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43
Light Linear Logic
"... The abuse of structural rules may have damaging complexity effects. ..."
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Cited by 625 (3 self)
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The abuse of structural rules may have damaging complexity effects.
Database Query Languages Embedded in the Typed Lambda Calculus
, 1993
"... We investigate the expressive power of the typed calculus when expressing computations over finite structures, i.e., databases. We show that the simply typed calculus can express various database query languages such as the relational algebra, fixpoint logic, and the complex object algebra. In ..."
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Cited by 25 (6 self)
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We investigate the expressive power of the typed calculus when expressing computations over finite structures, i.e., databases. We show that the simply typed calculus can express various database query languages such as the relational algebra, fixpoint logic, and the complex object algebra. In our embeddings, inputs and outputs are terms encoding databases, and a program expressing a query is a term which types when applied to an input and reduces to an output.
The Expressive Power of Higherorder Types or, Life without CONS
, 2001
"... Compare firstorder functional programs with higherorder programs allowing functions as function parameters. Can the the first program class solve fewer problems than the second? The answer is no: both classes are Turing complete, meaning that they can compute all partial recursive functions. In pa ..."
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Cited by 24 (1 self)
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Compare firstorder functional programs with higherorder programs allowing functions as function parameters. Can the the first program class solve fewer problems than the second? The answer is no: both classes are Turing complete, meaning that they can compute all partial recursive functions. In particular, higherorder values may be firstorder simulated by use of the list constructor ‘cons’ to build function closures. This paper uses complexity theory to prove some expressivity results about small programming languages that are less than Turing complete. Complexity classes of decision problems are used to characterize the expressive power of functional programming language features. An example: secondorder programs are more powerful than firstorder, since a function f of type &lsqb;Bool&rsqb;〉Bool is computable by a consfree firstorder functional program if and only if f is in PTIME, whereas f is computable by a consfree secondorder program if and only if f is in EXPTIME. Exact characterizations are given for those problems of type &lsqb;Bool&rsqb;〉Bool solvable by programs with several combinations of operations on data: presence or absence of constructors; the order of data values: 0, 1, or higher; and program control structures: general recursion, tail recursion, primitive recursion.
Functional Database Query Languages as Typed Lambda Calculi of Fixed Order (Extended Abstract)
 In Proceedings 13th PODS
, 1994
"... We present a functional framework for database query languages, which is analogous to the conventional logical framework of firstorder and fixpoint formulas over finite structures. We use atomic constants of order 0, equality among these constants, variables, application, lambda abstraction, and le ..."
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Cited by 12 (5 self)
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We present a functional framework for database query languages, which is analogous to the conventional logical framework of firstorder and fixpoint formulas over finite structures. We use atomic constants of order 0, equality among these constants, variables, application, lambda abstraction, and let abstraction; all typed using fixed order ( 5) functionalities. In this framework, proposed in [21] for arbitrary order functionalities, queries and databases are both typed lambda terms, evaluation is by reduction, and the main programming technique is list iteration. We define two families of languages: TLI = i or simplytyped list iteration of order i +3 with equality, and MLI = i or MLtyped list iteration of order i+3 with equality; we use i+3 since our list representation of databases requires at least order 3. We show that: FOqueries ` TLI = 0 ` MLI = 0 ` LOGSPACEqueries ` TLI = 1 = MLI = 1 = PTIMEqueries ` TLI = 2 , where equality is no longer a primitive in TLI = 2 . We also show that ML type inference, restricted to fixed order, is polynomial in the size of the program typed. Since programming by using low order functionalities and type inference is common in functional languages, our results indicate that such programs suffice for expressing efficient computations and that their MLtypes can be efficiently inferred.
Finite Model Theory In The Simply Typed Lambda Calculus
, 1994
"... Church's simply typed calculus is a very basic framework for functional programming language research. However, it is common to augment this framework with additional programming constructs, because its expressive power for functions over the domain of Church numerals is very limited. In this ..."
