Skip to main content

Ugo Dal Lago

Università di Bologna, Dipartimento di Scienze dell'Informazione, Faculty Member

Followers

40

Following

20

Public Views

Arizona State University

University of Maryland

Fausto Giunchiglia

Alessandro Cimatti

Rice University

The University of Manchester

Diego Calvanese

Free University of Bozen-Bolzano

Nicolas D'Ippolito

Imperial College London

InterestsView All (7)

Uploads

Papers by Ugo Dal Lago

Confluence Results for a Quantum Lambda Calculus with Measurements

Electronic Notes in Theoretical Computer …, Jan 1, 2011

A strong confluence result for Q * , a quantum λ-calculus with measurements, is proved. More prec... more A strong confluence result for Q * , a quantum λ-calculus with measurements, is proved. More precisely, confluence is shown to hold both for finite and infinite computations. The technique used in the confluence proof is syntactical but innovative. This makes Q * different from similar quantum lambda calculi, which are either measurement-free or provided with a reduction strategy.

General Ramified Recurrence is Sound for Polynomial Time

Arxiv preprint arXiv:1005.0521, Jan 1, 2010

Leivant's ramified recurrence is one of the earliest examples of an implicit characterization of ... more Leivant's ramified recurrence is one of the earliest examples of an implicit characterization of the polytime functions as a subalgebra of the primitive recursive functions. Leivant's result, however, is originally stated and proved only for word algebras, i.e. free algebras whose constructors take at most one argument. This paper presents an extension of these results to ramified functions on any free algebras, provided the underlying terms are represented as graphs rather than trees, so that sharing of identical subterms can be exploited.

Linear Dependent Types and Relative Completeness

A system of linear dependent types for the lambda calculus with full higher-order recursion, call... more A system of linear dependent types for the lambda calculus with full higher-order recursion, called d PCF, is introduced and proved sound and relatively complete. Completeness holds in a strong sense: d PCF is not only able to precisely capture the functional behaviour of PCF programs (i.e. how the output relates to the input) but also some of their intensional properties, namely the complexity of evaluating them with Krivine's Machine. d PCF is designed around dependent types and linear logic and is parametrized on the underlying language of index terms, which can be tuned so as to sacrifice completeness for tractability.

Proofs as Eﬃcient Programs

Deduction, computation, experiment: …, Jan 1, 2008

Proofs as Eﬃcient Programs Ugo Dal Lago and Simone Martini Dipartimento di Scienze dell&amp;#... more

Semantic Frameworks for Implicit Computational Complexity

The University of Bologna Department of Computer Science Research Technical Reports are available... more

Probabilistic Operational Semantics for the Lambda Calculus

Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studie... more Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics, inductively and coinductively defined, are given. Moreover, small-step and big-step semantics are shown to produce identical outcomes, both in call-by-value and in call-by-name. Plotkin's CPS translation is extended to accommodate the choice operator and shown correct with respect to the operational semantics. Finally, the expressive power of the obtained system is studied: the calculus is shown to be sound and complete with respect to computable probability distributions.

Type Inference for Sublinear Space Functional Programming

Programming Languages and Systems, Jan 1, 2010

We consider programming language aspects of algorithms that operate on data too large to fit into... more We consider programming language aspects of algorithms that operate on data too large to fit into memory. In previous work we have introduced IntML, a functional programming language with primitives that support the implementation of such algorithms. We have shown that IntML can express all LOGSPACE functions but have left open the question how easy it is in practice to program typical LOGSPACE algorithms in IntML. In this paper we develop algorithms for IntML type inference. We show that with type inference one can handle programs that could not be reasonably manipulated by hand. We do so by implementing in IntML a typical LOGSPACE algorithm, a test for acyclicity of undirected graphs. Thus we show that with type inference IntML can express typical algorithmic patterns of LOGSPACE easily and in a natural way.

Light logics and the call-by-value lambda calculus

Arxiv preprint arXiv:0809.0195, Jan 1, 2008

The so-called light logics have been introduced as logical systems enjoying quite remarkable norm... more The so-called light logics have been introduced as logical systems enjoying quite remarkable normalization properties. Designing a type assignment system for pure lambda calculus from these logics, however, is problematic, as discussed in . In this paper we show that shifting from usual call-by-name to call-by-value lambda calculus allows regaining strong connections with the underlying logic. This will be done in the context of Elementary Affine Logic (EAL), designing a type system in natural deduction style assigning EAL formulae to lambda terms.

