cross-modal retrieval system. All these methods achieve significant dimensionality reduction. Among them CFA gives the best retrieval performance. Finally, this paper addresses the use of cross-modal association to detect talking heads. The CFA method achieves 91.1 detection accuracy, while LSI and Canonical Correlation Analysis (CCA) achieve 66.1 and 73.9 accuracy, respectively. As shown by experiments, cross-modal association provides many useful benefits, such as robust noise resistance and effective feature selection. Compared to CCA and LSI, the proposed CFA shows several advantages in analysis performance and feature usage. Its capability in feature selection and noise resistance also makes CFA a promising tool for many multimedia analysis applications. Abstract In this paper, we outline a proof procedure which combines reasoning with defined predicates together with reasoning with undefined, abducible , predicates. Defined predicates are defined in if-and-only-if form. Abducible predicates are constrained by means of integrity constraints. Given an initial query, the task of the proof procedure is to construct a definition of the abducible predicates and a substitution for the variables in the query, such that both the resulting instance of the query and the integrity constraints are implied by the extended set of definitions. The iff proof procedure can be regarded as a hybrid of the proof procedure of and the SLDNFA procedure of Denecker and De Schreye. It consists of a number of inference rules which, starting from the initial query, rewrite a formula into an equivalent formula. These rules are: 1) unfolding , which replaces an atom by its definition; 2) propagation , which resolves an atom with an implication; 3) splitting , which uses distributivity to represent a goal as a disjunction of conjunctions; 4) case analysis for an equality X = t in the conditions of an implication, which considers the two cases X = t and X ≠ t ; 5) factoring of two abducible atoms, which considers the two cases, where the atoms are identical and where they are different, 6) rewrite rules for equality , which simulate the unification algorithm; and 7) logical simplifications , such as A ∧ false ↔ false . The proof procedure is both sound and complete relative to the three-valued completion semantics. These soundness and completeness results improve previous results obtained for other proof procedures.