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Proofs that count

2014
*
Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '14
*

*Counting*arguments are among the most basic

*proof*methods in mathematics. ... While

*counting*arguments are common in informal, hand-written

*proofs*of such programs, there are no fully automated techniques to construct

*counting*arguments. ... Recall

*that*our use of

*counting*

*proofs*is the following (informal)

*proof*rule: Suppose

*that*P is a program, A, ϕ is a

*counting*

*proof*which satisfies the specification ψpre/ψpost, and

*that*every trace of ...

##
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Vote Counting as Mathematical Proof
[chapter]

2015
*
Lecture Notes in Computer Science
*

Trust in the correctness of an election outcome requires

doi:10.1007/978-3-319-26350-2_41
fatcat:2uecfms56fghhoryhskvvzefzy
*proof*of the correctness of vote*counting*. ... By formalising particular voting protocols as rules, correctness of vote*counting*amounts to verifying*that*all rules have been applied correctly. ... By analogy, a tree or sequence of correctly applied vote*counting*rules*that*determine the outcome of an election is a*proof*of the correctness of the*count*. ...##
###
Logic and Complexity: Independence results and the complexity of propositional calculus
[chapter]

1995
*
Proceedings of the International Congress of Mathematicians
*

The

doi:10.1007/978-3-0348-9078-6_22
fatcat:th6np34k3fehvfyffm73lbha2a
*counting*principle*Count*q is the statement*that*the set {0,... ,n -1} cannot be decomposed into blocks of size q, for every n not divisible by q. ... . , # m are a*proof**that*(1) does not have a solution. Hubert's Nullstellensatz is a completeness result for this "calculus", it says*that*there is such a*proof*if (1) is unsolvable. ...##
###
System Description: CYNTHIA
[chapter]

1999
*
Lecture Notes in Computer Science
*

Selecting this term produces the nal program: tree -> int fun

doi:10.1007/3-540-48660-7_36
fatcat:nsfzcsxnqncdnja5lqbofdejta
*count*(leaf n) = 1 |*count*(node(x,xs,ys)) = 1 + (*count*ys) + (*count*xs); Representing ML De nitions as*Proofs*Each ML function is represented ... In C Y NTHIA, each ML function de nition is represented as a*proof*of a speci cation of*that*function, using the idea of*proofs*-as-programs 2]. ...##
###
A Framework for Proving Contract-Equipped Classes
[chapter]

2003
*
Lecture Notes in Computer Science
*

As part of a general effort to provide a new basis for software development through reuse of "Trusted Components", we outline a scheme for proving

doi:10.1007/3-540-36498-6_6
fatcat:6nrloh7hhfaszc4fv3lcxnq7t4
*that*classes equipped with contracts in the Eiffel style ... The approach takes advantage of the inheritance structure to separate*proof*obligations between deferred (abstract) classes, to be validated against a model, and their effective implementations, which ... The*proof*of the second postcondition clause,*count*= old*count*+ 1, is similar, using the postcondition of extended in SEQUENCE. ...##
###
Proof Complexity and the Kneser-Lovász Theorem
[chapter]

2014
*
Lecture Notes in Computer Science
*

This is a family of propositional tautologies, indexed by an nonnegative integer parameter k

doi:10.1007/978-3-319-09284-3_11
fatcat:viuf4mzmzvduhpqhrxvemokoma
*that*generalizes the Pigeonhole Principle (obtained for k = 1). ... On the other hand for the cases k = 2, 3 (for which combinatorial*proofs*of the Kneser-Lovász Theorem are known) we give polynomial size Frege (k = 2), respectively extended Frege (k = 3)*proofs*. ... Ant 2,n ∧ Onto 2,n U r + P (2) r ≤ q r · (n − 1) has poly-size LK*proofs*.*Proof*. U r*counts*sets {i, j} such*that*both i and j are special. ...##
###
Proving the monotonicity criterion for a plurality vote-counting program as a step towards verified vote-counting

2014
*
2014 6th International Conference on Electronic Voting: Verifying the Vote (EVOTE)
*

We show how modern interactive verification tools can be used to prove complex properties of vote-

doi:10.1109/evote.2014.7001138
dblp:conf/ev/GoreM14
fatcat:l6yrawg7mbc57k6scxwpyqdidm
*counting*software. ... The command "detex evote14.tex | wc" indicates*that*our souce-file contains 4426 words, including the appendix. ... The biggest advantage of this method is*that*we can trust the final*proof*completely. The disadvantage is*that*the user has to be expert in logic and formal*proof*. III. ...##
###
Graph-Based Proof Counting and Enumeration with Applications for Program Fragment Synthesis
[chapter]

2005
*
Lecture Notes in Computer Science
*

We formally study

doi:10.1007/11506676_17
fatcat:f6nkmwaegzd5pnar5cge3bd3ve
*proof*enumeration and*counting*in this calculus. We prove*that**proof**counting*is solvable and give an algorithm to solve it. This in turn yields a*proof*enumeration algorithm. ... To solve the problem, we use the Curry-Howard correspondence (propositions-as-types,*proofs*-as-programs) to transform it into a*proof*enumeration problem for an intuitionistic logic calculus. ... We believe our approach to*proof**counting*and enumeration is the first*that*has the following properties. ...##
###
Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