Taming Modal Impredicativity: Superlazy Reduction

Logical Foundations of Computer …, Jan 1, 2009

Pure, or type-free, Linear Logic proof nets are Turing complete once cut-elimination is 5 conside... more Pure, or type-free, Linear Logic proof nets are Turing complete once cut-elimination is 5 considered as computation. We introduce modal impredicativity as a new form of impredicativity causing the complexity of cut-elimination to be problematic from a complexity point of view. Modal impredicativity occurs when, during reduction, the conclusion of a residual of a box b interacts with a node that belongs to the proof net inside another residual of b. Technically speaking, Superlazy reduction is a new notion of reduction that allows to con-10 trol modal impredicativity. More specifically, superlazy reduction replicates a box only when all its copies are opened. This makes the overall cost of the reduction finite and predictable. Specifically, superlazy reduction applied to any pure proof nets takes primitive recursive time. Moreover, any primitive recursive function can be computed by a pure proof net via superlazy reduction.

Soft session types

We show how systems of sessions types can enforce interaction to be bounded for all typable proce... more We show how systems of sessions types can enforce interaction to be bounded for all typable processes. The type system we propose is based on Lafont's soft linear logic and is strongly inspired by recent works about session types as intuitionistic linear logic formulas. Our main result is the existence, for every typable process, of a polynomial bound on the length of any reduction sequence starting from it and on the size of any of its reducts.

Light Logics and Higher-Order Processes

We show that the techniques for resource control that have been developed in the so-called "light... more We show that the techniques for resource control that have been developed in the so-called "light logics" can be fruitfully applied also to process algebras. In particular, we present a restriction of Higher-Order π-calculus inspired by Soft Linear Logic. We prove that any soft process terminates in polynomial time. We argue that the class of soft processes may be naturally enlarged so that interesting processes are expressible, still maintaining the polynomial bound on executions.

Bounded linear logic, revisited

Typed Lambda Calculi and Applications, Jan 1, 2009

We present QBAL, an extension of Girard, Scedrov and Scott&amp;#x27;s bounded linear logic. T... more We present QBAL, an extension of Girard, Scedrov and Scott&amp;#x27;s bounded linear logic. The main novelty of the system is the possibility of quantifying over resource variables. This generalization makes bounded linear logic considerably more flexible, while preserving soundness and completeness for polynomial time. In particular, we provide compositional embeddings of Leivant&amp;#x27;s RRW and Hofmann&amp;#x27;s LFPL into QBAL.

Quantum implicit computational complexity

Theoretical Computer Science, Jan 1, 2010

We introduce a quantum lambda calculus inspired by Lafont's Soft Linear Logic and capturing the p... more We introduce a quantum lambda calculus inspired by Lafont's Soft Linear Logic and capturing the polynomial quantum complexity classes EQP, BQP and ZQP. The calculus is based on the ''classical control and quantum data'' paradigm. This is the first example of a formal system capturing quantum complexity classes in the spirit of implicit computational complexity -it is machine-free and no explicit bound (e.g., polynomials) appears in its syntax.

Functional programming in sublinear space

Programming Languages and Systems, Jan 1, 2010

We consider the problem of functional programming with data in external memory, in particular as ... more We consider the problem of functional programming with data in external memory, in particular as it appears in sublinear space computation. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot compose functions directly for lack of space for the intermediate result, but must instead compute and recompute small parts of the intermediate result on demand. In this paper, we study how the implementation of such techniques can be supported by functional programming languages. Our approach is based on modelling computation by interaction using the Int construction of Joyal, Street & Verity. We derive functional programming constructs from the structure obtained by applying the Int construction to a term model of a given functional language. The thus derived functional language is formulated by means of a type system inspired Baillot & Terui's Dual Light Affine Logic. We assess its expressiveness by showing that it captures LOGSPACE.

A Semantic Proof of Polytime Soundness of Light Affine Logic

Theory of Computing Systems, Jan 1, 2010

We define a denotational semantics for Light Affine Logic (LAL) which has the property that denot... more We define a denotational semantics for Light Affine Logic (LAL) which has the property that denotations of functions are polynomial time computable by construction of the model. This gives a new proof of polytime-soundness of LAL which is considerably simpler than the standard proof based on proof nets and also is entirely semantical in nature. The model construction uses a new instance of a resource monoid; a general method for interpreting variations of linear logic with complexity restrictions introduced earlier by the authors.