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Cited by 8 (5 self)
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Church's simply typed calculus is a very basic framework for functional programming language research. However, it is common to augment this framework with additional programming constructs, because its expressive power for functions over the domain of Church numerals is very limited. In this thesis: (1) We reexamine the expressive power of the "pure" simply typed calculus, but over encodings of finite relational structures, i. e., finite models or databases . In this novel framework the simply typed calculus expresses all elementary functions from finite models to finite models. In addition, many common database query languages, e. g., relational algebra, Datalog : , and the Abiteboul/Beeri complex object algebra, can be embedded into it. The embeddings are feasible in the sense that the terms corresponding to PTIME queries can be evaluated in polynomial time. (2) We examine fixedorder fragments of the simply typed calculus to determine machine independent characterizations of complexity classes. For this we augment the calculus with atomic constants and equality among atomic constants. We show that over ordered structures, the order 3, 4, 5, and 6 fragments express exactly the firstorder, PTIME, PSPACE, and EXPTIME queries, respectively, and we conjecture that for general k 1, order 2 k + 4 expresses exactly the kEXPTIME queries and order 2 k + 5 expresses exactly the kEXPSPACE queries. (3) We also reexamine other functional characterizations of PTIME and we show that method schemas with ordered objects express exactly PTIME. This is a firstorder framework proposed for objectoriented databasesas opposed to the above higherorder frameworks. In summary, this research provides a link between finite model theory (and thus computational complexity), dat...
A characterization of alternating log time by first order functional programs
 In LPAR 2006, volume 4246 of LNAI
, 2006
"... Abstract. We a give an intrinsic characterization of the class of functions which are computable in NC 1 that is by a uniform, logarithmic depth and polynomial size family circuit. Recall that the class of functions in ALogTime, that is in logarithmic time on an Alternating Turing Machine, is NC 1. ..."
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Cited by 7 (5 self)
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Abstract. We a give an intrinsic characterization of the class of functions which are computable in NC 1 that is by a uniform, logarithmic depth and polynomial size family circuit. Recall that the class of functions in ALogTime, that is in logarithmic time on an Alternating Turing Machine, is NC 1. Our characterization is in terms of first order functional programming languages. We define measuretools called Supinterpretations, which allow to give space and time bounds and allow also to capture a lot of program schemas. This study is part of a research on static analysis in order to predict program resources. It is related to the notion of Quasiinterpretations and belongs to the implicit computational complexity line of research. 1
On the expressive power of simply typed and letpolymorphic lambda calculi
 11th Annual IEEE Symp. on Logic in Computer Science (LICS'96)
, 1996
"... We present a functional framework for descriptive computational complexity, in which the Regular, Firstorder, Ptime, Pspace, kExptime, kExpspace (k 1), and Elementary sets have syntactic characterizations. In this framework, typed lambda terms represent inputs and outputs as well as programs. The ..."
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Cited by 6 (0 self)
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We present a functional framework for descriptive computational complexity, in which the Regular, Firstorder, Ptime, Pspace, kExptime, kExpspace (k 1), and Elementary sets have syntactic characterizations. In this framework, typed lambda terms represent inputs and outputs as well as programs. The lambda calculi describing the above computational complexity classes are simply or letpolymorphically typed with functionalities of fixed order. They consist of: order 0 atomic constants, order 1 equality among these constants, variables, application, and abstraction. Increasing functionality order by one for these languages corresponds to increasing the computational complexity by one alternation. This exact correspondence is established using a semantic evaluation of languages for each fixed order, which is the primary technical contribution of this paper.
The safe lambda calculus
 of Lecture Notes in Computer Science
, 2007
"... Abstract. Safety is a syntactic condition of higherorder grammars that constrains occurrences of variables in the production rules according to their typetheoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition ..."
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Cited by 5 (1 self)
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Abstract. Safety is a syntactic condition of higherorder grammars that constrains occurrences of variables in the production rules according to their typetheoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition to the setting of the simplytyped lambda calculus. In contrast to the original definition of safety, our calculus does not constrain types (to be homogeneous). We show that in the safe lambda calculus, there is no need to rename bound variables when performing substitution, as variable capture is guaranteed not to happen. We also propose an adequate notion of βreduction that preserves safety. In the same vein as Schwichtenberg’s 1976 characterization of the simplytyped lambda calculus, we show that the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a characterization of representable word functions. We then study the complexity of deciding betaeta equality of two safe simplytyped terms and show that this problem is PSPACEhard. Finally we give a gamesemantic analysis of safety: We show that safe terms are denoted by Pincrementally justified strategies. Consequently pointers in the game semantics of safe λterms are only necessary from order 4 onwards.
Polygraphic programs and polynomialtime functions
"... Abstract – We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of firstorder functional programs. We explain their semantics and prove that they form a Turingcomplete computational model. Their algebraic s ..."
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Abstract – We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of firstorder functional programs. We explain their semantics and prove that they form a Turingcomplete computational model. Their algebraic structure is used by analysis tools, called polygraphic interpretations, for complexity analysis. In particular, we delineate a subclass of polygraphic programs that compute exactly the functions that are Turingcomputable in polynomial time.