2015
*
Transactions of the American Mathematical Society
*

Recall: The "quasipolynomial simulation" means there is a 2 log O(1) n -time procedure to thusly convert AC 0 [p k ]

doi:10.1090/s0002-9947-2015-06233-3
fatcat:46rx4kfeqrbnbiv3lpevq2os2a
*proofs*. Open: Does this hold for composite m as well? ... Collapsing Modular*Counting*Propositional*proof*systems Constant depth*proofs*Constant depth PK ⊕ p and PCK i p Constant depth PK ⊕p*proofs*allow ∧'s, ∨'s, and ⊕ p gates to appear at any level. ... Jeřábek showed*that*APC 1 can*count*the size of polynomial time sets to within a constant fraction ǫ. Namely, let X , Y ⊆ 2 n be defined by Boolean circuits. ...##
###
Proof Complexity and the Kneser-Lovász Theorem
[article]

2018
*
arXiv
*
pre-print

This is a family of propositional tautologies, indexed by an nonnegative integer parameter k

arXiv:1402.4338v2
fatcat:vdtkqwrlrzb6thww7ku7brchoi
*that*generalizes the Pigeonhole Principle (obtained for k=1). ... On the other hand for the cases k=2,3 (for which combinatorial*proofs*of the Kneser-Lovász Theorem are known) we give polynomial size Frege (k=2), respectively extended Frege (k=3)*proofs*. ... Ant 2,n ∧ Onto 2,n U r + P (2) r ≤ q r · (n − 1) has poly-size LK*proofs*.*Proof*. U r*counts*sets {i, j} such*that*both i and j are special. ...##
###
Page 753 of The Insurance Law Journal Vol. 6, Issue 10
[page]

1877
*
The Insurance Law Journal
*

But the plaintiff insists

*that*this first*count*of the declaration was not intended as a special*count*on the policy, but as a*count*based upon the ad- justment only, and*that*the setting forth of the ... regulating this*proof*. ...##
###
Starling: Lightweight Concurrency Verification with Views
[chapter]

2017
*
Lecture Notes in Computer Science
*

Starling takes a

doi:10.1007/978-3-319-63387-9_27
fatcat:mcm7f74bbvbqdmovs2nytc2fza
*proof*outline written in an abstracted Hoare-logic style, and converts it into*proof*terms*that*can be discharged by a sequential solver. ... Shared-variable version of ARC, and*proof*. ... The final important properties represented in the*proof*are, first,*that*the ARC is not disposed until all references are removed; and, second,*that**count*accurately records the number of references. ...##
###
Lower Bounds on Hilbert's Nullstellensatz and Propositional Proofs

1996
*
Proceedings of the London Mathematical Society
*

The weak form of the Hilbert's Nullstellensatz says

doi:10.1112/plms/s3-73.1.1
fatcat:xfrhl4ig2jer7p657sfccu3dri
*that*a system of algebraic equations over a field, Q;(%) = 0, does not have a solution in the algebraic closure 8 1 is in the ideal generated by the ... Denote by FP the eflect of ap-*Proof*: Assume*that*II is a depth d , size N k ( N ) Frege*proof*of*Count*: from instances of some*Count*? (and thus Mi = automatically.) ... ) = br(Tg)3. i f 4 =*Count*: E r then brl(T+) = 0 # br(T+) Assume now*that*II is a short constant-depth F'rege*proof*of*Count*: from some instances of CountR;*that*is from some formulas where M E j (mod ...##
###
Page 4 of The Insurance Law Journal Vol. , Issue 300
[page]

1948
*
The Insurance Law Journal
*

as to

*that**count*. ...*Proof*of Loss Regarding the*count*in the declaration on the policy, there is authority*that*under the facts stated the*proof*of loss was waived by the insurer for two reasons: (1) “*Proof*of loss is waived ...##
###
Improved estimate for the prime counting function π(x)
[article]

2018
*
arXiv
*
pre-print

Using some simple combinatorial arguments, we establish some new estimates for the prime

arXiv:1805.10303v2
fatcat:qea2a5jnazd2zel3glz5tidb6q
*counting*function and its allied functions. ... In particular we show*that*π(x)=Θ(x)+O(1/ x), where Θ(x)=θ(x)/ x+x/2 x-1/4- 2/ x∑_n≤ x Ω(n)=k k≥ 2 2| n (x/n)/ 2. This is an improvement to the estimate π(x)=θ(x)/ x+O(x/^2 x) found in the literature. ... There exist some constant c > 0 such*that*θ(x) = x + O x e c √ log x .*Proof*. For a*proof*, see for instance [3] . Lemma 3.2. For all x ≥ 2 π(x) = θ(x) log x + x 2 θ(t) t log 2 t dt.*Proof*. ...
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