The weak lambda calculus as a reasonable machine

Theoretical Computer Science, Jan 1, 2008

We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis... more We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the call-by-value lambda-calculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties of usual beta-reduction, without any reference to a specific machine or evaluator. In particular, the cost of a single beta reduction is proportional to the difference between the size of the redex and the size of the reduct. In this way, the total cost of normalizing a lambda term will take into account the size of all intermediate results (as well as the number of steps to normal form).

On the expressive power of light affine logic

Theoretical Computer Science, Jan 1, 2003

Light affine logic (LAL) is a formal system derived from linear logic that is claimed to correspo... more Light affine logic (LAL) is a formal system derived from linear logic that is claimed to correspond, through the Curry-Howard Isomorphism, to the class PTIME of polytime functions. The completeness of the system with respect to PTIME has been proven by embedding different presentations of PTIME into LAL. The dual property of polytime soundness, on the other hand, has been stated and proven in a more debatable way, depending crucially on the underlying coding scheme. In this paper, we introduce two relevant classes of coding schemes, namely uniform and canonical coding schemes. We then investigate on the equality between PTIME and the classes of functions that are representable in LAL using these coding schemes, obtaining a positive and a negative result.

On a measurement-free quantum lambda calculus with classical control

Mathematical Structures in …, Jan 1, 2009

We study a measurement-free, untyped λ-calculus with quantum data and classical control. This wor... more We study a measurement-free, untyped λ-calculus with quantum data and classical control. This work arises from previous proposals by Selinger and Valiron, and Van Tonder. We focus on operational and expressiveness issues, rather than (denotational) semantics. We prove subject reduction and confluence, and a standardisation theorem. Moreover, we prove the computational equivalence of the proposed calculus with a suitable class of quantum circuit families.

Phase semantics and decidability of elementary affine logic

Theoretical computer science, Jan 1, 2004

Quantitative game semantics for linear logic

Computer Science Logic, Jan 1, 2008

We present a game-based semantic framework into which the time complexity of any IMELL proof can ... more We present a game-based semantic framework into which the time complexity of any IMELL proof can be read out of its interpretation. This gives a compositional view of the geometry of interaction framework introduced by the first author. In our model the time measure is given by means of slots, as introduced by Ghica in a recent paper. The cost associated to a strategy is polynomially related to the normalization time of the interpreted proof, in the style of a complexity-theoretical full abstraction result.

Confluence Results for a Quantum Lambda Calculus with Measurements

Electronic Notes in Theoretical Computer …, Jan 1, 2011

A strong confluence result for Q * , a quantum λ-calculus with measurements, is proved. More prec... more A strong confluence result for Q * , a quantum λ-calculus with measurements, is proved. More precisely, confluence is shown to hold both for finite and infinite computations. The technique used in the confluence proof is syntactical but innovative. This makes Q * different from similar quantum lambda calculi, which are either measurement-free or provided with a reduction strategy.

General Ramified Recurrence is Sound for Polynomial Time

Arxiv preprint arXiv:1005.0521, Jan 1, 2010

Leivant's ramified recurrence is one of the earliest examples of an implicit characterization of ... more Leivant's ramified recurrence is one of the earliest examples of an implicit characterization of the polytime functions as a subalgebra of the primitive recursive functions. Leivant's result, however, is originally stated and proved only for word algebras, i.e. free algebras whose constructors take at most one argument. This paper presents an extension of these results to ramified functions on any free algebras, provided the underlying terms are represented as graphs rather than trees, so that sharing of identical subterms can be exploited.

Linear Dependent Types and Relative Completeness

A system of linear dependent types for the lambda calculus with full higher-order recursion, call... more A system of linear dependent types for the lambda calculus with full higher-order recursion, called d PCF, is introduced and proved sound and relatively complete. Completeness holds in a strong sense: d PCF is not only able to precisely capture the functional behaviour of PCF programs (i.e. how the output relates to the input) but also some of their intensional properties, namely the complexity of evaluating them with Krivine's Machine. d PCF is designed around dependent types and linear logic and is parametrized on the underlying language of index terms, which can be tuned so as to sacrifice completeness for tractability.

Proofs as Eﬃcient Programs

Deduction, computation, experiment: …, Jan 1, 2008

Proofs as Eﬃcient Programs Ugo Dal Lago and Simone Martini Dipartimento di Scienze dell&amp;#... more

Semantic Frameworks for Implicit Computational Complexity

The University of Bologna Department of Computer Science Research Technical Reports are available... more

Probabilistic Operational Semantics for the Lambda Calculus

Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studie... more Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics, inductively and coinductively defined, are given. Moreover, small-step and big-step semantics are shown to produce identical outcomes, both in call-by-value and in call-by-name. Plotkin's CPS translation is extended to accommodate the choice operator and shown correct with respect to the operational semantics. Finally, the expressive power of the obtained system is studied: the calculus is shown to be sound and complete with respect to computable probability distributions.

Type Inference for Sublinear Space Functional Programming

Programming Languages and Systems, Jan 1, 2010

We consider programming language aspects of algorithms that operate on data too large to fit into... more We consider programming language aspects of algorithms that operate on data too large to fit into memory. In previous work we have introduced IntML, a functional programming language with primitives that support the implementation of such algorithms. We have shown that IntML can express all LOGSPACE functions but have left open the question how easy it is in practice to program typical LOGSPACE algorithms in IntML. In this paper we develop algorithms for IntML type inference. We show that with type inference one can handle programs that could not be reasonably manipulated by hand. We do so by implementing in IntML a typical LOGSPACE algorithm, a test for acyclicity of undirected graphs. Thus we show that with type inference IntML can express typical algorithmic patterns of LOGSPACE easily and in a natural way.

Light logics and the call-by-value lambda calculus

Arxiv preprint arXiv:0809.0195, Jan 1, 2008

The so-called light logics have been introduced as logical systems enjoying quite remarkable norm... more The so-called light logics have been introduced as logical systems enjoying quite remarkable normalization properties. Designing a type assignment system for pure lambda calculus from these logics, however, is problematic, as discussed in . In this paper we show that shifting from usual call-by-name to call-by-value lambda calculus allows regaining strong connections with the underlying logic. This will be done in the context of Elementary Affine Logic (EAL), designing a type system in natural deduction style assigning EAL formulae to lambda terms.

Taming Modal Impredicativity: Superlazy Reduction

Logical Foundations of Computer …, Jan 1, 2009

Pure, or type-free, Linear Logic proof nets are Turing complete once cut-elimination is 5 conside... more Pure, or type-free, Linear Logic proof nets are Turing complete once cut-elimination is 5 considered as computation. We introduce modal impredicativity as a new form of impredicativity causing the complexity of cut-elimination to be problematic from a complexity point of view. Modal impredicativity occurs when, during reduction, the conclusion of a residual of a box b interacts with a node that belongs to the proof net inside another residual of b. Technically speaking, Superlazy reduction is a new notion of reduction that allows to con-10 trol modal impredicativity. More specifically, superlazy reduction replicates a box only when all its copies are opened. This makes the overall cost of the reduction finite and predictable. Specifically, superlazy reduction applied to any pure proof nets takes primitive recursive time. Moreover, any primitive recursive function can be computed by a pure proof net via superlazy reduction.

Soft session types

We show how systems of sessions types can enforce interaction to be bounded for all typable proce... more We show how systems of sessions types can enforce interaction to be bounded for all typable processes. The type system we propose is based on Lafont's soft linear logic and is strongly inspired by recent works about session types as intuitionistic linear logic formulas. Our main result is the existence, for every typable process, of a polynomial bound on the length of any reduction sequence starting from it and on the size of any of its reducts.

Light Logics and Higher-Order Processes

We show that the techniques for resource control that have been developed in the so-called "light... more We show that the techniques for resource control that have been developed in the so-called "light logics" can be fruitfully applied also to process algebras. In particular, we present a restriction of Higher-Order π-calculus inspired by Soft Linear Logic. We prove that any soft process terminates in polynomial time. We argue that the class of soft processes may be naturally enlarged so that interesting processes are expressible, still maintaining the polynomial bound on executions.

Bounded linear logic, revisited

Typed Lambda Calculi and Applications, Jan 1, 2009

We present QBAL, an extension of Girard, Scedrov and Scott&amp;#x27;s bounded linear logic. T... more We present QBAL, an extension of Girard, Scedrov and Scott&amp;#x27;s bounded linear logic. The main novelty of the system is the possibility of quantifying over resource variables. This generalization makes bounded linear logic considerably more flexible, while preserving soundness and completeness for polynomial time. In particular, we provide compositional embeddings of Leivant&amp;#x27;s RRW and Hofmann&amp;#x27;s LFPL into QBAL.

Quantum implicit computational complexity

Theoretical Computer Science, Jan 1, 2010

We introduce a quantum lambda calculus inspired by Lafont's Soft Linear Logic and capturing the p... more We introduce a quantum lambda calculus inspired by Lafont's Soft Linear Logic and capturing the polynomial quantum complexity classes EQP, BQP and ZQP. The calculus is based on the ''classical control and quantum data'' paradigm. This is the first example of a formal system capturing quantum complexity classes in the spirit of implicit computational complexity -it is machine-free and no explicit bound (e.g., polynomials) appears in its syntax.

Functional programming in sublinear space

Programming Languages and Systems, Jan 1, 2010

We consider the problem of functional programming with data in external memory, in particular as ... more We consider the problem of functional programming with data in external memory, in particular as it appears in sublinear space computation. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot compose functions directly for lack of space for the intermediate result, but must instead compute and recompute small parts of the intermediate result on demand. In this paper, we study how the implementation of such techniques can be supported by functional programming languages. Our approach is based on modelling computation by interaction using the Int construction of Joyal, Street & Verity. We derive functional programming constructs from the structure obtained by applying the Int construction to a term model of a given functional language. The thus derived functional language is formulated by means of a type system inspired Baillot & Terui's Dual Light Affine Logic. We assess its expressiveness by showing that it captures LOGSPACE.

A Semantic Proof of Polytime Soundness of Light Affine Logic

Theory of Computing Systems, Jan 1, 2010

We define a denotational semantics for Light Affine Logic (LAL) which has the property that denot... more We define a denotational semantics for Light Affine Logic (LAL) which has the property that denotations of functions are polynomial time computable by construction of the model. This gives a new proof of polytime-soundness of LAL which is considerably simpler than the standard proof based on proof nets and also is entirely semantical in nature. The model construction uses a new instance of a resource monoid; a general method for interpreting variations of linear logic with complexity restrictions introduced earlier by the authors.

The weak lambda calculus as a reasonable machine

Theoretical Computer Science, Jan 1, 2008

We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis... more We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the call-by-value lambda-calculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties of usual beta-reduction, without any reference to a specific machine or evaluator. In particular, the cost of a single beta reduction is proportional to the difference between the size of the redex and the size of the reduct. In this way, the total cost of normalizing a lambda term will take into account the size of all intermediate results (as well as the number of steps to normal form).

On the expressive power of light affine logic

Theoretical Computer Science, Jan 1, 2003

Light affine logic (LAL) is a formal system derived from linear logic that is claimed to correspo... more Light affine logic (LAL) is a formal system derived from linear logic that is claimed to correspond, through the Curry-Howard Isomorphism, to the class PTIME of polytime functions. The completeness of the system with respect to PTIME has been proven by embedding different presentations of PTIME into LAL. The dual property of polytime soundness, on the other hand, has been stated and proven in a more debatable way, depending crucially on the underlying coding scheme. In this paper, we introduce two relevant classes of coding schemes, namely uniform and canonical coding schemes. We then investigate on the equality between PTIME and the classes of functions that are representable in LAL using these coding schemes, obtaining a positive and a negative result.

On a measurement-free quantum lambda calculus with classical control

Mathematical Structures in …, Jan 1, 2009

We study a measurement-free, untyped λ-calculus with quantum data and classical control. This wor... more We study a measurement-free, untyped λ-calculus with quantum data and classical control. This work arises from previous proposals by Selinger and Valiron, and Van Tonder. We focus on operational and expressiveness issues, rather than (denotational) semantics. We prove subject reduction and confluence, and a standardisation theorem. Moreover, we prove the computational equivalence of the proposed calculus with a suitable class of quantum circuit families.

Phase semantics and decidability of elementary affine logic

Theoretical computer science, Jan 1, 2004

Quantitative game semantics for linear logic

Computer Science Logic, Jan 1, 2008

We present a game-based semantic framework into which the time complexity of any IMELL proof can ... more We present a game-based semantic framework into which the time complexity of any IMELL proof can be read out of its interpretation. This gives a compositional view of the geometry of interaction framework introduced by the first author. In our model the time measure is given by means of slots, as introduced by Ghica in a recent paper. The cost associated to a strategy is polynomially related to the normalization time of the interpreted proof, in the style of a complexity-theoretical full abstraction result.